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Spherical Vacuum Model

1 · Prerequisites

2 · Summary

This page builds the spherical vacuum solution of the field law and its immediate mathematical consequences. Sixteen items are mathematics — the exterior datum, its connection, curvature and geodesic data, the two smooth extensions of the chart across the horizon, and the local warped-product argument that forces the Schwarzschild form — alongside three physics theorems and two physics-domain remarks. The theorems comprise the model result that adopts the vacuum prescription and two restricted predictions about ideal clocks, rays and test bodies. No item reports a measurement; no empirical premise or uncertainty occurs anywhere on the page, and the physics theorems state explicitly that they assert nothing about nature.

The exterior datum and its geometry. The datum is fixed in The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation: the manifold $R \times (s, \infty) \times S^{2}$ with the metric $g = - f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ , $f (r) = 1 - s / r$ , $s = 2 G_{N} m_{*} / c^{2}$ , the time orientation $\partial_{0}$ and product charts from an atlas of the round sphere. The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold proves that this is a smooth time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ . Christoffel symbols of the Schwarzschild exterior metric in product charts lists the Christoffel symbols of its metric-selected connection, and Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric computes the curvature: the Ricci tensor and Einstein tensor vanish identically, while the curvature tensor is nowhere zero. The frame components and the invariant $R_{a b c d} R^{a b c d} = 12 s^{2} / r^{6}$ are computed in Orthonormal-frame curvature components and the Kretschmann scalar 12s^2/r^6 of the Schwarzschild exterior metric, and the isotropic form of the metric with the areal-radius conversion $r = ρ (1 + s / (4 ρ))^{2}$ is established in The isotropic coordinate form of the Schwarzschild exterior metric and the isothermal radius shift.

Horizon and extensions. Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart constructs the advanced-chart metric on $R \times (0, \infty) \times S^{2}$ , smooth and nondegenerate across $r = s$ , isometric to the datum on $r > s$ , with a time orientation that restricts to the datum's future cone and with the same curvature invariant. The Kruskal model datum: domain, implicit radial function, metric and time orientation fixes the full Kruskal datum $K = {U V < 1} \times S^{2}$ with the implicit radial function $U V = - (r / s - 1) e^{r / s}$ ; Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar proves its properties, the four-region isometries with the exterior datum and the vanishing of its Ricci tensor, and The causal cone, the escape horizon and decreasing areal radius in the Kruskal model derives the exact causal cone criterion, the monotonicity of $U$ and $V$ , the radial null geodesics, the one-way character of the horizon in the precise escape sense, and the decreasing of $r$ inside the trapped region. The distinction between the chart breakdown at $r = s$ and the curvature obstruction at $r = 0$ is stated in Horizon versus singularity in the Schwarzschild model: a chart breakdown, a smooth null surface and a curvature obstruction, without any maximality, completeness or singularity-theorem claim.

Local rigidity. The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor computes the Christoffel symbols and Ricci tensor of an arbitrary warped product of a two-dimensional Lorentzian base with the round sphere. Two-dimensional base identities for the warped spherical metric: pure-trace Ricci, the vacuum reduction and the extra Killing field proves the two-dimensional pure-trace identity, the vacuum reduction $Hess r = \frac{1}{2} Δ r q$ and $r (1 - z) = s$ , and the existence of the extra Killing field $K_{A} = ε_{A B} \partial^{B} r$ . Local Birkhoff theorem: a vacuum warped product with a round sphere factor is locally Schwarzschild with an extra Killing field then proves the precise local Birkhoff theorem: a smooth vacuum metric on an open region which already has the warped form with $z \neq 0$ is locally Schwarzschild in coordinates $(T, r)$ , with the extra Killing field $\partial_{T}$ . The statement is local, and $s$ need not be positive.

The model and its restricted readings. The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter adopts the framework's postulates with $T = 0$ , $Λ = 0$ and proves that the exterior datum satisfies the field law, together with the static clock identity $Δ τ = \sqrt{f (r_{0})} Δ t$ and the mass-parameter reading through the Kepler-like law and the isotropic far-field coefficient. On that model, Restricted redshift prediction of the Schwarzschild vacuum model: the observer-energy ratio between static observers gives the ideal stationary redshift ratio $E (λ_{R}) / E (λ_{E}) = \sqrt{f (r_{E}) / f (r_{R})}$ for a future null geodesic joining two static observers, and Restricted geodesic predictions of the Schwarzschild vacuum model: circular free-fall orbits, the stability threshold and the photon sphere reads the circular timelike geodesics, the existence threshold $r_{0} > 3 s / 2$ , the stability threshold $r = 3 s$ and the photon sphere $r = 3 s / 2$ as restricted predictions for ideal test bodies and ideal rays. The geodesic mathematics underlying both is Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model and Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2; the static observer congruence is The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios. Scope and status of the spherical vacuum model page records exactly what the page does not assert: no observation, no collapse, no singularity theorem, no ADM mass, no maximality and no global uniqueness.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation

Definition

This item assumes the axiom of countable choice ${A C}_{ω}$ through the tangent-bundle structure used by Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c) (The Axiom of Countable Choice ( ${A C}_{ω}$ )); no further choice principle is used.

(a) Parameters and the Schwarzschild radius. Let $c > 0$ and $G_{N} > 0$ be real constants and let $m_{*} > 0$ be a real mass parameter. Put

$s : = \frac{2 G_{N} m_{*}}{c^{2}} > 0.$

The number $s$ is the Schwarzschild radius of the datum. Neither $c$ , $G_{N}$ nor $m_{*}$ is measured here; the physical reading of $c$ and $G_{N}$ as the framework's conversion and Newton constants and of $m_{*}$ as a mass is fixed by the physical conventions of the framework, not by this definition.

(b) The underlying manifold and its product charts. Let $S^{2} : = S_{2} (0, 1) = {y \in R^{3} : ∥ y ∥_{2} = 1}$ be the unit Euclidean sphere with the subspace topology of Euclidean spheres and closed balls as subspaces of $R^{n}$ . Specify its smooth structure as the maximal atlas generated by the following two charts (Smooth manifolds and their smooth charts). For $ε \in {+ 1, - 1}$ put $p_{ε} = (0, 0, ε)$ and $U_{ε} = S^{2} ∖ {p_{ε}}$ , and define

$σ_{ε} : U_{ε} \to R^{2}, σ_{ε} (y) = \frac{(y_{1}, y_{2})}{1 - ε y_{3}} .$

These open domains cover $S^{2}$ . Each chart is a homeomorphism onto $R^{2}$ , with inverse, for $u = (u_{1}, u_{2})$ and $q = u_{1}^{2} + u_{2}^{2}$ ,

$σ_{ε}^{- 1} (u) = (\frac{2 u_{1}}{1 + q}, \frac{2 u_{2}}{1 + q}, ε \frac{q - 1}{1 + q}) .$

The two inverse identities follow by substitution; all displayed denominators are nonzero on their domains. On the overlap both transition maps are $u \mapsto u / (u_{1}^{2} + u_{2}^{2})$ on $R^{2} ∖ {0}$ , hence smooth. The subspace is Hausdorff and has a countable basis given by intersections with Euclidean balls of rational centres and positive rational radii. Thus these charts specify a smooth two-manifold without boundary. This is the standard sphere structure used for the round metric below: the inclusion has the displayed smooth local parametrizations, and their differentials are injective because composition with $σ_{ε}$ is the identity. Let $M$ be the product manifold

$M : = R \times (s, \infty) \times S^{2}$

with the product smooth structure (Products of smooth manifolds have a canonical product smooth structure); it is a smooth four-manifold without boundary. Write $x^{0} : M \to R$ and $r : M \to (s, \infty)$ for the first and second projections and $ι : M \to S^{2}$ for the third. A product chart of $M$ is a smooth chart on the product whose angular part is a smooth chart of $S^{2}$ . In a spherical-polar product chart, $0 < ϑ < π$ and the components of the round metric $γ_{S^{2}}$ of The round metric on the sphere as an induced metric are $γ_{22} = 1$ , $γ_{23} = γ_{32} = 0$ , $γ_{33} = \sin^{2} ϑ$ . These polar charts omit the two poles; ordinary smooth sphere charts around the poles complete the product atlas.

(c) The metric. Let $f : (s, \infty) \to (0, \infty)$ be the smooth function

$f (r) : = 1 - \frac{s}{r} .$

Define $g$ globally by the coordinate-free expression

$g = - f d (x^{0}) \otimes d (x^{0}) + f^{- 1} d r \otimes d r + r^{2} ι^{*} γ_{S^{2}} .$

In every spherical-polar product chart its component functions are

$g_{00} : = - f (r), g_{11} : = \frac{1}{f (r)}, g_{22} : = r^{2}, g_{33} : = r^{2} \sin^{2} ϑ,$

and $g_{a b} : = 0$ whenever $a \neq b$ , $a, b \in {0, 1, 2, 3}$ , with the index order $(x^{0}, x^{1}, x^{2}, x^{3}) = (x^{0}, r, ϑ, φ)$ . This is a smooth symmetric covariant $2$ -tensor field on all of $M$ (Smooth covariant 2-tensor fields on a smooth manifold); the component display is local to spherical-polar charts, and $f > 0$ because $r > s$ . The pair $(M, g)$ is the Schwarzschild exterior metric of mass parameter $m_{*}$ ; the datum of this definition is the quadruple $(c, G_{N}, m_{*}, g)$ . With these parameters fixed, write $(M, g, N)$ for its associated geometric datum, with $N$ as in clause (d). The geometric datum depends on the parameters only through $s$ and does not retain $c$ , $G_{N}$ and $m_{*}$ separately. That $(M, g, N)$ is a time-oriented smooth Lorentzian four-manifold of signature $(-, +, +, +)$ is not asserted here; it is proved in the companion metric-properties lemma of this page (its justification record in the frontmatter), which also records the equivalent coordinate-free description $g = - f d (x^{0}) \otimes d (x^{0}) + f^{- 1} d r \otimes d r + r^{2} ι^{*} γ_{S^{2}}$ .

(d) Time orientation and the Schwarzschild chart. The field $N : = \partial_{0}$ is the smooth product vector field on $M$ (Derivations at a point and the tangent space); the chart components of $g$ are independent of $x^{0}$ , so $N$ is the restriction of a global vector field, and $g (N, N) = - f (r) < 0$ at every point. The datum carries the time orientation $N$ . Spherical-polar product charts are Schwarzschild charts of the datum; at the poles the same metric is represented in the smooth sphere charts of the product atlas. We write $t$ for the coordinate $x^{0} / c$ when the datum is read in the framework's unit conventions.

(e) Scope. The domain of the datum is the exterior region $r > s$ ; the definition asserts nothing about $r = s$ , about $r \leq 0$ , about any matter, or about any equation satisfied by $g$ . The horizon $r = s$ is not part of the datum, and a continuation of the metric to $r = s$ requires the separate constructions recorded on this page, the regularity lemma for the advanced chart and the Kruskal model-properties lemma. Whether the datum is a solution of the framework's field law is proved, not defined, in the Ricci-and-Einstein-tensor lemma of this page.

Remarks

Units and the physical reading. In the framework's conventions the coordinate $x^{0} = c t$ and the radial coordinate $r$ have dimension of length in metres, the angular coordinates are dimensionless, and the metric coefficients $g_{00}$ and $g_{11}$ are dimensionless, while $g_{22}$ and $g_{33}$ have units of square metres; $m_{*}$ is read in kilograms and $s$ in metres. No numerical value of any constant is asserted. The reading of $x^{0}$ as a clock coordinate of static observers is part of the framework's postulates and is not part of this geometric definition (see the static-congruence and clock-rate lemma and the exterior vacuum-model theorem of this page).
The sphere factor. The function $r$ is the areal radius: on the level set ${x^{0} = c_{0}, r = c_{1}}$ the induced metric is $c_{1}^{2} γ_{S^{2}}$ , so the area of the sphere is $4 π c_{1}^{2}$ . Nothing in this definition identifies $r$ with a Euclidean radial coordinate of any ambient space.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold

Statement

Let $c > 0$ , $G_{N} > 0$ , $m_{*} > 0$ and let $(M, g, N)$ be the Schwarzschild exterior datum of mass parameter $m_{*}$ with Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation). Then:

(a) Global metric and local components. $M$ is a smooth four-manifold without boundary and $g$ is the smooth symmetric covariant $2$ -tensor field globally defined by

$g = - f d (x^{0}) \otimes d (x^{0}) + f^{- 1} d r \otimes d r + r^{2} ι^{*} γ_{S^{2}} .$

In every spherical-polar product chart $(x^{0}, r, ϑ, φ)$ its components are

$g_{00} = - f (r), g_{11} = \frac{1}{f (r)}, g_{22} = r^{2}, g_{33} = r^{2} \sin^{2} ϑ, g_{a b} = 0 (a \neq b),$

where $f (r) = 1 - s / r$ , $ι : M \to S^{2}$ is the third projection and $γ_{S^{2}}$ is the round metric of the unit sphere. The component display is local to the spherical-polar charts; the global tensor formula includes the poles.

(b) Lorentzian signature. At every $p \in M$ the form $g_{p}$ is nondegenerate with inertia $(3, 1, 0)$ ; consequently $(M, g, N)$ is a smooth Lorentzian four-manifold of signature $(-, +, +, +)$ .

(c) Time orientation. The global product field $N = \partial_{0}$ is smooth and nowhere vanishing, and $g_{p} (N_{p}, N_{p}) = - f (r (p)) < 0$ at every $p$ ; it is therefore a time orientation of $(M, g)$ . The globally defined base fields $\partial_{0}, \partial_{r}$ are orthogonal with squared norms $- f$ and $f^{- 1}$ . In a spherical-polar chart the angular coordinate fields are orthogonal with positive squared norms $r^{2}$ and $r^{2} \sin^{2} ϑ$ ; at a pole use any angular chart and the positive-definite round metric instead.

(d) Domain of the coordinates. Product charts form an atlas covering $M$ . Spherical-polar charts are defined on the regular angular locus $0 < ϑ < π$ and omit the poles; pole charts are ordinary smooth charts of $S^{2}$ . The functions $f$ and $f^{- 1}$ are smooth on $r > s$ , and the spherical-polar component functions are smooth on their chart domains.

(e) Conventions. The global tensor identity holds on all of $M$ ; the displayed component identities are in spherical-polar product charts. In pole charts the sphere block is $r^{2}$ times the round-metric component matrix.

Facts & Assumptions

Given: Constants $c > 0$ , $G_{N} > 0$ , $m_{*} > 0$ , the Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ , the product manifold $M = R \times (s, \infty) \times S^{2}$ , the assignments of clause (c) of The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation in product charts, and the field $N = \partial_{0}$ .

[L1]

The metric is globally $g = - f d (x^{0}) \otimes d (x^{0}) + f^{- 1} d r \otimes d r + r^{2} ι^{*} γ_{S^{2}}$ ; on the dense regular angular locus its spherical-polar components are $g_{00} = - f (r)$ , $g_{11} = f (r)^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ and all mixed components vanish. The round metric is smooth and positive definite also at the two poles; $f (r) = 1 - s / r$ is smooth and strictly positive on $(s, \infty)$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clauses (b)-(c), The round metric on the sphere as an induced metric).

[L2]

$M$ carries the product smooth structure: it is a Hausdorff, second-countable smooth four-manifold without boundary, and its smooth atlas is generated by products of charts of the factors; the projections $x^{0}$ , $r$ , $ι$ are smooth (Products of smooth manifolds have a canonical product smooth structure, Smooth manifolds and their smooth charts, Euclidean spheres and closed balls as subspaces of $R^{n}$ , $C^{r}$ and smooth maps between smooth manifolds).

[L3]

A covariant $2$ -tensor field is a family of bilinear forms whose components in every chart are $C^{\infty}$ ; specifying a symmetric matrix of component functions in the charts of one atlas defines a smooth symmetric covariant $2$ -tensor field when the assigned components transform by the covariant $2$ -tensor law on overlaps (Smooth covariant 2-tensor fields on a smooth manifold, Chart independence and the transformation law of a smooth covariant 2-tensor field).

[L4]

A smooth symmetric covariant $2$ -tensor field $g$ on a smooth four-manifold $M$ is a Lorentzian metric of signature $(-, +, +, +)$ when $g_{p}$ is nondegenerate and has inertia $(3, 1, 0)$ at every $p$ , that is, some (equivalently every) basis of $T_{p} M$ diagonalizing $g_{p}$ has exactly one negative and three positive diagonal entries (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (a), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space); a smooth vector field $N$ with $g_{p} (N_{p}, N_{p}) < 0$ for every $p$ is a time orientation (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c)).

[L5]

On the overlap of two smooth charts, coordinates and component functions of a smooth tensor transform by the chain rule; in particular the transition maps of the standard spherical atlas of $S^{2}$ are smooth and the round metric components transform as a $(0, 2)$ -tensor (Chart independence and the transformation law of a smooth covariant 2-tensor field, Coordinate derivations form a basis of the tangent space, $C^{r}$ and smooth maps between smooth manifolds).

[L6]

The functions $\sin ϑ$ and $\sin^{2} ϑ$ are smooth on $0 < ϑ < π$ and strictly positive there; $r^{2}$ , $r^{4} \sin^{2} ϑ$ , $f (r)$ and $f (r)^{- 1}$ are smooth and positive on the ranges stated (Rational powers $a^{r}$ of a positive base, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · use the global pullback formula, the product tangent-space splitting and positive definiteness of the round metric

1.1L1L2L3L5L6algebra

(Global smooth tensor.) The product manifold atlas covers $M$ , including charts about both sphere poles, by [L2]. The projections $x^{0}, r, ι$ are smooth, $f$ and $f^{- 1}$ are smooth on $r > s$ , and $γ_{S^{2}}$ is a smooth covariant tensor on all of $S^{2}$ by [L1]-[L3]. Therefore each term of the coordinate-free expression in [L1] is a smooth tensor field on $M$ , their sum is a smooth symmetric covariant $2$ -tensor field, and on the regular angular locus its polar components are the display of clause (a). By [L5], smooth changes of angular coordinates re-express this same tensor by the covariant component transformation law in the corresponding coordinate bases, including pole charts. This proves the global assertion and the chart-local statement.

1.2L1L4algebra

(Nondegeneracy and inertia, including the poles.) At $p = (x^{0}, r, ω)$ the product tangent space splits as $T_{p} M ≅ R \oplus R \oplus T_{ω} S^{2}$ . For $v = (a, b, w)$ in this splitting, $g_{p} (v, v) = - f (r) a^{2} + f (r)^{- 1} b^{2} + r^{2} γ_{S^{2}, ω} (w, w)$ . Here $f (r) > 0$ , and $γ_{S^{2}, ω}$ is positive definite at every sphere point, including the poles. The first two summands thus have one negative and one positive direction, and the sphere summand has two positive directions; $g_{p}$ is nondegenerate with inertia $(3, 1, 0)$ at every $p$ . This proves (b).

2.1L1L4step 1.2algebra

(Time orientation.) The product field $N = \partial_{0}$ is smooth and nowhere zero. By [L1] and the global metric formula, $g (N, N) = - f (r) < 0$ everywhere, so $N$ is a time orientation. The base-vector signs and the polar angular signs in (c) follow from that formula and the positive-definite round metric.

3.1step 1.1step 1.2step 2.1given∎

(Conclusion.) Step 1.1 proves (a), step 1.2 proves (b) at all sphere points, and step 2.1 proves (c). Clauses (d)-(e) record the polar-chart domain of the component display and the global scope of the tensor formula. No property outside $r > s$ is asserted.

Remarks

Why the poles cause no degeneracy. The global product tangent-space splitting separates the Lorentzian time-radial block from the positive round metric on $S^{2}$ . Polar coordinates are only a local component display; their failure at the poles does not affect the metric or its inertia.
No extension is claimed. The product-chart frame remains a basis at every point with $r > s$ , while its component $g_{11} = f^{- 1}$ diverges as $r ↓ s$ . The chart has no point at $r = s$ , so it has no frame limit there; this lemma asserts nothing at that boundary. The extension and Kruskal items of this page study the metric there in different charts.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Christoffel symbols of the Schwarzschild exterior metric in product charts

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum of mass parameter $m_{*}$ with Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), and let $\nabla$ be the metric-selected connection of $g$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields). In every spherical-polar product chart $(x^{0}, r, ϑ, φ)$ with $0 < ϑ < π$ and $f^{'} = \frac{d f}{d r} = s / r^{2}$ , the Christoffel symbols $Γ^{a}_{b c}$ of $\nabla$ (Christoffel symbols of an affine connection) are:

(a) $Γ^{0}_{01} = Γ^{0}_{10} = \frac{f^{'}}{2 f}$ , and $Γ^{0}_{00} = Γ^{0}_{11} = Γ^{0}_{i j} = 0$ for all remaining index pairs;

(b) $Γ^{1}_{00} = \frac{f f^{'}}{2}$ , $Γ^{1}_{11} = - \frac{f^{'}}{2 f}$ , $Γ^{1}_{22} = - r f$ , $Γ^{1}_{33} = - r f \sin^{2} ϑ$ , and $Γ^{1}_{01} = Γ^{1}_{02} = Γ^{1}_{03} = Γ^{1}_{12} = Γ^{1}_{13} = Γ^{1}_{23} = 0$ ;

(c) $Γ^{2}_{12} = Γ^{2}_{21} = \frac{1}{r}$ , $Γ^{2}_{33} = - \sin ϑ \cos ϑ$ , and every other symbol with upper index $2$ vanishes;

(d) $Γ^{3}_{13} = Γ^{3}_{31} = \frac{1}{r}$ , $Γ^{3}_{23} = Γ^{3}_{32} = \cot ϑ$ , and every other symbol with upper index $3$ vanishes.

All components not listed vanish; equivalently, the only nonzero symbols are those displayed. For the model value $f (r) = 1 - s / r$ , so that $f^{'} = s / r^{2}$ , the listed symbols read

$Γ^{0}_{01} = Γ^{0}_{10} = \frac{s}{2 r^{2} f}, Γ^{1}_{00} = \frac{s f}{2 r^{2}}, Γ^{1}_{11} = - \frac{s}{2 r^{2} f},$

$Γ^{1}_{22} = - r f, Γ^{1}_{33} = - r f \sin^{2} ϑ, Γ^{2}_{12} = Γ^{3}_{13} = \frac{1}{r}, Γ^{2}_{33} = - \sin ϑ \cos ϑ, Γ^{3}_{23} = \cot ϑ .$

Facts & Assumptions

Given: The Schwarzschild exterior datum with Schwarzschild radius $s$ , the metric $g$ and its metric-selected connection $\nabla$ , and a product chart $(x^{0}, r, ϑ, φ)$ with $0 < ϑ < π$ .

[L1]

In the product chart the metric components are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ and $g_{a b} = 0$ for $a \neq b$ , with $f = f (r) > 0$ smooth; the metric-selected connection exists and is the unique torsion-free $g$ -compatible affine connection (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

The inverse-metric components are characterised by $\sum_{b} g^{a b} g_{b c} = δ^{a}_{c}$ ; for the diagonal matrix of [L1] they are $g^{00} = - f^{- 1}$ , $g^{11} = f$ , $g^{22} = r^{- 2}$ , $g^{33} = r^{- 2} \sin^{- 2} ϑ$ , all off-diagonal components vanishing (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L3]

In every smooth chart the Christoffel symbols of the metric-selected connection are $Γ^{a}_{b c} = \frac{1}{2} \sum_{d} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ , and they are symmetric in the lower pair, $Γ^{a}_{b c} = Γ^{a}_{c b}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (a), Christoffel symbols of an affine connection, Torsion freedom and metric compatibility in chart components).

[L4]

All metric components of [L1] are independent of $x^{0}$ and $φ$ , so $\partial_{0} g_{b c} = 0$ and $\partial_{3} g_{b c} = 0$ for all indices; the coordinate partial derivatives of the components are those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , and $\frac{d}{d r} f = f^{'} = \frac{s}{r^{2}}$ for the model function $f (r) = 1 - s / r$ by The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set and the power rule.

[L5]

On the chart domain $0 < ϑ < π$ , so $\sin ϑ > 0$ , $\sin^{2} ϑ > 0$ and $r > s > 0$ ; the displayed expressions for $Γ$ are smooth there (Rational powers $a^{r}$ of a positive base).

Proof

technique · substitute the metric components of [L1] and the inverse components of [L2] into the Koszul formula of [L3], organised by the upper index

1.1L1L2L4algebra

(Metric derivatives and inverse components.) The only nonconstant components of [L1] are $g_{00} = - f (r)$ , $g_{11} = f (r)^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ , and their nonzero coordinate derivatives are $\partial_{1} g_{00} = - f^{'}$ , $\partial_{1} g_{11} = - f^{'} / f^{2}$ , $\partial_{1} g_{22} = 2 r$ , $\partial_{1} g_{33} = 2 r \sin^{2} ϑ$ , $\partial_{2} g_{33} = 2 r^{2} \sin ϑ \cos ϑ$ ; by [L4] every $\partial_{0}$ or $\partial_{3}$ derivative vanishes. The inverse components are $g^{00} = - 1 / f$ , $g^{11} = f$ , $g^{22} = 1 / r^{2}$ , $g^{33} = 1 / (r^{2} \sin^{2} ϑ)$ , as the diagonal matrix inverse of [L2].

2.1L3step 1.1algebra

(The family $Γ^{0}_{\cdot \cdot}$ .) By [L3], $Γ^{0}_{b c} = \frac{1}{2} g^{00} (\partial_{b} g_{0 c} + \partial_{c} g_{0 b} - \partial_{0} g_{b c})$ . If ${b, c}$ contains an index in ${2, 3}$ then $g_{0 c}$ or $g_{0 b}$ vanishes identically and $\partial_{0} g_{b c} = 0$ ; the cases $(b, c) = (0, 0)$ and $(1, 1)$ give $\partial_{0} g_{00} = 0$ and $\partial_{0} g_{11} = 0$ ; hence only $(b, c) \in {(0, 1), (1, 0)}$ can contribute, with value $\frac{1}{2} g^{00} \partial_{1} g_{00} = \frac{1}{2} (- 1 / f) (- f^{'}) = f^{'} / (2 f)$ . This is claim (a).

2.2L3step 1.1algebra

(The family $Γ^{1}_{\cdot \cdot}$ .) By [L3], $Γ^{1}_{b c} = \frac{1}{2} g^{11} (\partial_{b} g_{1 c} + \partial_{c} g_{1 b} - \partial_{1} g_{b c})$ . If $g_{1 c}$ or $g_{1 b}$ is nonzero the corresponding term contributes; since $g_{1 d} = 0$ unless $d = 1$ , only $b = c = 1$ retains a contribution and $Γ^{1}_{11} = \frac{1}{2} g^{11} \partial_{1} g_{11} = \frac{1}{2} f (- f^{'} / f^{2}) = - f^{'} / (2 f)$ . The cases $(b, c) = (0, 0)$ , $(2, 2)$ and $(3, 3)$ have $Γ^{1}_{b c} = - \frac{1}{2} g^{11} \partial_{1} g_{b c}$ , giving $Γ^{1}_{00} = \frac{1}{2} f f^{'} = f f^{'} / 2$ , $Γ^{1}_{22} = - \frac{1}{2} f (2 r) = - r f$ and $Γ^{1}_{33} = - r f \sin^{2} ϑ$ . Every mixed case $(b \neq c)$ has $\partial_{b} g_{1 c} + \partial_{c} g_{1 b} = 0$ and $\partial_{1} g_{b c} = 0$ for the off-diagonal components, hence vanishes. This is claim (b).

2.3L3L4step 1.1algebra

(The families $Γ^{2}_{\cdot \cdot}$ and $Γ^{3}_{\cdot \cdot}$ .) For the upper index $2$ , [L3] gives $Γ^{2}_{b c} = \frac{1}{2} g^{22} (\partial_{b} g_{2 c} + \partial_{c} g_{2 b} - \partial_{2} g_{b c})$ with $g^{22} = 1 / r^{2}$ . The only nonzero $g_{2 c}$ is $c = 2$ , so only $b = c = 2$ and the pairs with $g_{b c}$ depending on $ϑ$ contribute: $Γ^{2}_{22} = \frac{1}{2} r^{- 2} \partial_{2} g_{22} = 0$ and $Γ^{2}_{33} = - \frac{1}{2} r^{- 2} \partial_{2} g_{33} = - \frac{1}{2} r^{- 2} (2 r^{2} \sin ϑ \cos ϑ) = - \sin ϑ \cos ϑ$ , while $Γ^{2}_{12} = Γ^{2}_{21} = \frac{1}{2} r^{- 2} \partial_{1} g_{22} = r^{- 2} \cdot r = 1 / r$ and the $(0, 0)$ case vanishes. For the upper index $3$ , $g^{33} = 1 / (r^{2} \sin^{2} ϑ)$ and only $g_{33}$ is nonzero among the $g_{3 c}$ : $Γ^{3}_{13} = Γ^{3}_{31} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} \partial_{1} g_{33} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} (2 r \sin^{2} ϑ) = 1 / r$ , $Γ^{3}_{23} = Γ^{3}_{32} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} \partial_{2} g_{33} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} (2 r^{2} \sin ϑ \cos ϑ) = \cot ϑ$ , and $Γ^{3}_{33} = \frac{1}{2} g^{33} \partial_{3} g_{33} = 0$ by [L4]; all cases with an index $0$ or $1$ in $b, c$ other than the displayed ones vanish because the corresponding $g_{3 c}$ or $g_{b c}$ is constant. Together with the lower-pair symmetry of [L3], these are claims (c) and (d).

3.1L3L4L5step 2.1step 2.2step 2.3∎

(Exhaustiveness and the model values.) The index triples $(a, b, c)$ with $a, b, c \in {0, 1, 2, 3}$ split into the families treated in step 1.1 and steps 2.1-2.3; each family was computed from the same unique connection $\nabla$ of [L1], so the displays (a)-(d) list exactly the nonzero symbols of $\nabla$ in the chart. Substituting $f = 1 - s / r$ and $f^{'} = s / r^{2}$ of [L4] into the general expressions (a)-(d) gives the second display of the Statement. All symbols are smooth on the chart domain by [L5], and the expressions are those of the metric-selected connection, not of an arbitrary connection.

Remarks

Sign check at the leading order. For large $r$ one has $f = 1 - s / r + O (r^{- 2})$ and $f^{'} = s / r^{2}$ , so $Γ^{1}_{00} = \frac{1}{2} f f^{'} = \frac{s}{2 r^{2}} + O (r^{- 3})$ ; this is the coefficient $+ c^{- 2} \partial_{r} Φ$ with the Newtonian potential $Φ = - G_{N} m_{*} / r$ of the flat and Newtonian bridge page, whose spatial acceleration reads $- \partial_{r} Φ = - G_{N} m_{*} / r^{2}$ and matches the geodesic acceleration $- c^{2} Γ^{1}_{00}$ computed in the weak-field bridge lemma of the flat-and-Newtonian page in the overlapping regime. The comparison is made with the round metric of the same chart and is not a statement of this lemma.
Distance to the Newtonian form. The symbols here are written with $x^{0} = c t$ as the time coordinate; in the coordinate $t = x^{0} / c$ the components acquire factors of $c$ by the chain rule and the invariant statements of the following lemmas are used rather than the raw symbols.

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Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum of mass parameter $m_{*}$ , Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), with metric-selected connection $\nabla$ and curvature data $R^{a}_{b c d}$ , $R_{a b c d}$ , ${Ric}_{a b}$ , $S$ , $G_{a b}$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). In every spherical-polar product chart $(x^{0}, r, ϑ, φ)$ , writing $f^{'} = \frac{d f}{d r} = s / r^{2}$ and $f^{''} = \frac{d^{2} f}{d r^{2}} = - 2 s / r^{3}$ :

(a) Curvature components. All nonzero components of the lowered curvature tensor, up to the symmetries of The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors, are

$R_{0101} = \frac{f^{''}}{2}, R_{0202} = \frac{r f f^{'}}{2}, R_{0303} = \frac{r f f^{'} \sin^{2} ϑ}{2},$

$R_{1212} = - \frac{r f^{'}}{2 f}, R_{1313} = - \frac{r f^{'} \sin^{2} ϑ}{2 f}, R_{2323} = r^{2} (1 - f) \sin^{2} ϑ .$

Every component whose index pairs do not coincide as unordered pairs, and every component with an index repeated inside a pair, vanishes.

(b) Ricci tensor. The Ricci tensor has the diagonal components

${Ric}_{00} = f (\frac{f^{''}}{2} + \frac{f^{'}}{r}), {Ric}_{11} = - \frac{1}{f} (\frac{f^{''}}{2} + \frac{f^{'}}{r}),$

${Ric}_{22} = 1 - f - r f^{'}, {Ric}_{33} = (1 - f - r f^{'}) \sin^{2} ϑ,$

and all mixed components ${Ric}_{a b}$ with $a \neq b$ vanish.

(c) Vanishing for the model. For $f (r) = 1 - s / r$ one has $\frac{f^{''}}{2} + \frac{f^{'}}{r} = 0$ and $1 - f - r f^{'} = 0$ ; consequently

${Ric}_{a b} = 0, S = 0, G_{a b} = 0$

on the regular angular locus and, as tensor identities, on all of $M$ by smoothness and density. In particular the Schwarzschild exterior metric is a Ricci-flat metric, and $G = 0$ holds as an identity of smooth tensor fields.

(d) The curvature does not vanish. The model is not locally flat. The global base-field contraction $R_{0101} : = g (R (\partial_{0}, \partial_{r}) \partial_{r}, \partial_{0})$ equals $- s / r^{3}$ on the regular angular locus, hence everywhere by smoothness and density. It is therefore nowhere zero even though the Ricci and Einstein tensors vanish identically.

Facts & Assumptions

Given: The Schwarzschild exterior datum, its metric and connection, a spherical-polar product chart $(x^{0}, r, ϑ, φ)$ with $0 < ϑ < π$ , and the functions $f (r) = 1 - s / r$ , $f^{'} = s / r^{2}$ , $f^{''} = - 2 s / r^{3}$ .

[L1]

On a spherical-polar product chart the metric components are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ , off-diagonal components vanish, and the inverse components are $g^{00} = - f^{- 1}$ , $g^{11} = f$ , $g^{22} = r^{- 2}$ , $g^{33} = r^{- 2} \sin^{- 2} ϑ$ (The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, The inverse metric, musical isomorphisms and index raising); all components and all Christoffel symbols are independent of $x^{0}$ and $φ$ .

[L2]

The nonzero Christoffel symbols of $\nabla$ in the chart are $Γ^{0}_{01} = Γ^{0}_{10} = f^{'} / (2 f)$ , $Γ^{1}_{00} = f f^{'} / 2$ , $Γ^{1}_{11} = - f^{'} / (2 f)$ , $Γ^{1}_{22} = - r f$ , $Γ^{1}_{33} = - r f \sin^{2} ϑ$ , $Γ^{2}_{12} = Γ^{2}_{21} = 1 / r$ , $Γ^{2}_{33} = - \sin ϑ \cos ϑ$ , $Γ^{3}_{13} = Γ^{3}_{31} = 1 / r$ and $Γ^{3}_{23} = Γ^{3}_{32} = \cot ϑ$ (Christoffel symbols of the Schwarzschild exterior metric in product charts).

[L3]

In the chart, $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ , and $R$ is a smooth tensor field whose components transform under changes of chart. Tensor vanishing and contractions are intrinsic; the component formulas in (a)-(b) are computed separately in each spherical-polar product chart with the same metric form (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clauses (a)-(b), The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (b)).

[L4]

The lowered curvature satisfies $R_{a b c d} = - R_{b a c d} = - R_{a b d c} = R_{c d a b}$ and the first Bianchi identity $R_{a b c d} + R_{a c d b} + R_{a d b c} = 0$ ; the Ricci tensor, scalar curvature and Einstein tensor are ${Ric}_{b d} = g^{a c} R_{a b c d}$ , $S = g^{b d} {Ric}_{b d}$ and $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ , with ${Ric}_{a b} = {Ric}_{b a}$ (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(d), The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (c)).

[L5]

The model function satisfies $f^{'} = s / r^{2}$ , $f^{''} = - 2 s / r^{3}$ , so $f^{''} / 2 + f^{'} / r = 0$ and $1 - f - r f^{'} = 1 - (1 - s / r) - s / r = 0$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Rational powers $a^{r}$ of a positive base); in the chart domain $r > s > 0$ and $\sin ϑ > 0$ .

Proof

technique · exhaust the independent curvature components by the curvature symmetries and a finite pair-index table, contract to Ricci, and extend the resulting smooth tensor identities from the dense polar-chart locus to the poles

1.1L2L3L4

(Exhaustive pair-index table.) By [L4], a lowered component with a repeated index inside either pair vanishes, and the two skew-symmetries and pair exchange reduce all other components to a symmetric $6 \times 6$ array indexed by the six choices $01, 02, 03, 12, 13, 23$ . Substitution of every pair-pair choice into [L3] gives this complete array; entries below the diagonal equal their transposes, and reversing the order within either pair changes the sign: [L2, L3, L4] $\begin{array}{crrrrrr} 01 & 02 & 03 & 12 & 13 & 23 \\ 01 & f^{''} / 2 & 0 & 0 & 0 & 0 & 0 \\ 02 & r f f^{'} / 2 & 0 & 0 & 0 & 0 \\ 03 & r f f^{'} \sin^{2} ϑ / 2 & 0 & 0 & 0 \\ 12 & - r f^{'} / (2 f) & 0 & 0 \\ 13 & - r f^{'} \sin^{2} ϑ / (2 f) & 0 \\ 23 & r^{2} (1 - f) \sin^{2} ϑ \end{array}$ Thus all fifteen unequal-pair entries are explicitly zero; the six diagonal entries are computed in steps 1.2, 1.3 and 2.1. This exhausts the $21$ independent pair-pair entries and, by [L4], every curvature component.

1.2L2L3algebra

(The $01$ and $23$ components.) For $R_{0101} = g_{00} R^{0}_{101}$ : by [L3], [L2] and $\partial_{0} Γ^{0}_{11} = 0$ , $R^{0}_{101} = - \partial_{1} Γ^{0}_{01} + Γ^{0}_{01} Γ^{1}_{11} - Γ^{0}_{10} Γ^{0}_{01} = - \partial_{1} (f^{'} / (2 f)) - \frac{(f^{'})^{2}}{4 f^{2}} - \frac{(f^{'})^{2}}{4 f^{2}} = - \frac{f^{''}}{2 f} + \frac{(f^{'})^{2}}{2 f^{2}} - \frac{(f^{'})^{2}}{2 f^{2}} = - \frac{f^{''}}{2 f}$ , and multiplication by $g_{00} = - f$ gives $R_{0101} = f^{''} / 2$ . For $R_{2323} = g_{22} R^{2}_{323}$ : $R^{2}_{323} = \partial_{2} Γ^{2}_{33} + Γ^{2}_{21} Γ^{1}_{33} - Γ^{2}_{33} Γ^{3}_{23} = (\sin^{2} ϑ - \cos^{2} ϑ) - f \sin^{2} ϑ + \cos^{2} ϑ = (1 - f) \sin^{2} ϑ$ , using [L2]; hence $R_{2323} = r^{2} (1 - f) \sin^{2} ϑ$ .

1.3L2L3algebra

(The $02$ and $12$ components.) For $R_{0202} = g_{00} R^{0}_{202}$ : $R^{0}_{202} = Γ^{0}_{01} Γ^{1}_{22} = - (r f^{'}) / 2$ by [L2] and [L3] (all remaining terms contain a vanishing symbol or a $\partial_{0}$ -derivative), so $R_{0202} = (- f) (- r f^{'} / 2) = r f f^{'} / 2$ . For $R_{1212} = g_{11} R^{1}_{212}$ : $R^{1}_{212} = \partial_{1} Γ^{1}_{22} + Γ^{1}_{11} Γ^{1}_{22} - Γ^{1}_{22} Γ^{2}_{12} = - (f + r f^{'}) + \frac{r f^{'}}{2} + f = - \frac{r f^{'}}{2}$ , using [L2], hence $R_{1212} = f^{- 1} (- r f^{'} / 2) = - r f^{'} / (2 f)$ .

2.1L2L3step 1.1algebra

(The remaining $03$ and $13$ components.) From [L2]-[L3], $R^{0}_{303} = Γ^{0}_{01} Γ^{1}_{33} = - (r f^{'} \sin^{2} ϑ) / 2$ , so $R_{0303} = g_{00} R^{0}_{303} = r f f^{'} \sin^{2} ϑ / 2$ . Also $R^{1}_{313} = \partial_{1} Γ^{1}_{33} + Γ^{1}_{11} Γ^{1}_{33} - Γ^{1}_{33} Γ^{3}_{13} = - (r f^{'} \sin^{2} ϑ) / 2$ , hence $R_{1313} = g_{11} R^{1}_{313} = - r f^{'} \sin^{2} ϑ / (2 f)$ . These verify the final two diagonal entries of the table in step 1.1.

3.1step 1.1step 1.2step 1.3step 2.1L4

(Mixed and zero components.) The fifteen unequal-pair entries are exactly the off-diagonal entries of the exhaustive array in step 1.1. Every component with a repeated index inside a pair vanishes by [L4]; all remaining sign and order variants are determined by its skew-symmetries and pair exchange. Thus clause (a) is exhaustive.

4.1step 1.2step 1.3step 3.1L4L1algebra

(Contraction to the Ricci tensor.) By [L4] and clause (a), ${Ric}_{00} = g^{11} R_{1010} + g^{22} R_{2020} + g^{33} R_{3030} = f \cdot \frac{f^{''}}{2} + \frac{1}{r^{2}} \cdot \frac{r f f^{'}}{2} + \frac{1}{r^{2} \sin^{2} ϑ} \cdot \frac{r f f^{'} \sin^{2} ϑ}{2} = f (\frac{f^{''}}{2} + \frac{f^{'}}{r})$ . Similarly ${Ric}_{11} = g^{00} R_{0101} + g^{22} R_{2121} + g^{33} R_{3131} = - \frac{f^{''}}{2 f} - \frac{f^{'}}{2 r f} - \frac{f^{'}}{2 r f} = - \frac{1}{f} (\frac{f^{''}}{2} + \frac{f^{'}}{r})$ , ${Ric}_{22} = g^{00} R_{0202} + g^{11} R_{1212} + g^{33} R_{3232} = - \frac{r f^{'}}{2} - \frac{r f^{'}}{2} + (1 - f) = 1 - f - r f^{'}$ , and ${Ric}_{33} = g^{00} R_{0303} + g^{11} R_{1313} + g^{22} R_{2323} = - \frac{r f^{'} \sin^{2} ϑ}{2} - \frac{r f^{'} \sin^{2} ϑ}{2} + (1 - f) \sin^{2} ϑ = (1 - f - r f^{'}) \sin^{2} ϑ$ . For a mixed pair, say $(0, 1)$ , every term $g^{a a} R_{a 0 a 1}$ of [L4] contains a component with unequal index pairs or a repeated index, which vanishes by step 3.1; hence ${Ric}_{01} = 0$ , and all mixed components vanish likewise. This is clause (b).

5.1L3L4L5step 4.1algebra

(Specialisation and the Einstein tensor.) For $f = 1 - s / r$ the identities of [L5] give $f^{''} / 2 + f^{'} / r = 0$ and $1 - f - r f^{'} = 0$ . Clause (b) therefore gives $Ric = 0$ on the dense open regular angular locus $M^{\circ} = R \times (s, \infty) \times (S^{2} ∖ {two poles})$ . The Ricci tensor is smooth on all of $M$ by [L3]; in any chart about a pole its components are continuous and vanish on the dense subset from $M^{\circ}$ , hence vanish at the pole too. Thus $Ric = 0$ globally. Its contractions then give $S = 0$ and $G = 0$ by [L4]. This proves (c).

6.1L1L3L5step 1.1algebra∎

(Nonvanishing curvature.) On $M^{\circ}$ , the $01$ entry of step 1.1 and $f^{''} = - 2 s / r^{3}$ give the global base-field contraction $g (R (\partial_{0}, \partial_{r}) \partial_{r}, \partial_{0}) = - s / r^{3}$ . This contraction is continuous on $M$ and equals the smooth function $- s / r^{3}$ on the dense subset $M^{\circ}$ , so the identity holds everywhere. Since $s, r > 0$ , the contraction is nowhere zero and therefore the curvature tensor is nowhere zero. This proves (d).

Remarks

What is proved and what is not. The computation proves the tensor identities $Ric = 0$ , $S = 0$ , $G = 0$ on the exterior domain. It does not assert that $g$ is a solution of the framework's field law: that reading requires the field law and the vacuum prescription and is the content of the exterior vacuum-model theorem of this page. It also does not assert that the Schwarzschild metric is the only Ricci-flat metric or the only spherically symmetric one; uniqueness statements are the local Birkhoff theorem of this page.
The curvature invariant. Contracting the components of clause (a) against the inverse metric gives the Kretschmann scalar of the model, a computation performed separately in the curvature-and-Kretschmann lemma of this page; the result $12 s^{2} / r^{6}$ is finite at $r = s$ and diverges as $r ↓ 0$ , the basis of the singularity discussion of this page.

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Orthonormal-frame curvature components and the Kretschmann scalar 12s^2/r^6 of the Schwarzschild exterior metric

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum of mass parameter $m_{*}$ , Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), with curvature data as in Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric. On each spherical-polar product chart $(x^{0}, r, ϑ, φ)$ define the local frame field

$e_{\hat{0}} : = f^{- 1 / 2} \partial_{0}, e_{\hat{1}} : = f^{1 / 2} \partial_{1}, e_{\hat{2}} : = r^{- 1} \partial_{2}, e_{\hat{3}} : = (r \sin ϑ)^{- 1} \partial_{3},$

and for the curvature tensor define the frame components

${\hat{R}}_{\hat{a} \hat{b} \hat{c} \hat{d}} : = g (R (e_{\hat{c}}, e_{\hat{d}}) e_{\hat{b}}, e_{\hat{a}}) .$

Then:

(a) The frame field. $(e_{\hat{0}}, e_{\hat{1}}, e_{\hat{2}}, e_{\hat{3}})$ is a smooth $g$ -orthonormal frame (tetrad field) on the chart domain, with $g (e_{\hat{a}}, e_{\hat{b}}) = diag (- 1, 1, 1, 1)_{\hat{a} \hat{b}}$ , and $e_{\hat{0}}$ is future-directed timelike; it is a tetrad field in the sense of Local frame fields, coordinate frames and tetrad fields. It is not a coordinate frame.

(b) Frame components. The nonzero frame components, up to the symmetries $R_{a b c d} = - R_{b a c d} = - R_{a b d c} = R_{c d a b}$ of The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors, are

${\hat{R}}_{\hat{0} \hat{1} \hat{0} \hat{1}} = \frac{f^{''}}{2}, {\hat{R}}_{\hat{0} \hat{2} \hat{0} \hat{2}} = {\hat{R}}_{\hat{0} \hat{3} \hat{0} \hat{3}} = \frac{f^{'}}{2 r},$

${\hat{R}}_{\hat{1} \hat{2} \hat{1} \hat{2}} = {\hat{R}}_{\hat{1} \hat{3} \hat{1} \hat{3}} = - \frac{f^{'}}{2 r}, {\hat{R}}_{\hat{2} \hat{3} \hat{2} \hat{3}} = \frac{1 - f}{r^{2}},$

with $f^{'} = s / r^{2}$ and $f^{''} = - 2 s / r^{3}$ ; all other frame components vanish.

(c) Kretschmann scalar. Define the global smooth scalar field $K : = R_{a b c d} R^{a b c d}$ , with indices raised by the inverse metric. On the spherical-polar chart locus its component expression is

$K = f^{'' 2} + \frac{4 (f^{'})^{2}}{r^{2}} + \frac{4 (1 - f)^{2}}{r^{4}},$

and for the model function $f = 1 - s / r$ this is

$K = \frac{12 s^{2}}{r^{6}} = \frac{48 G_{N}^{2} m_{*}^{2}}{c^{4} r^{6}} .$

The scalar identity is first computed on the regular angular locus and holds on all of $M$ by smoothness and density. Since $r > s$ on $M$ , one has $0 < K = 12 s^{2} / r^{6} < 12 / s^{4}$ at every point; as $r ↓ s$ the expression has the finite boundary value $12 / s^{4}$ , and the expression $12 s^{2} / r^{6}$ , read as a function of $r > 0$ , grows without bound as $r ↓ 0$ .

(d) Scope. The lemma computes curvature data of the exterior datum on $r > s$ . The limit $r ↓ s$ is the boundary value on $M$ of the smooth expression $12 s^{2} / r^{6}$ ; no value of the metric at $r = s$ or at $r = 0$ is used or asserted here, those regions being outside the datum (the regular continuation across $r = s$ is the separate extension lemma of this page).

Facts & Assumptions

Given: The Schwarzschild exterior datum, a spherical-polar product chart $(x^{0}, r, ϑ, φ)$ with $0 < ϑ < π$ , the functions $f = 1 - s / r$ , $f^{'} = s / r^{2}$ , $f^{''} = - 2 s / r^{3}$ , and the local frame field $e_{\hat{a}}$ of the statement.

[L1]

The metric components and inverse components in the chart are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ , $g^{00} = - f^{- 1}$ , $g^{11} = f$ , $g^{22} = r^{- 2}$ , $g^{33} = r^{- 2} \sin^{- 2} ϑ$ , off-diagonal components zero (The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, The inverse metric, musical isomorphisms and index raising).

[L2]

The nonzero lowered curvature components in the chart are $R_{0101} = f^{''} / 2$ , $R_{0202} = r f f^{'} / 2$ , $R_{0303} = r f f^{'} \sin^{2} ϑ / 2$ , $R_{1212} = - r f^{'} / (2 f)$ , $R_{1313} = - r f^{'} \sin^{2} ϑ / (2 f)$ , $R_{2323} = r^{2} (1 - f) \sin^{2} ϑ$ , and every component whose index pairs do not coincide as unordered pairs, or with a repeated index inside a pair, vanishes (Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric clauses (a) and (d)).

[L3]

Curvature components transform as those of a $(0, 4)$ -tensor: if $e_{\hat{a}} = d_{\hat{a}}^{μ} \partial_{μ}$ are the coefficients of a frame field, then the frame components of the lowered curvature are ${\hat{R}}_{\hat{a} \hat{b} \hat{c} \hat{d}} = d_{\hat{a}}^{α} d_{\hat{b}}^{β} d_{\hat{c}}^{γ} d_{\hat{d}}^{δ} R_{α β γ δ}$ evaluated on the same chart domain; the tensor $R$ is smooth, its components transform by this law, and tensor identities and scalar contractions are intrinsic (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clauses (a)-(b), Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors).

[L4]

The scalar $R_{a b c d} R^{a b c d}$ with $R^{a b c d} = g^{a α} g^{b β} g^{c γ} g^{d δ} R_{α β γ δ}$ is frame independent. In an orthonormal Lorentz frame each ordered component contributes $(- 1)^{m} ({\hat{R}}_{\hat{a} \hat{b} \hat{c} \hat{d}})^{2}$ , where $m$ is the number of time indices among $\hat{a}, \hat{b}, \hat{c}, \hat{d}$ . The only nonzero frame components listed in clause (b), and all their symmetry variants, have either zero or two time indices, so every nonzero contribution has positive sign (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, Local frame fields, coordinate frames and tetrad fields, Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric clause (a)).

[L5]

The model functions satisfy $f^{'} = s / r^{2}$ , $f^{''} = - 2 s / r^{3}$ ; on the chart domain $r > s > 0$ and $\sin ϑ > 0$ , and the displayed expressions are smooth and positive or negative as indicated (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Rational powers $a^{r}$ of a positive base).

Proof

technique · verify the frame is orthonormal, rescale the coordinate curvature components of the preceding lemma into the frame, and add the squares of the frame components

1.1L1L5givenalgebra

(Orthonormality.) By [L1], $g (e_{\hat{0}}, e_{\hat{0}}) = f^{- 1} g_{00} = - 1$ , $g (e_{\hat{1}}, e_{\hat{1}}) = f g_{11} = 1$ , $g (e_{\hat{2}}, e_{\hat{2}}) = r^{- 2} g_{22} = 1$ and $g (e_{\hat{3}}, e_{\hat{3}}) = (r \sin ϑ)^{- 2} g_{33} = 1$ . Each $e_{\hat{a}}$ is a smooth multiple of a coordinate field by a smooth nonvanishing function on the chart domain, so the $e_{\hat{a}}$ are smooth vector fields and form a basis at every point; the mixed pairings vanish because the coordinate frame is $g$ -orthogonal by [L1]. Hence the frame is $g$ -orthonormal with the stated diagonal pairing. Since $e_{\hat{0}} = f^{- 1 / 2} \partial_{0}$ with $f^{- 1 / 2} > 0$ and $\partial_{0}$ is future-directed by the time orientation of the datum, $e_{\hat{0}}$ is future-directed. This proves (a).

2.1L2L3step 1.1algebra

(Frame components.) The coefficient matrix $d_{\hat{a}}^{μ}$ of [L3] is diagonal with entries $d_{\hat{0}}^{0} = f^{- 1 / 2}$ , $d_{\hat{1}}^{1} = f^{1 / 2}$ , $d_{\hat{2}}^{2} = r^{- 1}$ , $d_{\hat{3}}^{3} = (r \sin ϑ)^{- 1}$ . Applying [L3] to the nonzero components of [L2]: ${\hat{R}}_{\hat{0} \hat{1} \hat{0} \hat{1}} = R_{0101} (d_{\hat{0}}^{0})^{2} (d_{\hat{1}}^{1})^{2} = (f^{''} / 2) f^{- 1} f = f^{''} / 2$ ; ${\hat{R}}_{\hat{0} \hat{2} \hat{0} \hat{2}} = R_{0202} (d_{\hat{0}}^{0})^{2} (d_{\hat{2}}^{2})^{2} = (r f f^{'} / 2) f^{- 1} r^{- 2} = f^{'} / (2 r)$ ; ${\hat{R}}_{\hat{0} \hat{3} \hat{0} \hat{3}} = R_{0303} (d_{\hat{0}}^{0})^{2} (d_{\hat{3}}^{3})^{2} = (r f f^{'} \sin^{2} ϑ / 2) f^{- 1} (r \sin ϑ)^{- 2} = f^{'} / (2 r)$ ; ${\hat{R}}_{\hat{1} \hat{2} \hat{1} \hat{2}} = R_{1212} (d_{\hat{1}}^{1})^{2} (d_{\hat{2}}^{2})^{2} = (- r f^{'} / (2 f)) f r^{- 2} = - f^{'} / (2 r)$ ; ${\hat{R}}_{\hat{1} \hat{3} \hat{1} \hat{3}} = R_{1313} (d_{\hat{1}}^{1})^{2} (d_{\hat{3}}^{3})^{2} = (- r f^{'} \sin^{2} ϑ / (2 f)) f (r \sin ϑ)^{- 2} = - f^{'} / (2 r)$ ; and ${\hat{R}}_{\hat{2} \hat{3} \hat{2} \hat{3}} = R_{2323} (d_{\hat{2}}^{2})^{2} (d_{\hat{3}}^{3})^{2} = r^{2} (1 - f) \sin^{2} ϑ r^{- 2} (r \sin ϑ)^{- 2} = (1 - f) / r^{2}$ . Every other frame component is a diagonal coefficient multiple of a coordinate component that vanishes by [L2], hence vanishes. This proves (b) and identifies all nonzero frame components.

3.1L2L3L4step 2.1algebra

(The Kretschmann scalar from the frame components.) By [L4] and the fact that every nonzero component has zero or two time indices, each nonzero ordered quadruple contributes the positive square of its component. The pair symmetries of [L2] and [L3] show that for each of the six unordered pairs $P$ of distinct frame indices, the four ordered quadruples realizing $P$ in both pairs contribute four times the square of the corresponding component of step 2.1. Hence $K = 4 (f^{''} / 2)^{2} + 4 (f^{'} / (2 r))^{2} \cdot 4 + 4 ((1 - f) / r^{2})^{2}$ : the first term is the pair ${0, 1}$ ; the factor $4 \cdot 4$ collects the four pairs ${0, 2}, {0, 3}, {1, 2}, {1, 3}$ each contributing $4 (f^{'} / (2 r))^{2}$ ; and the last term is the pair ${2, 3}$ . Simplifying, $K = f^{'' 2} + 4 (f^{'})^{2} / r^{2} + 4 (1 - f)^{2} / r^{4}$ . The contraction is a smooth scalar on all of $M$ , and this expression is smooth there; they agree on the dense regular angular locus, so they agree at the poles by continuity. This proves (c) up to the model specialisation.

4.1L5step 3.1algebra∎

(Specialisation and limits.) For $f = 1 - s / r$ one has $f^{'} = s / r^{2}$ and $f^{''} = - 2 s / r^{3}$ , so $f^{'' 2} = 4 s^{2} / r^{6}$ , $4 (f^{'})^{2} / r^{2} = 4 s^{2} / r^{6}$ and $4 (1 - f)^{2} / r^{4} = 4 s^{2} / r^{6}$ ; the sum is $12 s^{2} / r^{6} = 48 G_{N}^{2} m_{*}^{2} / (c^{4} r^{6})$ , using $s = 2 G_{N} m_{*} / c^{2}$ . On $M$ the identity $K = 12 s^{2} / r^{6}$ holds at every point with $r > s > 0$ , so $0 < K < 12 / s^{4}$ ; as $r ↓ s$ the right-hand side tends to the finite value $12 / s^{4}$ , as $r \to + \infty$ it tends to $0$ , and as a function of $r > 0$ the expression $12 s^{2} / r^{6}$ tends to $+ \infty$ as $r ↓ 0$ . These are statements about the displayed expression, not about the metric at $r = s$ or $r = 0$ ; no value of the metric outside $M$ is used or asserted, which is clause (d).

Remarks

Conventions and the source value. In the literature the invariant is often written $48 G^{2} M^{2} / r^{6}$ in units $G = c = 1$ ; with our $s = 2 G_{N} m_{*} / c^{2}$ this is the same number as $12 s^{2} / r^{6}$ . The displayed value $12 G^{2} M^{2} / r^{6}$ appearing in one source cited above is a factor-4 discrepancy from the direct computation performed here and is not inherited; the other cited source gives the consistent value.
What the invariant shows. The boundary value $12 / s^{4}$ of $K$ at $r ↓ s$ shows that the divergence of the chart component $g_{11}$ at $r = s$ is not accompanied by a divergence of this curvature invariant; on the smooth extension of the next lemma, where $r$ ranges over $(0, \infty)$ and $K = 12 s^{2} / r^{6}$ holds there, the same expression grows without bound as $r ↓ 0$ . Both facts are used in the horizon-and-singularity remark of this page.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum with Schwarzschild radius $s > 0$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation). Let $M_{e x t} : = R \times (0, \infty) \times S^{2}$ carry the product smooth structure and let $ι_{e x t} : M_{e x t} \to S^{2}$ be the third projection. Its product atlas includes ordinary sphere charts at the poles (Products of smooth manifolds have a canonical product smooth structure). Define the tortoise function

$r_{*} (r) : = r + s \log ∣ \frac{r}{s} - 1 ∣, r > 0, r \neq s,$

on each of the two intervals $(0, s)$ and $(s, \infty)$ , and define the advanced-chart metric globally on $M_{e x t}$ by

$g_{e x t} = - f (r) d v \otimes d v + d v \otimes d r + d r \otimes d v + r^{2} ι_{e x t}^{*} γ_{S^{2}} .$

In spherical-polar product charts its components are

$g_{e x t} = - f (r) d v^{2} + 2 d v d r + r^{2} d Ω^{2},$

namely $g_{v v} = - f$ , $g_{v r} = g_{r v} = 1$ , $g_{ϑ ϑ} = r^{2}$ , $g_{φ φ} = r^{2} \sin^{2} ϑ$ and all other components zero. In another sphere chart the angular block is $r^{2} (γ_{i j})$ . Then:

(a) Smooth Lorentzian metric. $g_{e x t}$ is a smooth symmetric covariant $2$ -tensor field on $M_{e x t}$ , and at every point it is nondegenerate with inertia $(3, 1, 0)$ ; hence $(M_{e x t}, g_{e x t})$ is a smooth Lorentzian four-manifold of signature $(-, +, +, +)$ on all of $r > 0$ , including $r = s$ .

(b) Time orientation. The smooth vector field

$N_{e x t} : = A (r) \partial_{v} - \partial_{r}, A (r) : = \frac{f (r) + \sqrt{f (r)^{2} + 4}}{2} > 0,$

satisfies $g_{e x t} (N_{e x t}, N_{e x t}) < 0$ everywhere and $g_{e x t} (N_{e x t}, \partial_{v}) < 0$ everywhere; hence $N_{e x t}$ is a time orientation of $g_{e x t}$ , and on the region $r > s$ it determines the same future cone as the datum's time orientation $\partial_{v} = \partial_{0}$ .

(c) Isometric embedding of the exterior. The map $Φ : M \to M_{e x t}$ , $(x^{0}, r, ω) \mapsto (x^{0} + r_{*} (r), r, ω)$ restricts to a diffeomorphism of $M$ onto the open subset ${r > s}$ of $M_{e x t}$ , and $Φ^{*} g_{e x t} = g$ ; the exterior datum is therefore isometric to the region $r > s$ of the extension.

(d) Vacuum and Kretschmann scalar of the extension. The metric-selected connection of $g_{e x t}$ exists and is unique, its Ricci tensor, scalar curvature and Einstein tensor vanish identically on $M_{e x t}$ , and its Kretschmann scalar equals

$R_{a b c d} R^{a b c d} = \frac{12 s^{2}}{r^{6}}$

on all of $M_{e x t}$ . In particular the scalar is smooth and finite on the horizon $r = s$ , where it takes the value $12 / s^{4}$ .

(e) The horizon is a null hypersurface. The level set ${r = s}$ is a smooth null hypersurface of $(M_{e x t}, g_{e x t})$ whose null generator direction is $\partial_{v}$ ; every smooth product-chart frame remains a basis across $r = s$ . In spherical-polar charts $\det (g_{e x t}) = - r^{4} \sin^{2} ϑ$ ; in an arbitrary sphere chart the determinant is $- r^{4} \det (γ_{i j})$ , which is nonzero.

(f) Scope. The extension constructed here is a smooth Lorentzian manifold containing the exterior isometrically and regular across $r = s$ ; it is not claimed to be maximal, and no uniqueness or global structure theorem is asserted. The region $r \leq 0$ and the surfaces relevant to the Kruskal model of this page are outside this statement.

Facts & Assumptions

Given: The Schwarzschild exterior datum with $s > 0$ , $f = 1 - s / r$ ; the product manifold $M_{e x t} = R \times (0, \infty) \times S^{2}$ ; the function $r_{*}$ of the statement; the assigned components of $g_{e x t}$ ; and the maps $Φ$ and $N_{e x t}$ .

[L1]

On each of $(0, s)$ and $(s, \infty)$ the positive function $∣ r / s - 1 ∣$ is smooth. The natural logarithm is smooth on $(0, \infty)$ because it is the inverse of the smooth exponential with nonzero derivative, and its derivative is $1 / x$ ; hence $r_{*} (r) = r + s \log ∣ r / s - 1 ∣$ is smooth on each branch and $r_{*}^{'} = 1 + s / (r - s) = r / (r - s) = 1 / f (r)$ there (The natural logarithm as the inverse of the exponential function, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , Choice-free smooth inverse function theorem in Euclidean space, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Rational powers $a^{r}$ of a positive base). The function $f$ is smooth on $(0, \infty)$ and is positive exactly for $r > s$ .

[L2]

Globally $g = - f d (x^{0}) \otimes d (x^{0}) + f^{- 1} d r \otimes d r + r^{2} ι^{*} γ_{S^{2}}$ ; in a spherical-polar product chart its angular components are $γ_{22} = 1$ , $γ_{33} = \sin^{2} ϑ$ . The map $Φ$ is the identity on the sphere factor, so its pullback preserves $r^{2} γ_{S^{2}}$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clauses (b)-(c), The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, The round metric on the sphere as an induced metric).

[L3]

The base block $(\begin{array}{cc} - f & 1 \\ 1 & 0 \end{array})$ has determinant $- 1$ , inverse $(\begin{array}{cc} 0 & 1 \\ 1 & f \end{array})$ , and one positive and one negative eigenvalue; the sphere block is $r^{2} γ_{S^{2}}$ , positive definite at every sphere point. Their orthogonal product therefore has inertia $(3, 1, 0)$ everywhere. In a polar chart the full determinant is $- r^{4} \sin^{2} ϑ$ , while in any angular chart it is $- r^{4} \det (γ_{i j})$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, The round metric on the sphere as an induced metric).

[L4]

A smooth vector field $N$ with $g (N, N) < 0$ everywhere is a time orientation; if two timelike fields $N, N^{'}$ satisfy $g (N, N^{'}) < 0$ at every point of a connected set, they determine the same future cone there (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c), Two time orientations agree or are opposite on each connected set).

[L5]

The global tensor $g_{e x t} = - f d v \otimes d v + d v \otimes d r + d r \otimes d v + r^{2} ι_{e x t}^{*} γ_{S^{2}}$ is smooth because all scalar functions, one-forms and the round metric are smooth; thus the product chart atlas, including pole charts, gives continuous Ricci and curvature-scalar components (Smooth covariant 2-tensor fields on a smooth manifold, Chart independence and the transformation law of a smooth covariant 2-tensor field, $C^{r}$ and smooth maps between smooth manifolds, The round metric on the sphere as an induced metric).

[L6]

If two continuous maps from a topological space into a Hausdorff space agree on a dense subset, they agree everywhere (Two continuous maps into a Hausdorff space that agree on a dense subset are equal).

[L7]

The Ricci tensor, scalar curvature and Einstein tensor of the metric $g_{f} = - f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} d Ω^{2}$ with $f = 1 - s / r$ vanish identically on $r > s$ ; the computation of (a)-(c) of Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric is algebraic in $f, f^{'}, f^{''}$ and the identities $f^{''} / 2 + f^{'} / r = 0$ , $1 - f - r f^{'} = 0$ hold for $f = 1 - s / r$ at every $r \neq 0$ (Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric, Christoffel symbols of the Schwarzschild exterior metric in product charts).

[L8]

The Kretschmann scalar of the exterior metric is $12 s^{2} / r^{6}$ on $r > s$ (Orthonormal-frame curvature components and the Kretschmann scalar 12s^2/r^6 of the Schwarzschild exterior metric); a fully contracted curvature scalar of a smooth nondegenerate metric is a smooth function in every chart.

Proof

technique · verify smoothness and signature from the explicit component matrix, verify the pullback identity by the chain rule, obtain the vacuum property on $r\ne s$ algebraically and extend by continuity, and verify the horizon and orientation claims directly

1.1L3L5L1algebra

(Smoothness and signature.) The global tensor expression in [L5] is smooth on the product manifold. At any point its base block on the $(v, r)$ factor is $(\begin{array}{cc} - f & 1 \\ 1 & 0 \end{array})$ , with determinant $- 1$ and inertia $(1, 1, 0)$ , while its angular block is $r^{2} γ_{S^{2}}$ , positive definite on the entire sphere including the poles. The product splitting therefore gives a smooth nondegenerate metric of inertia $(3, 1, 0)$ everywhere. In a spherical-polar chart the determinant is $- r^{4} \sin^{2} ϑ$ ; in an arbitrary sphere chart it is $- r^{4} \det (γ_{i j}) \neq 0$ . This proves (a) and the determinant claims of (e).

1.2L1L3L4algebra

(Time orientation.) For $N_{e x t} = A \partial_{v} - \partial_{r}$ with $A = (f + \sqrt{f^{2} + 4}) / 2$ , one has $g_{e x t} (N_{e x t}, N_{e x t}) = - f A^{2} - 2 A = - A (f A + 2)$ and $g_{e x t} (N_{e x t}, \partial_{v}) = - f A - 1$ . By [L1], $A > 0$ because $\sqrt{f^{2} + 4} > ∣ f ∣ \geq - f$ . Next $f A + 1 = \frac{f^{2} + f \sqrt{f^{2} + 4} + 2}{2} > 0$ : for $f \geq 0$ its numerator is at least $2$ ; for $f < 0$ , writing $u = - f > 0$ , the inequality $u \sqrt{u^{2} + 4} < u^{2} + 2$ follows by squaring the two positive sides, $u^{2} (u^{2} + 4) < (u^{2} + 2)^{2}$ , so the numerator is positive again. Hence $f A + 2 > f A + 1 > 0$ , and therefore $g_{e x t} (N_{e x t}, N_{e x t}) = - A (f A + 2) < 0$ and $g_{e x t} (N_{e x t}, \partial_{v}) = - (f A + 1) < 0$ . Thus $N_{e x t}$ is a smooth nowhere-zero timelike field; by [L4] it is a time orientation of $g_{e x t}$ , and on the region $r > s$ , where $f > 0$ and $\partial_{v} = \partial_{0}$ , the negative pairing $g_{e x t} (N_{e x t}, \partial_{v}) < 0$ shows by [L4] that it determines the same future cone as the datum's orientation. This proves (b).

1.3L1L2algebra

(Pullback identity.) The map $Φ$ and its inverse $(v, r, ω) \mapsto (v - r_{*} (r), r, ω)$ are smooth on $r > s$ by [L1], so $Φ$ is a diffeomorphism onto that open subset. Pull back the base terms using $d v = d (x^{0}) + f^{- 1} d r$ : the cross terms cancel and the $d r^{2}$ coefficient becomes $- f (f^{- 1})^{2} + 2 f^{- 1} = f^{- 1}$ . The map is the identity on the sphere factor, so it pulls $r^{2} ι_{e x t}^{*} γ_{S^{2}}$ back to $r^{2} ι^{*} γ_{S^{2}}$ . Thus $Φ^{*} g_{e x t} = - f d (x^{0}) \otimes d (x^{0}) + f^{- 1} d r \otimes d r + r^{2} ι^{*} γ_{S^{2}} = g$ by [L2], proving (c).

2.1L6L7step 1.1step 1.3algebra

(Vacuum property on $r \neq s$ .) On $r > s$ the pullback identity of step 1.3 expresses $g_{e x t}$ as the datum metric of [L7], which is Ricci-flat there. On $0 < r < s$ the same coordinate calculation applies on the other branch of $r_{*}$ ; its Christoffel and Ricci formulas are algebraic in $f, f^{'}, f^{''}$ and $r$ and do not use the sign of $f$ , so the metric is Ricci-flat there too. Thus the Ricci tensor vanishes on the dense open set $D = R \times ((0, s) \cup (s, \infty)) \times (S^{2} ∖ {two poles})$ . The global metric is smooth and nondegenerate by step 1.1, so its Ricci tensor is smooth. In any smooth product chart, each Ricci component is continuous and vanishes on the dense subset $D$ of that chart; it therefore vanishes on the chart, including at $r = s$ and at the sphere poles. The scalar curvature and Einstein tensor vanish by their definitions. This proves the first part of (d).

3.1L6L8step 2.1algebra

(Kretschmann scalar of the extension.) By [L8] and the algebraic calculation on the two non-horizon branches, the global scalar $K = R_{a b c d} R^{a b c d}$ equals $12 s^{2} / r^{6}$ on $D$ defined in step 2.1. Both sides are smooth on all of $M_{e x t}$ : the left by the smooth nondegenerate global metric and the right because $r > 0$ . They therefore agree everywhere by continuity and density, including at the horizon, the sphere poles, and $r = s$ , where the value is $12 / s^{4}$ . This proves the remaining part of (d).

4.1step 1.1L3L5given∎

(The horizon and the conclusion.) At $r = s$ one has $f = 0$ , so the inverse base block of [L3] is $(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ and the covector $d r$ has squared norm $g_{e x t}^{- 1} (d r, d r) = 0$ . Since $d r \neq 0$ and the level set ${r = s}$ has codimension one, it is a null hypersurface. Its tangent space is spanned by $\partial_{v}$ and the sphere tangent space; $\partial_{v}$ is null at $r = s$ because $g_{e x t} (\partial_{v}, \partial_{v}) = - f = 0$ , and it is the null generator direction. The global block decomposition of step 1.1 is nondegenerate at the horizon, so every smooth chart frame there is a basis; in polar charts the determinant is $- s^{4} \sin^{2} ϑ$ , and in pole charts it is $- s^{4} \det (γ_{i j}) \neq 0$ . Clause (f) records that no maximality or uniqueness is claimed. This proves (e) and completes the proof.

Remarks

What the extension achieves and what it does not. The construction shows that the apparent breakdown of the Schwarzschild chart at $r = s$ is a property of that chart: the same metric is smooth and nondegenerate in the advanced chart across the horizon, and its curvature invariant is finite there. It does not identify the maximal extension of the model; that involves the Kruskal model of this page, and it does not show that every geodesic extends, a statement requiring global arguments not made here.
The two branches. The tortoise function is defined separately on $(0, s)$ and $(s, \infty)$ and is not continuous across $r = s$ ; it is only its derivative $1 / f$ that is used in the pullback computation, and the metric components of $g_{e x t}$ do not involve $r_{*}$ at all. This is why the extension is smooth even though the auxiliary coordinate $x^{0}$ is not defined across the horizon as a single-valued function (only its combinations $v$ and $u = v - 2 r_{*}$ are).

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The isotropic coordinate form of the Schwarzschild exterior metric and the isothermal radius shift

Statement

$R : (s / 4, \infty) \to (s, \infty), R (ρ) : = ρ (1 + \frac{s}{4 ρ})^{2},$

and define the isotropic radius of a point $p \in M$ by $ρ (p) : = ρ_{0} (r (p))$ , where $ρ_{0} : (s, \infty) \to (s / 4, \infty)$ is the inverse of $R$ . Write $a : = s / (4 ρ)$ . Then:

(a) The coordinate change. $R$ is a smooth bijection with smooth inverse and $R^{'} (ρ) = 1 - \frac{s^{2}}{16 ρ^{2}} > 0$ ; its inverse is the smooth function

$ρ_{0} (r) = \frac{(r - \frac{s}{2}) + \sqrt{r (r - s)}}{2},$

and the identity $R (ρ) = ρ + \frac{s}{2} + \frac{s^{2}}{16 ρ}$ holds on $(s / 4, \infty)$ .

(b) The metric in isotropic coordinates. In the chart $(x^{0}, ρ, ϑ, φ)$ obtained by replacing the coordinate $r$ by $ρ (p)$ (so that $r = R (ρ)$ in the chart), the metric of the datum has the components

$g = - (\frac{1 - a}{1 + a})^{2} d (x^{0})^{2} + (1 + a)^{4} (d ρ^{2} + ρ^{2} (d ϑ^{2} + \sin^{2} ϑ d φ^{2})),$

that is, $g_{00} = - (1 - a)^{2} / (1 + a)^{2}$ , $g_{ρ ρ} = (1 + a)^{4}$ , $g_{ϑ ϑ} = (1 + a)^{4} ρ^{2}$ , $g_{φ φ} = (1 + a)^{4} ρ^{2} \sin^{2} ϑ$ and all other components zero.

(c) Conformal flatness and the large-radius expansion. On the region where $(ρ, ϑ, φ)$ are the standard spherical coordinates of $R^{3} ∖ {0}$ , the spatial part of $g$ is the conformally flat metric $(1 + a)^{4} δ$ , with $δ = d ρ^{2} + ρ^{2} (d ϑ^{2} + \sin^{2} ϑ d φ^{2})$ in Cartesian coordinates $y = ρ \hat{ω}$ , $∣ y ∣ = ρ$ . Moreover, for every lower bound $ρ_{*} > s / 4$ there is a constant $C (ρ_{*})$ with

$∣ (1 + \frac{s}{4 ρ})^{4} - 1 - \frac{s}{ρ} ∣ \leq \frac{C (ρ_{*}) s^{2}}{ρ^{2}} for ρ \geq ρ_{*},$

and $ρ_{0} (r) = r - \frac{s}{2} - \frac{s^{2}}{16 r} + O (s^{3} / r^{2})$ as $r / s \to \infty$ with fixed $s > 0$ ; in particular $ρ$ is not equal to the areal radius $r$ .

(d) The comparison warning. The areal-radius coefficient $f (r)^{- 1} = (1 - s / r)^{- 1} = 1 + s / r + O (r^{- 2})$ and the isotropic coefficient $(1 + s / (4 ρ))^{4} = 1 + s / ρ + O (ρ^{- 2})$ agree at leading order but differ at next order when one substitutes $ρ = r$ ; consequently any comparison of the Schwarzschild coefficients with weak-field coefficients stated in another chart must first convert to a single chart by the explicit map $R$ and must carry the resulting error terms. Matching coefficient functions in two different radial coordinates without the conversion is not a valid identification.

Facts & Assumptions

Given: The Schwarzschild exterior datum with $s > 0$ , the functions $R$ and $a$ of the statement, and a spherical-polar product chart of the datum.

[L1]

The function $f (r) = 1 - s / r$ is smooth and positive on $(s, \infty)$ , and in a spherical-polar product chart the metric components are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold).

[L2]

The function $x \mapsto x^{1 / 2}$ agrees with the positive square root on $[0, \infty)$ and is smooth with derivative $\frac{1}{2} x^{- 1 / 2}$ on $(0, \infty)$ ; consequently $\sqrt{r (r - s)}$ is smooth on $(s, \infty)$ because $r (r - s)$ is smooth and strictly positive there. For $h (t) : = (1 - t)^{1 / 2}$ on $[0, 1 / 2]$ , $h (0) = 1$ , $h^{'} (0) = - 1 / 2$ , $h^{''} (0) = - 1 / 4$ , and $∣ h^{'''} (t) ∣ = \frac{3}{8} (1 - t)^{- 5 / 2} \leq \frac{3}{8} 2^{5 / 2}$ . Set $R_{3} (t) : = h (t) - 1 + t / 2 + t^{2} / 8$ . For fixed $0 < t \leq 1 / 2$ , the function $G (u) : = R_{3} (u) - u^{3} R_{3} (t) / t^{3}$ has $G (0) = G^{'} (0) = G^{''} (0) = G (t) = 0$ . Three applications of Rolle's theorem, which follows from the mean value theorem, give $ξ \in (0, t)$ with $G^{'''} (ξ) = 0$ , hence $R_{3} (t) = h^{'''} (ξ) t^{3} / 6$ . Therefore $h (t) = 1 - \frac{t}{2} - \frac{t^{2}}{8} + R_{3} (t), ∣ R_{3} (t) ∣ \leq \frac{(3 / 8) 2^{5 / 2}}{6} t^{3} (0 \leq t \leq 1 / 2) .$ Sums, compositions and quotients of smooth functions are smooth (The exponential definition of real powers agrees with the existing rational powers, Continuity and derivatives of positive-base real powers, Rational powers $a^{r}$ of a positive base, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{r}$ and smooth maps between smooth manifolds, The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[L3]

Derivative rules: $(R)^{'} = 1 - \frac{s^{2}}{16 ρ^{2}}$ for the function $R (ρ) = ρ (1 + \frac{s}{4 ρ})^{2}$ , obtained by expanding $R (ρ) = ρ + \frac{s}{2} + \frac{s^{2}}{16 ρ}$ and differentiating termwise; the derivative is continuous on $(s / 4, \infty)$ and strictly positive there because $16 ρ^{2} > s^{2}$ precisely when $ρ > s / 4$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Rational powers $a^{r}$ of a positive base, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L4]

A continuous strictly monotone function on an interval has an inverse on its image, and a continuous function on an interval whose one-sided limits at the endpoints are $s$ and $+ \infty$ attains every intermediate value; the intervals $(s / 4, \infty)$ and $(s, \infty)$ are connected (A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values, A subset of $R$ is connected if and only if it is order-convex, that is, an interval, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L5]

For a map that is a diffeomorphism onto its image, the components of a tensor field pull back by the chain rule; in particular replacing the coordinate $r$ by $ρ$ with $r = R (ρ)$ multiplies $d r$ by $R^{'} (ρ)$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Smooth covariant 2-tensor fields on a smooth manifold).

Proof

technique · verify the explicit inverse and positivity of $R'$, substitute $r=R(\rho)$ in the metric components, and expand the resulting coefficients

1.1L2L3L4algebra

(The coordinate change and its inverse.) Expanding $R (ρ) = ρ (1 + \frac{s}{4 ρ})^{2} = ρ + \frac{s}{2} + \frac{s^{2}}{16 ρ}$ gives the algebraic identity of clause (a), and differentiating termwise gives $R^{'} (ρ) = 1 - \frac{s^{2}}{16 ρ^{2}} = (16 ρ^{2} - s^{2}) / (16 ρ^{2}) > 0$ for $ρ > s / 4$ by [L3]. Hence $R$ is strictly increasing and continuous on the interval $(s / 4, \infty)$ and $R (ρ) \to s$ as $ρ ↓ s / 4$ , while $R (ρ) \to + \infty$ as $ρ \to \infty$ ; by [L4] the image of $R$ is the whole interval $(s, \infty)$ and $R$ is a bijection onto it. For the displayed inverse, substitution gives $2 ρ_{0} (r) - r + \frac{s}{2} = \sqrt{r (r - s)}$ , so $ρ_{0} (r)$ satisfies the quadratic identity $ρ_{0} (r)^{2} - ρ_{0} (r) (r - \frac{s}{2}) + \frac{s^{2}}{16} = 0$ ; dividing this identity by $ρ_{0} (r) > 0$ gives $ρ_{0} (r) + \frac{s^{2}}{16 ρ_{0} (r)} = r - \frac{s}{2}$ , hence $R (ρ_{0} (r)) = ρ_{0} (r) + \frac{s}{2} + \frac{s^{2}}{16 ρ_{0} (r)} = r$ . Thus $ρ_{0}$ is a left inverse of $R$ , hence the inverse of the bijection $R$ ; it is smooth on $(s, \infty)$ by [L2], and $ρ_{0} (r) > s / 4$ because $2 ρ_{0} (r) = r - \frac{s}{2} + \sqrt{r (r - s)} > \frac{s}{2}$ for $r > s$ .

1.2L1L3L5algebra

(Metric coefficients in the new chart.) In the chart with coordinates $(x^{0}, ρ, ϑ, φ)$ , where $r = R (ρ)$ , the components of $g$ are determined by the transformation $r \mapsto ρ$ : the coefficient of $d ρ^{2}$ is $g_{11} (R^{'} (ρ))^{2}$ and the angular coefficient $ρ^{2}$ acquires the factor $(R / ρ)^{2}$ in addition to $g_{22}$ and $g_{33}$ , because $r^{2} = R (ρ)^{2} = ρ^{2} (1 + \frac{s}{4 ρ})^{4}$ ; the time component $g_{00}$ is unchanged since $x^{0}$ is not transformed. Writing $a = s / (4 ρ)$ , i.e. $R = ρ (1 + a)^{2}$ and $R^{'} = 1 - a^{2}$ by [L3], we obtain $1 - f (R) = s / R = 4 a / (1 + a)^{2}$ , hence $f (R) = 1 - \frac{4 a}{(1 + a)^{2}} = \frac{(1 - a)^{2}}{(1 + a)^{2}}, \frac{R^{' 2}}{f (R)} = \frac{(1 - a^{2})^{2} (1 + a)^{2}}{(1 - a)^{2}} = (1 + a)^{4} .$ Thus $g_{00} = - (1 - a)^{2} / (1 + a)^{2}$ , $g_{ρ ρ} = (1 + a)^{4}$ and the angular components are $R^{2} γ_{i j} = (1 + a)^{4} ρ^{2} γ_{i j}$ with $γ_{22} = 1$ , $γ_{33} = \sin^{2} ϑ$ . All off-diagonal components remain zero. This is clause (b).

2.1L2step 1.1algebra

(Conformal flatness and the expansion.) Setting $Φ : = (1 + a)^{4}$ , the spatial metric is $Φ δ$ , whose Cartesian components are $Φ δ_{i j}$ ; in $(ρ, ϑ, φ)$ it is conformal to $δ = d ρ^{2} + ρ^{2} (d ϑ^{2} + \sin^{2} ϑ d φ^{2})$ in spherical coordinates on $R^{3}$ ; this is the conformally flat form of clause (c). By the binomial identity, $(1 + \frac{s}{4 ρ})^{4} = 1 + \frac{s}{ρ} + \frac{3 s^{2}}{8 ρ^{2}} + \frac{s^{3}}{16 ρ^{3}} + \frac{s^{4}}{256 ρ^{4}},$ so the stated inequality holds with $C (ρ_{*}) = \frac{3}{8} + \frac{s}{16 ρ_{*}} + \frac{s^{2}}{256 ρ_{*}^{2}}$ . For $r \geq 2 s$ set $t = s / r \in [0, 1 / 2]$ . The explicit inverse in clause (a) and the estimate in [L2] give $ρ_{0} (r) = \frac{r - s / 2 + r h (t)}{2} = r - \frac{s}{2} - \frac{s^{2}}{16 r} + \frac{r}{2} R_{3} (t) .$ The displayed bound on $R_{3}$ implies $∣ r R_{3} (t) / 2 ∣ \leq C s^{3} / r^{2}$ for a constant $C$ independent of $r$ ; this proves the expansion in clause (c) as $r / s \to \infty$ with fixed $s$ .

3.1step 2.1L1algebra∎

(The comparison warning.) The areal coefficient is $f (r)^{- 1} = (1 - s / r)^{- 1} = 1 + \frac{s}{r} + \frac{(s / r)^{2}}{1 - s / r}$ , an exact identity whose last term is $O (r^{- 2})$ , while the isotropic coefficient has the exact expansion $Φ = 1 + \frac{s}{ρ} + O (ρ^{- 2})$ of step 2.1; the two expressions are both $1 + s / (radius)$ at leading order but involve different radial coordinates related by $ρ = r - \frac{s}{2} + O (1 / r)$ . Substituting $ρ = r$ in the first-order terms would replace $s / r$ by $s / ρ = s / (r - \frac{s}{2} + \dots) = s / r + O (r^{- 2})$ , so the mismatch appears at relative order $(s / r)^{2}$ ; the comparison of coefficients must therefore be performed after the conversion $r = R (ρ)$ (or $ρ = ρ_{0} (r)$ ) with the error terms retained. This is clause (d); no statement is made here about the numerical size of $s$ or about any observed system.

Remarks

What the lemma is for. The isotropic form is the coordinate system in which the spatial part of the Schwarzschild metric is conformally flat and decays to the Euclidean metric as $ρ \to \infty$ ; it is used to compare with weak-field ansätze whose spatial part is written as a conformal factor times the Euclidean metric. The warning of clause (d) is the exact content of the source's remark (cited above) that the comparison must be made in one and the same coordinate system; the coordinates $r$ and $ρ$ differ by the isothermal shift at order $s$ .

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The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum of mass parameter $m_{*}$ with Schwarzschild radius $s > 0$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), with metric-selected connection $\nabla$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields). Define the smooth vector field

$n : = f^{- 1 / 2} \partial_{0} = (1 - \frac{s}{r})^{- 1 / 2} \partial_{0}$

on $M$ . Then:

(a) Observer congruence. $n$ is a smooth observer congruence on $M$ in the sense of Observer congruences, their observer curves and adapted charts: $g (n, n) = - 1$ and $g (n, N) < 0$ at every point, so $n$ is a future-directed unit timelike field.

(b) Integral curves. For any $p = (c_{0}, r_{0}, ω_{0}) \in M$ , including sphere poles, the maximal integral curve of $n$ through $p$ , with length-valued flow parameter $λ \in R$ , is $λ \mapsto (c_{0} + f (r_{0})^{- 1 / 2} λ, r_{0}, ω_{0}) .$ It is defined on all of $R$ and is the static observer worldline at radius $r_{0}$ . In a spherical-polar chart it has constant angular coordinates $(ϑ_{0}, φ_{0})$ ; in any angular chart the sphere point is constant, and the global coordinate $x^{0}$ increases monotonically.

(c) Proper time and clock rate. Along each static observer worldline the proper time $τ$ is an affine function of the flow parameter $λ$ , with $\frac{d τ}{d λ} = \frac{1}{c}$ , and

$Δ τ = \frac{\sqrt{f (r_{0})}}{c} Δ x^{0} = \sqrt{f (r_{0})} Δ t,$

where $Δ x^{0}$ is the coordinate difference in metres between two events of the worldline and $Δ t = Δ x^{0} / c$ is the same difference in seconds. In particular the proper-time increment is strictly positive for $Δ x^{0} > 0$ , and it is smaller than $Δ t$ for $Δ x^{0} > 0$ at every finite radius because $0 < f (r_{0}) < 1$ .

(d) Clock-rate ratio. For two static observers at radii $r_{1}, r_{2} \in (s, \infty)$ compared over the same coordinate interval $Δ x^{0} \neq 0$ , the proper-time increments satisfy

$\frac{Δ τ_{1}}{Δ τ_{2}} = \sqrt{\frac{f (r_{1})}{f (r_{2})}} = \sqrt{\frac{1 - s / r_{1}}{1 - s / r_{2}}} .$

(e) Acceleration. The acceleration field of the congruence is $a : = \nabla_{n} n = \frac{f^{'} (r)}{2} \partial_{1} = \frac{s}{2 r^{2}} \partial_{r}$ ; it is spatial, nowhere zero, and points in the direction of increasing $r$ , so the static observers are not geodesics.

(f) Scope. These are identities of the geometric datum. The reading of $τ$ as the time measured by an ideal clock and of the congruence as a family of static observers belongs to the framework's postulates and is not asserted here; no measurement and no comparison with any observed rate is made.

Facts & Assumptions

Given: The Schwarzschild exterior datum with $s > 0$ , a spherical-polar product chart $(x^{0}, r, ϑ, φ)$ , the metric $g$ of the datum, its connection $\nabla$ and the field $n = f^{- 1 / 2} \partial_{0}$ .

[L1]

In a spherical-polar product chart the metric components are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ with $f (r) = 1 - s / r > 0$ on $r > s$ , and the nonzero Christoffel symbols are those of Christoffel symbols of the Schwarzschild exterior metric in product charts, in particular $Γ^{1}_{00} = f f^{'} / 2$ with $f^{'} = s / r^{2}$ (The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Christoffel symbols of the Schwarzschild exterior metric in product charts).

[L2]

An observer congruence on an open set is a smooth field $n$ with $g (n, n) = - 1$ and $g (n, N) < 0$ everywhere; its observer curves are the maximal integral curves of $n$ , and the field is called the acceleration field $a = \nabla_{n} n$ , which is spatial; the flow parameter of $n$ is not by itself a clock reading (Observer congruences, their observer curves and adapted charts, Observers, rest spaces and the spatial projection at an event, Acceleration, expansion and the along-congruence derivative of an observer congruence, Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative).

[L3]

For a $C^{1}$ future timelike worldline the proper-time density is $c^{- 1} \sqrt{- g (γ^{'}, γ^{'})}$ , the proper time is strictly increasing with the parameter, and the four-velocity $U = γ_{τ}^{'}$ satisfies $g (U, U) = - c^{2}$ (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).

[L4]

The function $f^{- 1 / 2}$ is smooth on $(s, \infty)$ because $f$ is smooth, positive and never zero there, and $x \mapsto x^{- 1 / 2}$ is smooth on $(0, \infty)$ ; the product and chain rules apply (Continuity and derivatives of positive-base real powers, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Rational powers $a^{r}$ of a positive base, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[L5]

The maximal integral curve of a smooth vector field through a point is unique and defined on an open interval; for the coordinate field $\partial_{0}$ on a product domain the integral curves are obtained by translating the coordinate $x^{0}$ (Observer congruences, their observer curves and adapted charts clause (b), A smooth vector field is a smooth section of the tangent bundle, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · evaluate the metric on $n$, integrate the flow explicitly, and apply the proper-time and acceleration definitions

1.1L1L2L4algebra

(Unit timelike and future-directed.) By [L1] and [L4], $g (n, n) = f^{- 1} g_{00} = f^{- 1} (- f) = - 1$ , and $g (n, N) = f^{- 1 / 2} g (\partial_{0}, \partial_{0}) = - f^{1 / 2} < 0$ ; hence $n$ is a unit timelike future-directed smooth field, i.e. an observer congruence, and $f^{1 / 2}$ is smooth and positive. This proves (a).

1.2L1L4L5algebra

(Integral curves.) Because $n = f (r)^{- 1 / 2} \partial_{0}$ and $f$ depends only on $r$ , in any angular chart at the fixed sphere point, the components of $n$ are $n^{0} = f (r_{0})^{- 1 / 2}$ , $n^{1} = n^{2} = n^{3} = 0$ along the curve at constant $(r_{0}, ω_{0})$ ; the curve of clause (b) has tangent $n$ at every value of the parameter $λ$ and satisfies the initial condition at $λ = 0$ , so by uniqueness of maximal integral curves [L5] it is the maximal integral curve, defined for all $λ \in R$ because the explicit formula is defined there. This proves (b).

2.1L1L3step 1.2algebra

(Proper time.) The tangent $γ^{'} (λ) = n_{γ (λ)}$ has $g (γ^{'}, γ^{'}) = - 1$ by step 1.1, so by [L3] the proper-time density is $c^{- 1} \sqrt{1} = 1 / c$ , constant; integrating over a parameter interval of length $Δ λ$ gives $Δ τ = Δ λ / c$ . Since $x^{0}$ increases by $Δ x^{0} = f (r_{0})^{- 1 / 2} Δ λ$ over that interval by step 1.2, we obtain $Δ λ = f (r_{0})^{1 / 2} Δ x^{0}$ and hence $Δ τ = f (r_{0})^{1 / 2} Δ x^{0} / c = f (r_{0})^{1 / 2} Δ t$ ; the increment is positive for $Δ x^{0} > 0$ , and $0 < f < 1$ on $r > s$ . This proves (c).

3.1L1L2step 1.1algebra

(Clock-rate ratio and acceleration.) Since the common increment $Δ x^{0} \neq 0$ and $f (r_{2}) > 0$ , step 2.1 gives $Δ τ_{2} \neq 0$ . Dividing the identity of step 2.1 for $r_{1}$ by the same identity for $r_{2}$ over a common coordinate interval gives $Δ τ_{1} / Δ τ_{2} = \sqrt{f (r_{1}) / f (r_{2})}$ , which is (d). For the acceleration, by [L2] and [L1], $\nabla_{n} n = f^{- 1 / 2} \nabla_{\partial_{0}} (f^{- 1 / 2} \partial_{0}) = f^{- 1 / 2} (\partial_{0} (f^{- 1 / 2}) \partial_{0} + f^{- 1 / 2} Γ^{a}_{00} \partial_{a}) = f^{- 1} Γ^{1}_{00} \partial_{1} = \frac{1}{2} f^{'} \partial_{1}$ , where the first term vanishes because $f$ is independent of $x^{0}$ ; substituting $f^{'} = s / r^{2}$ gives $a = \frac{s}{2 r^{2}} \partial_{1}$ , which is spatial, nonzero and along $\partial_{1}$ . This computation holds on the dense polar-chart locus. Both $\nabla_{n} n$ and $s (2 r^{2})^{- 1} \partial_{r}$ are smooth global vector fields, so their equality extends to the poles by continuity. This proves (e) globally.

4.1step 1.1step 1.2step 2.1step 3.1given∎

(Conclusion.) Steps 1.1, 1.2, 2.1 and 3.1 prove clauses (a)-(e) as identities of the datum, using only its geometric data and the cited definitions; clause (f) records that the clock reading and the observer interpretation belong to the framework's postulates and that no measurement or empirical comparison is made.

Remarks

Static does not mean geodesic. The acceleration $a = \frac{s}{2 r^{2}} \partial_{1}$ is nonzero at every finite radius: the static observers are held at fixed $r$ by an external constraint and are not free-fall worldlines. The magnitude $∣ a ∣_{g} = \frac{f^{'}}{2 \sqrt{f}} = \frac{s}{2 r^{2} \sqrt{1 - s / r}}$ diverges as $r ↓ s$ ; this is a statement about the congruence, not about the curvature.
Relation to the redshift. The ratio of clause (d) is a ratio of proper times of static observers over the same coordinate interval; the related but distinct statement about the observer energies of a null geodesic between two static observers is proved in the physics theorem of this page, and it is the latter that involves the light-ray postulate.

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Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum with $s > 0$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), with metric-selected connection $\nabla$ . Let $γ : I \to M$ be a future timelike geodesic, parametrized by proper time $τ$ , with tangent $U = γ^{'}$ , so that $g (U, U) = - c^{2}$ and $U$ is future-directed (Geodesic of an affine connection, Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed clause 3, Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold). Fix a spherical-polar product chart $(x^{0}, r, ϑ, φ)$ containing the image of $γ$ and write a dot for $\frac{d}{d τ}$ . Define

$e : = - g (U, \partial_{0}) = f {\dot{x}}^{0}, ℓ : = g (U, \partial_{φ}) = r^{2} \sin^{2} ϑ \dot{φ} .$

Then:

(a) Conserved quantities. $\partial_{0}$ and $\partial_{φ}$ are Killing fields of $g$ , and $e$ and $ℓ$ are constant along $γ$ ; moreover $e > 0$ .

(b) Equatorial reduction. Suppose $ϑ (τ_{0}) = \frac{π}{2}$ and $\dot{ϑ} (τ_{0}) = 0$ at some $τ_{0} \in I$ . Then $ϑ (τ) = \frac{π}{2}$ for all $τ \in I$ ; the geodesic is equatorial.

(c) Radial reduction. If $γ$ is equatorial, define the effective potential

$W_{ℓ} (r) : = f (r) (c^{2} + \frac{ℓ^{2}}{r^{2}}) .$

Then the normalization identity

${\dot{r}}^{2} = e^{2} - W_{ℓ} (r)$

holds at every $τ \in I$ , and, given this identity, the $r$ -component of the geodesic equation is equivalent to the one-dimensional equation

$\ddot{r} = - \frac{1}{2} W_{ℓ}^{'} (r), W_{ℓ}^{'} (r) = f^{'} (r) (c^{2} + \frac{ℓ^{2}}{r^{2}}) - 2 f (r) \frac{ℓ^{2}}{r^{3}} .$

(d) Scope. The reduction is local: it is stated on the interval $I$ on which the geodesic stays in one product chart with $r > s$ and $0 < ϑ < π$ ; no completeness, no extension of $γ$ beyond $I$ , and no statement about null geodesics is made. The quantities $e$ and $ℓ$ are constants of the given geodesic, not functions of the model.

Facts & Assumptions

Given: The Schwarzschild exterior datum, a spherical-polar product chart $(x^{0}, r, ϑ, φ)$ containing the image of the future timelike geodesic $γ : I \to M$ with proper-time tangent $U$ , and the constants $e, ℓ$ of the statement.

[L1]

In the spherical-polar product chart the metric components are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ , off-diagonal components zero, with $f = 1 - s / r$ , $f^{'} = s / r^{2}$ ; the nonzero Christoffel symbols are $Γ^{0}_{01} = Γ^{0}_{10} = f^{'} / (2 f)$ , $Γ^{1}_{00} = f f^{'} / 2$ , $Γ^{1}_{11} = - f^{'} / (2 f)$ , $Γ^{1}_{22} = - r f$ , $Γ^{1}_{33} = - r f \sin^{2} ϑ$ , $Γ^{2}_{12} = Γ^{2}_{21} = 1 / r$ , $Γ^{2}_{33} = - \sin ϑ \cos ϑ$ , $Γ^{3}_{13} = Γ^{3}_{31} = 1 / r$ and $Γ^{3}_{23} = Γ^{3}_{32} = \cot ϑ$ (The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Christoffel symbols of the Schwarzschild exterior metric in product charts, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

A smooth vector field $ξ$ is Killing precisely when $\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = 0$ , and then $g (ξ, γ^{'})$ is constant along every geodesic $γ$ (Killing fields of a Lorentzian metric, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clauses (a)-(b)).

[L3]

For a geodesic $γ$ , the components satisfy ${\ddot{γ}}^{a} + \sum_{b, c} Γ^{a}_{b c} {\dot{γ}}^{b} {\dot{γ}}^{c} = 0$ in every chart (Coordinate geodesic equation, Geodesic of an affine connection); a geodesic with given initial event and tangent is unique on its maximal interval, so two geodesics agreeing on an open set of parameters agree wherever both are defined (The geodesic initial-value problem and the exponential map of the metric-selected connection).

[L4]

Proper time is an affine parameter of a future timelike geodesic, and $U = γ^{'}$ satisfies $g (U, U) = - c^{2}$ (Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed clause 3, Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).

[L5]

The static congruence of the datum is the future-directed unit timelike field $n = f^{- 1 / 2} \partial_{0}$ ; for a future timelike vector $U$ , $- g (n, U) > 0$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios, Pairing of an observer with future causal vectors, Observers, rest spaces and the spatial projection at an event, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L6]

The derivative identities used below are the chain rule and product rule for $R$ -valued functions of $τ$ , and $f^{'} = s / r^{2} > 0$ on $r > s$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Rational powers $a^{r}$ of a positive base).

[L7]

A first-order system with continuous right-hand side locally Lipschitz in the state variable has a local solution for each initial datum; smooth right-hand sides have this property (Picard-Lindelöf local existence and uniqueness for first-order systems).

Proof

technique · verify the two Killing fields, reduce the $\vartheta$-equation by uniqueness, and combine the normalization with the $r$-equation

1.1L1L2L5algebra

(The fields $\partial_{0}$ and $\partial_{φ}$ are Killing.) For $ξ = \partial_{0}$ the lowered field is $ξ_{b} = g_{0 b} = - f δ_{b}^{0}$ , and the Killing equation reads $\partial_{a} g_{0 b} + \partial_{b} g_{0 a} - 2 Γ^{c}_{a b} g_{0 c} = 0$ ; substituting [L1], the only potentially nonzero components are $(a, b) = (0, 1)$ : $\partial_{1} g_{00} + 2 f Γ^{0}_{01} = - f^{'} + f^{'} = 0$ , and $(a, b) = (0, 0)$ : $2 \partial_{0} g_{00} = 0$ ; all other components involve derivatives of vanishing components or vanishing symbols, so they vanish. For $ξ = \partial_{φ}$ the lowered field is $ξ_{b} = r^{2} \sin^{2} ϑ δ_{b}^{3}$ , and the equation is $\partial_{a} g_{3 b} + \partial_{b} g_{3 a} - 2 r^{2} \sin^{2} ϑ Γ^{3}_{a b} = 0$ ; its potentially nonzero components are $(a, b) = (1, 3)$ : $2 r \sin^{2} ϑ - 2 r^{2} \sin^{2} ϑ \cdot (1 / r) = 0$ , $(a, b) = (2, 3)$ : $2 r^{2} \sin ϑ \cos ϑ - 2 r^{2} \sin^{2} ϑ \cdot \cot ϑ = 0$ , and $(a, b) = (3, 3)$ : $2 \partial_{3} g_{33} = 0$ ; all other components vanish. Hence both fields are Killing by [L2], and $e = - g (U, \partial_{0})$ and $ℓ = g (U, \partial_{φ})$ are constant along $γ$ . For positivity: $n = f^{- 1 / 2} \partial_{0}$ is a future unit timelike field by [L5], and $- g (n, U) > 0$ ; since $- g (n, U) = f^{- 1 / 2} e$ , we get $e > 0$ . This proves (a).

1.2L1L3L7algebra

(Equatorial reduction.) The coordinate geodesic system [L3] and [L1] has the equations ${\ddot{x}}^{0} + \frac{f^{'}}{f} {\dot{x}}^{0} \dot{r} = 0, \ddot{r} + \frac{f f^{'}}{2} ({\dot{x}}^{0})^{2} - \frac{f^{'}}{2 f} {\dot{r}}^{2} - r f {\dot{ϑ}}^{2} - r f \sin^{2} ϑ {\dot{φ}}^{2} = 0,$ $\ddot{ϑ} + \frac{2}{r} \dot{r} \dot{ϑ} - \sin ϑ \cos ϑ {\dot{φ}}^{2} = 0, \ddot{φ} + \frac{2}{r} \dot{r} \dot{φ} + 2 \cot ϑ \dot{ϑ} \dot{φ} = 0.$ Set $ϑ = π / 2$ and $\dot{ϑ} = 0$ in the $x^{0}, r, φ$ equations. The resulting smooth three-equation subsystem, written as a first-order system for positions and velocities, has an open state domain containing the initial data because $r (τ_{0}) > s$ and the chosen product chart is open. Its right-hand side is smooth and hence locally Lipschitz, so [L7] gives a local solution with the same positions and velocities as $γ$ at $τ_{0}$ . Adjoining the constant coordinate $ϑ = π / 2$ gives a solution of the full four equations, because the $ϑ$ equation vanishes and the other equations reduce to that subsystem. This constructed geodesic has the same event and tangent as $γ$ at $τ_{0}$ , so uniqueness in [L3] makes them agree on a neighborhood of $τ_{0}$ . Let $E : = {τ \in I : ϑ (τ) = π / 2 and \dot{ϑ} (τ) = 0} .$ It is nonempty, closed by continuity, and open by the same local construction at each of its points. Since $I$ is an interval, $E = I$ . Thus $ϑ \equiv π / 2$ on $I$ , proving (b); in particular $\sin^{2} ϑ = 1$ there.

1.3L4givenalgebra

(Normalization identity.) By [L4], $- c^{2} = g (U, U) = - f ({\dot{x}}^{0})^{2} + f^{- 1} {\dot{r}}^{2} + r^{2} {\dot{φ}}^{2}$ on the equatorial geodesic. Substituting ${\dot{x}}^{0} = e / f$ and $\dot{φ} = ℓ / r^{2}$ from the definitions of $e, ℓ$ and multiplying by $f$ gives $- c^{2} f = - e^{2} + {\dot{r}}^{2} + f ℓ^{2} / r^{2}$ , that is, ${\dot{r}}^{2} = e^{2} - f (c^{2} + ℓ^{2} / r^{2}) = e^{2} - W_{ℓ} (r)$ . This is the normalization identity of (c).

2.1L1L3L6step 1.3algebra

(Radial equation.) The $r$ -component of the geodesic equation [L3] is, by [L1], $\ddot{r} + \frac{f f^{'}}{2} ({\dot{x}}^{0})^{2} - \frac{f^{'}}{2 f} {\dot{r}}^{2} - r f {\dot{φ}}^{2} = 0$ . Substituting ${\dot{x}}^{0} = e / f$ and $\dot{φ} = ℓ / r^{2}$ turns this into $\ddot{r} + \frac{f^{'} e^{2}}{2 f} - \frac{f^{'}}{2 f} {\dot{r}}^{2} - \frac{f ℓ^{2}}{r^{3}} = 0$ . The claimed equation reads, after substituting $W_{ℓ}^{'}$ , $\ddot{r} + \frac{f^{'}}{2} (c^{2} + \frac{ℓ^{2}}{r^{2}}) - \frac{f ℓ^{2}}{r^{3}} = 0$ . The difference between the two left-hand sides is $\frac{f^{'}}{2 f} (e^{2} - {\dot{r}}^{2} - f (c^{2} + ℓ^{2} / r^{2})) = \frac{f^{'}}{2 f} (e^{2} - {\dot{r}}^{2} - W_{ℓ} (r))$ , which vanishes precisely by the normalization identity of step 1.3; the substitution of $W_{ℓ}^{'}$ and the rearrangement use the derivative rules of [L6]. Hence, given that identity, the $r$ -equation and $\ddot{r} = - \frac{1}{2} W_{ℓ}^{'}$ are equivalent. This proves (c).

3.1step 1.1step 1.2step 1.3step 2.1given∎

(Conclusion.) Steps 1.1, 1.2, 1.3 and 2.1 prove clauses (a)-(c) for the given geodesic on its chart interval; clause (d) records that the reduction is local and asserts nothing about completeness, extension or null geodesics.

Remarks

Units of the constants. With $τ$ measured in seconds and $x^{0} = c t$ measured in metres, the definitions give $e = f d x^{0} / d τ$ in metres per second and, on an equatorial geodesic, $ℓ = r^{2} d φ / d τ$ in square metres per second. The displayed identities fix the parameter convention used below; no numerical comparison with a source convention is made.
No field equation was used. The reduction uses only the metric form, the time and axial Killing fields and the geodesic equation; it holds for every value of $s > 0$ and does not use the vacuum property $Ric = 0$ of the model.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2

Statement

For arbitrary $x_{0}^{0}, φ_{0} \in R$ , put $ω_{0} = (\cos φ_{0}, \sin φ_{0}, 0) \in S^{2}$ . For $α \in R$ define the rotation $R_{α} (y_{1}, y_{2}, y_{3}) : = (y_{1} \cos α - y_{2} \sin α, y_{1} \sin α + y_{2} \cos α, y_{3}),$ and the global axial vector field $Φ (y_{1}, y_{2}, y_{3}) : = (- y_{2}, y_{1}, 0)$ on $S^{2}$ ; it is tangent because $y \cdot Φ (y) = 0$ for $y \in S^{2}$ . In every spherical-polar chart near the equator, $Φ = \partial_{φ}$ ; longitude expressions below are local chart expressions for the global sphere curve $R_{α} ω_{0}$ .

In this item, a timelike circular geodesic means a future timelike geodesic whose areal radius $r$ is constant on an interval of positive length; its angular orientation is unrestricted.

(a) Existence of circular timelike geodesics. Let $r_{0} > 3 s / 2$ and choose $ℓ \in R$ with

$ℓ^{2} = \frac{c^{2} s r_{0}^{2}}{2 r_{0} - 3 s} > 0, e : = (f (r_{0}) (c^{2} + \frac{ℓ^{2}}{r_{0}^{2}}))^{1 / 2} = \frac{c (r_{0} - s) \sqrt{2}}{\sqrt{r_{0} (2 r_{0} - 3 s)}} > 0.$

Then the globally defined curve $γ : R \to M$ is

$γ (τ) : = (x_{0}^{0} + \frac{e}{f (r_{0})} τ, r_{0}, R_{ℓ τ / r_{0}^{2}} ω_{0}) .$ In each local polar chart along it, $ϑ = π / 2$ and the longitude is $φ_{0} + ℓ τ / r_{0}^{2}$ up to a constant multiple of $2 π$ .

It is a future timelike geodesic of the datum, with $g (γ^{'}, γ^{'}) = - c^{2}$ and with the conserved quantities $e = - g (γ^{'}, \partial_{0})$ and $ℓ = g (γ^{'}, Φ)$ of Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model; it is a circular orbit at radius $r_{0}$ . Its angular velocity in the coordinate $x^{0}$ is

$\frac{d φ}{d x^{0}} = \frac{ℓ f (r_{0})}{r_{0}^{2} e} = sgn (ℓ) (\frac{s}{2 r_{0}^{3}})^{1 / 2}, so (\frac{d φ}{d t})^{2} = \frac{G_{N} m_{*}}{r_{0}^{3}},$

$t = x^{0} / c$ .

(b) The circular-orbit condition is necessary. Conversely, if $γ : I \to M$ is any future timelike geodesic parametrized by proper time with $r (τ) \equiv r_{0} > s$ constant on an interval of positive length, then $r_{0} > \frac{3 s}{2}$ . If $γ$ is equatorial, its conserved angular momentum additionally satisfies $ℓ^{2} = \frac{c^{2} s r_{0}^{2}}{2 r_{0} - 3 s}$ . Consequently no timelike circular geodesic, in any angular orientation, exists with $s < r_{0} \leq \frac{3 s}{2}$ .

(c) Stability threshold of the radial potential. For each radius $r > \frac{3 s}{2}$ set $ℓ_{r}^{2} : = c^{2} s r^{2} / (2 r - 3 s)$ and regard $ℓ_{r}$ as a constant in the function $V_{ℓ_{r}} (x) : = \frac{1}{2} f (x) (c^{2} + ℓ_{r}^{2} / x^{2})$ . Differentiate with respect to $x$ at fixed $ℓ_{r}$ , then evaluate at $x = r$ . The result, evaluated at the circular value $ℓ_{r}$ , is

$V_{ℓ_{r}}^{''} (r) = \frac{c^{2} s (r - 3 s)}{r^{3} (2 r - 3 s)}, r > \frac{3 s}{2},$

so $V_{ℓ}$ has a strict local minimum at the circular-orbit radius exactly for $r > 3 s$ , a strict local maximum exactly for $\frac{3 s}{2} < r < 3 s$ , and $V_{ℓ}^{''} (3 s) = 0$ ; the radius $r = 3 s$ is the innermost stable circular orbit radius of the restricted model in this potential sense.

(d) The photon sphere at $r = 3 s / 2$ . Let $ℓ \neq 0$ and put $r_{0} = \frac{3 s}{2}$ and $e_{0} : = \frac{∣ ℓ ∣ \sqrt{f (r_{0})}}{r_{0}} = \frac{2 ∣ ℓ ∣}{3 \sqrt{3} s} > 0$ . Then the globally defined curve $η (τ) : = (x_{0}^{0} + \frac{e_{0}}{f (r_{0})} τ, r_{0}, R_{ℓ τ / r_{0}^{2}} ω_{0}), τ \in R,$ whose local polar expressions have $ϑ = π / 2$ and longitude $φ_{0} + ℓ τ / r_{0}^{2}$ up to chart shifts, is a future null geodesic (a null circular orbit or photon sphere orbit) with conserved quantities $e_{0} = - g (η^{'}, \partial_{0})$ and $ℓ = g (η^{'}, Φ)$ , and its angular velocity is

$\frac{d φ}{d x^{0}} = \frac{ℓ f (r_{0})}{r_{0}^{2} e_{0}} = sgn (ℓ) \frac{\sqrt{f (r_{0})}}{r_{0}}, (\frac{d φ}{d x^{0}})^{2} = \frac{f (r_{0})}{r_{0}^{2}} = \frac{4}{27 s^{2}} .$

The null effective potential $W_{ℓ}^{n} (r) : = f (r) ℓ^{2} / r^{2}$ satisfies $(W_{ℓ}^{n})^{'} (3 s / 2) = 0$ and $(W_{ℓ}^{n})^{''} (3 s / 2) = - \frac{32 ℓ^{2}}{81 s^{4}} < 0$ , so the photon-sphere radius is a strict local maximum of the null potential.

(e) Scope. All statements are exact data of the restricted model: the orbits are ideal geodesics of the metric on the exterior domain, the stability claim is the sign of the second derivative of the stated potential, and no claim is made about nonlinear stability, about orbital decay, about observations or about the interior region. The case $r_{0} = s$ is excluded, and the reduction is stated only where $r > s$ .

Facts & Assumptions

Given: The Schwarzschild exterior datum with $s > 0$ , the global rotations $R_{α}$ and axial field $Φ$ of the statement, the function $f = 1 - s / r$ with $f^{'} = s / r^{2}$ , and the reduction lemma of this page.

[L1]

In local spherical-polar product charts the metric components, Christoffel symbols and geodesic equations are those computed on this page; in particular the $ϑ$ -equation is $\ddot{ϑ} + 2 r^{- 1} \dot{r} \dot{ϑ} - \sin ϑ \cos ϑ {\dot{φ}}^{2} = 0$ , the $φ$ -equation is $\ddot{φ} + 2 r^{- 1} \dot{r} \dot{φ} + 2 \cot ϑ \dot{ϑ} \dot{φ} = 0$ , the $x^{0}$ -equation is ${\ddot{x}}^{0} + 2 Γ^{0}_{01} {\dot{x}}^{0} \dot{r} = 0$ , and the $r$ -equation is $\ddot{r} + \frac{f f^{'}}{2} ({\dot{x}}^{0})^{2} - \frac{f^{'}}{2 f} {\dot{r}}^{2} - r f ({\dot{ϑ}}^{2} + \sin^{2} ϑ {\dot{φ}}^{2}) = 0$ (Christoffel symbols of the Schwarzschild exterior metric in product charts, The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Coordinate geodesic equation, Geodesic of an affine connection).

[L2]

For a future timelike geodesic parametrized by proper time, $g (U, U) = - c^{2}$ and the constants $e = - g (U, \partial_{0}) = f {\dot{x}}^{0} > 0$ , $ℓ = g (U, \partial_{φ}) = r^{2} \sin^{2} ϑ \dot{φ}$ are conserved; along an equatorial geodesic ${\dot{r}}^{2} = e^{2} - W_{ℓ} (r)$ and the $r$ -equation is equivalent to $\ddot{r} = - \frac{1}{2} W_{ℓ}^{'} (r)$ (Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model).

[L3]

A curve is a geodesic with affine parameter $τ$ precisely when its components satisfy the four geodesic equations; a nonzero tangent that is future timelike (respectively null) makes the curve a future timelike (respectively null) geodesic (Geodesic of an affine connection, Coordinate geodesic equation, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L4]

The derivative rules for smooth functions of $r$ and the positivity of $f$ , $r$ and $\sqrt{\cdot}$ on the stated domains are as on the preceding items of this page (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Rational powers $a^{r}$ of a positive base, Continuity and derivatives of positive-base real powers).

[L5]

If a smooth one-parameter group acts by isometries, its infinitesimal generator is a Killing field; the pairing of a Killing field with the tangent of an affinely parametrized geodesic is constant (Killing fields of a Lorentzian metric, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clause (b)). The product time translations also act by isometries because the global metric is independent of $x^{0}$ .

[L6]

The sine and cosine functions are smooth, their derivatives are cosine and minus sine, their addition formulas hold, and $\sin^{2} α + \cos^{2} α = 1$ (Sine and cosine defined by their real power series, The derivatives of sine and cosine are cosine and minus sine, The addition formulas for sine and cosine, Pythagorean and parity identities for all six trigonometric functions on their natural domains). Thus $R_{α}$ is a smooth one-parameter group of orthogonal transformations of $R^{3}$ , its restriction preserves the induced round metric, and its derivative at $α = 0$ is $Φ$ .

[L7]

In any product chart $(x^{0}, r, ξ^{2}, ξ^{3})$ , the coordinate-free metric formula of The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation has components $g_{00} = - f$ , $g_{r r} = f^{- 1}$ , $g_{0 r} = g_{0 i} = g_{r i} = 0$ and $g_{i j} = r^{2} γ_{i j}$ , with $(γ_{i j})$ the positive-definite round-metric matrix (The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold). The block inverse has $g^{r r} = f$ and $g^{r 0} = g^{r i} = 0$ by matrix multiplication; substituting it and $g_{i j} = r^{2} γ_{i j}$ into the Christoffel formula of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields gives $Γ^{r}_{00} = - \frac{1}{2} g^{r r} \partial_{r} g_{00} = f f^{'} / 2$ and $Γ^{r}_{i j} = - \frac{1}{2} g^{r r} \partial_{r} g_{i j} = - r f γ_{i j}$ , while $Γ^{r}_{0 i} = 0$ .

Proof

technique · solve the stationarity equation of the effective potential, verify the constructed circular curves against the four geodesic equations directly, and differentiate the potential twice

1.1L2L4algebra

(The stationarity equation.) By [L2] and [L4], $W_{ℓ}^{'} (r) = f^{'} (r) (c^{2} + ℓ^{2} / r^{2}) - 2 f (r) ℓ^{2} / r^{3} = \frac{s}{r^{2}} (c^{2} + \frac{ℓ^{2}}{r^{2}}) - 2 (1 - \frac{s}{r}) \frac{ℓ^{2}}{r^{3}}$ . Multiplying by $r^{3} / s$ gives $r c^{2} + \frac{ℓ^{2}}{r} - 2 \frac{ℓ^{2}}{s} (1 - \frac{s}{r}) = r c^{2} + \frac{3 ℓ^{2}}{r} - \frac{2 ℓ^{2}}{s}$ , so $W_{ℓ}^{'} (r) = 0$ if and only if $ℓ^{2} (\frac{2}{s} - \frac{3}{r}) = r c^{2}$ , that is, $ℓ^{2} = \frac{c^{2} s r^{2}}{2 r - 3 s}$ ; this has a positive solution for $ℓ^{2}$ precisely when $r > 3 s / 2$ , and the value is unique. This is the condition displayed in clauses (a) and (b).

2.1L1L2L3L4L5L6step 1.1algebra

(The timelike circular geodesic.) Let $r_{0} > 3 s / 2$ , choose $ℓ$ as in (a) and $e = \sqrt{W_{ℓ} (r_{0})} > 0$ , and define the global curve $γ$ of clause (a). At any parameter value choose a spherical-polar chart near its angular point; the local tangent components are $U^{a} = (e / f (r_{0}), 0, 0, ℓ / r_{0}^{2})$ . Then $g (U, U) = - f (r_{0}) (e / f (r_{0}))^{2} + r_{0}^{2} (ℓ / r_{0}^{2})^{2} = - \frac{e^{2}}{f (r_{0})} + \frac{ℓ^{2}}{r_{0}^{2}} = - (c^{2} + \frac{ℓ^{2}}{r_{0}^{2}}) + \frac{ℓ^{2}}{r_{0}^{2}} = - c^{2}$ by the choice of $e$ , so $U$ is timelike; and $g (U, \partial_{0}) = - e < 0$ , so $U$ is future-directed. In that chart the $x^{0}$ -equation reduces to ${\ddot{x}}^{0} = 0$ because $\dot{r} = 0$ ; the $ϑ$ -equation reads $0 + 0 - \sin (π / 2) \cos (π / 2) (ℓ / r_{0}^{2})^{2} = 0$ ; the $φ$ -equation reduces to $\ddot{φ} = 0$ because $\dot{r} = \dot{ϑ} = 0$ ; and the $r$ -equation reads $\frac{f f^{'}}{2} (\frac{e}{f})^{2} - r f (\frac{ℓ}{r_{0}^{2}})^{2} = \frac{f^{'} e^{2}}{2 f} - \frac{f ℓ^{2}}{r_{0}^{3}} = \frac{f^{'}}{2} (c^{2} + \frac{ℓ^{2}}{r_{0}^{2}}) - \frac{f ℓ^{2}}{r_{0}^{3}} = \frac{1}{2} W_{ℓ}^{'} (r_{0}) = 0$ , using the choice of $e$ in the third equality and step 1.1 in the last. These equations hold in every local chart interval along the curve; longitude charts differ by constant shifts, so the global curve is a geodesic on $R$ by [L3]. By [L5] and [L6], the global time translations and rotations are isometries, so $\partial_{0}$ and $Φ$ are Killing; their pairings with $U$ are $- e$ and $ℓ$ . The Killing-pairing identity in [L5] proves these quantities are constant on the connected parameter interval. The local angular velocity is $\frac{d φ}{d x^{0}} = \frac{\dot{φ}}{{\dot{x}}^{0}} = \frac{ℓ f (r_{0})}{r_{0}^{2} e}$ ; squaring and substituting $ℓ^{2}$ and $e^{2} = f (r_{0}) (c^{2} + ℓ^{2} / r_{0}^{2})$ gives $(\frac{ℓ f}{r_{0}^{2} e})^{2} = \frac{ℓ^{2} f^{2}}{r_{0}^{4} f (c^{2} + ℓ^{2} / r_{0}^{2})} = \frac{ℓ^{2} f}{r_{0}^{4} c^{2} + r_{0}^{2} ℓ^{2}} = \frac{s}{2 r_{0}^{3}}$ , using $ℓ^{2} = c^{2} s r_{0}^{2} / (2 r_{0} - 3 s)$ ; with $t = x^{0} / c$ this is $(\frac{d φ}{d t})^{2} = c^{2} \frac{s}{2 r_{0}^{3}} = \frac{G_{N} m_{*}}{r_{0}^{3}}$ . This proves (a).

2.2L2L3L7step 1.1algebra

(Necessity in any angular orientation.) Let $γ : I \to M$ be a future timelike geodesic parametrized by proper time with $r \equiv r_{0}$ on an interval of positive length, and choose a product chart about any parameter value. Write $\dot{ξ} = ({\dot{ξ}}^{2}, {\dot{ξ}}^{3})$ for the angular tangent. Since $\dot{r} = \ddot{r} = 0$ , [L3, L7] give $0 = \frac{f f^{'}}{2} ({\dot{x}}^{0})^{2} - r f γ_{i j} {\dot{ξ}}^{i} {\dot{ξ}}^{j}$ , so $γ_{i j} {\dot{ξ}}^{i} {\dot{ξ}}^{j} = \frac{f^{'}}{2 r} ({\dot{x}}^{0})^{2}$ . Timelike normalization gives $- c^{2} = - f ({\dot{x}}^{0})^{2} + r^{2} γ_{i j} {\dot{ξ}}^{i} {\dot{ξ}}^{j} = - (f - r f^{'} / 2) ({\dot{x}}^{0})^{2}$ . The value ${\dot{x}}^{0}$ cannot be zero, since the radial equation and positive definiteness of $γ$ would then force $\dot{ξ} = 0$ , contradicting timelikeness. Hence $f - r f^{'} / 2 = 1 - 3 s / (2 r_{0}) > 0$ and $r_{0} > 3 s / 2$ . For an equatorial geodesic, [L2] additionally gives $0 = \ddot{r} = - \frac{1}{2} W_{ℓ}^{'} (r_{0})$ , and step 1.1 yields $ℓ^{2} = c^{2} s r_{0}^{2} / (2 r_{0} - 3 s)$ . This proves (b) and its equatorial angular-momentum claim.

2.3L4step 1.1algebra

(The second derivative at fixed angular momentum.) Differentiate $W_{ℓ} (x) = f (x) (c^{2} + ℓ^{2} / x^{2})$ twice while holding $ℓ$ constant. This gives $W_{ℓ}^{''} (x) = f^{''} (x) (c^{2} + ℓ^{2} / x^{2}) - 4 f^{'} (x) ℓ^{2} / x^{3} + 6 f (x) ℓ^{2} / x^{4}$ . Only after this differentiation set $x = r$ and $ℓ^{2} = ℓ_{r}^{2} = c^{2} s r^{2} / (2 r - 3 s)$ . Substituting $f^{'} = s / r^{2}$ , $f^{''} = - 2 s / r^{3}$ and simplifying gives $W_{ℓ_{r}}^{''} (r) = \frac{2 c^{2} s (r - 3 s)}{r^{3} (2 r - 3 s)}$ , hence $V_{ℓ_{r}}^{''} (r) = \frac{c^{2} s (r - 3 s)}{r^{3} (2 r - 3 s)}$ , whose sign is that of $r - 3 s$ . At $r = 3 s$ , the circular value is $ℓ^{2} = 3 c^{2} s^{2}$ , and direct factorization gives $V_{ℓ}^{'} (x) = c^{2} s (x - 3 s)^{2} / (2 x^{4})$ for $x > s$ . This is positive on both sides of $3 s$ ; thus $V_{ℓ}$ has neither a local minimum nor a local maximum there. Together with the nonzero-second-derivative test away from $3 s$ , this proves both "exactly" assertions in (c).

3.1L1L3L4L5L6algebra

(The photon sphere.) Let $ℓ \neq 0$ , $r_{0} = 3 s / 2$ and $e_{0} = ∣ ℓ ∣ \sqrt{f (r_{0})} / r_{0} > 0$ ; define the global curve $η$ of clause (d), with local tangent components $U^{a} = (e_{0} / f (r_{0}), 0, 0, ℓ / r_{0}^{2})$ . Then $g (U, U) = - \frac{e_{0}^{2}}{f (r_{0})} + \frac{ℓ^{2}}{r_{0}^{2}} = - \frac{ℓ^{2}}{r_{0}^{2}} + \frac{ℓ^{2}}{r_{0}^{2}} = 0$ , so $U$ is null; and $g (U, \partial_{0}) = - e_{0} < 0$ , so $U$ is future-directed. The local $x^{0}$ -, $ϑ$ - and $φ$ -equations hold as in step 2.1; the $r$ -equation reads $\frac{f^{'} e_{0}^{2}}{2 f} - r_{0} f \frac{ℓ^{2}}{r_{0}^{4}} = \frac{ℓ^{2}}{r_{0}^{2}} (\frac{f^{'}}{2} - \frac{f}{r_{0}}) = \frac{ℓ^{2}}{r_{0}^{4}} (\frac{3 s}{2} - r_{0}) = 0$ at $r_{0} = 3 s / 2$ . These local equations establish a future null geodesic on the entire domain $R$ by [L3]. By [L5] and [L6], the global Killing fields have pairings $- e_{0}$ and $ℓ$ , constant along $η$ . Its angular velocity is $\frac{ℓ f (r_{0})}{r_{0}^{2} e_{0}} = sgn (ℓ) \frac{\sqrt{f (r_{0})}}{r_{0}}$ , equal to $+ \frac{2}{3 \sqrt{3} s}$ for $ℓ > 0$ and to its negative for $ℓ < 0$ , with square $4 / (27 s^{2})$ . Finally $W_{ℓ}^{n} (r) = f ℓ^{2} / r^{2}$ has $(W_{ℓ}^{n})^{'} (r) = ℓ^{2} \frac{- 2 r + 3 s}{r^{4}}$ (that is, $ℓ^{2} (3 s - 2 r) / r^{4}$ ), which vanishes at $r = 3 s / 2$ , and $(W_{ℓ}^{n})^{''} (r) = ℓ^{2} \frac{6 (r - 2 s)}{r^{5}}$ , which at $r = 3 s / 2$ equals $- \frac{32 ℓ^{2}}{81 s^{4}} < 0$ . This proves (d).

4.1step 1.1step 2.1step 2.2step 2.3step 3.1given∎

(Conclusion.) Steps 1.1-3.1 prove clauses (a)-(d) from the metric data and the geodesic equations; clause (e) records the exact scope: the statements are about ideal geodesics of the model on $r > s$ , with stability understood as the sign of the potential's second derivative, and no empirical or nonlinear claim is made.

Remarks

The innermost stable circular orbit. Clause (c) shows that the potential's second derivative changes sign at $r = 3 s$ , so in this reduced one-dimensional description the circular orbits with $\frac{3 s}{2} < r < 3 s$ are maxima of the potential and those with $r > 3 s$ are minima. This is a statement about the reduced potential only; the physical interpretation of the threshold as the innermost stable circular orbit of the model, and any account of how it would be observed, belong to the physics theorem and its scope remark, not to this lemma.
Null geodesics are not particle orbits. The photon-sphere curve of clause (d) has null tangent and no proper-time parametrization; its parameter $τ$ is an affine parameter, and only the ratio $d φ / d x^{0}$ of the tangent components is asserted, the tangent of a null geodesic being determined only up to positive scale.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The Kruskal model datum: domain, implicit radial function, metric and time orientation

Definition

This item assumes the axiom of countable choice ${A C}_{ω}$ through the tangent-bundle convention of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c); the product-smooth-structure supplier has no choice hypothesis (The Axiom of Countable Choice ( ${A C}_{ω}$ )); no further choice principle is used.

(a) Parameters. Let $c > 0$ , $G_{N} > 0$ and $m_{*} > 0$ be the constants and mass parameter of the Schwarzschild exterior datum, and put $s : = 2 G_{N} m_{*} / c^{2} > 0$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clause (a)). The Kruskal model datum of mass parameter $m_{*}$ is constructed from these data as follows.

(b) Domain and smooth structure. Let $Ω : = {(U, V) \in R^{2} : U V < 1}$ and let $K : = Ω \times S^{2}$ , where $S^{2} \subset R^{3}$ is the unit Euclidean sphere, with the product smooth structure (Products of smooth manifolds have a canonical product smooth structure, Euclidean spheres and closed balls as subspaces of $R^{n}$ ). It is a smooth four-manifold without boundary. Write $U, V : K \to R$ for the first two projections and $ι : K \to S^{2}$ for the third. A Kruskal product chart uses any smooth chart on $S^{2}$ . In a spherical-polar Kruskal chart the angular coordinates satisfy $0 < ϑ < π$ and $γ_{S^{2}} = diag (1, \sin^{2} ϑ)_{i j}$ ; these charts omit the two poles, which are covered by ordinary sphere charts.

(c) The radial function. The datum carries a function $r : K \to (0, \infty)$ , the Kruskal radial function, characterized by the implicit relation

$U V = - (\frac{r}{s} - 1) e^{r / s} at every point of K .$

That for every $(U, V) \in Ω$ there is exactly one $r (U, V) > 0$ satisfying this relation, and that $r$ is smooth, is not asserted here; it is proved in Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar ↗ on this page, the justification record of this definition. The value $r = 0$ is not attained on $K$ , and $r = s$ exactly on the axes $U = 0$ or $V = 0$ .

(d) The metric. Define the metric globally by

$g_{K} = - \frac{2 s^{3}}{r} e^{- r / s} (d U \otimes d V + d V \otimes d U) + r^{2} ι^{*} γ_{S^{2}} .$

In every spherical-polar Kruskal chart its component functions are

$g_{U V} : = g_{V U} : = - \frac{2 s^{3}}{r} e^{- r / s}, g_{ϑ ϑ} : = r^{2}, g_{φ φ} : = r^{2} \sin^{2} ϑ, g_{U U} : = g_{V V} : = g_{U ϑ} : = g_{U φ} : = g_{V ϑ} : = g_{V φ} : = 0,$

where $r$ is the function of clause (c) evaluated at the point. This global expression defines a smooth symmetric covariant $2$ -tensor field on $K$ once the smoothness of $r$ is established by Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar ↗. In another sphere chart the angular block is $r^{2} (γ_{i j})$ . That $g_{K}$ is Lorentzian is proved, not asserted, in the same lemma.

(e) Time orientation. Define the field $N_{K} : = \partial_{U} + \partial_{V}$ on $K$ . Its component expression is the same in every Kruskal chart; that it is a smooth timelike field and hence a time orientation of $g_{K}$ is proved in Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar ↗.

(f) Scope. The datum is a geometric object: the definition asserts nothing about whether $g_{K}$ satisfies any field equation, about its causal structure, about the maximality of the extension it provides, or about nature. The comparison with the exterior datum, the vacuum property and the causal structure are the content of the two following items on this page. The region ${U V = 1}$ , where the implicit relation of clause (c) has no solution with $r > 0$ , is not part of $K$ and is not appended to it by this definition.

Remarks

Four regions. The set ${U V \neq 0}$ has four connected components (the quadrants $U < 0 < V$ , $U > 0, V > 0$ , $U > 0 > V$ and $U < 0, V < 0$ ), separated by the null axes $U = 0$ and $V = 0$ . On components with $U V < 0$ the radial function exceeds $s$ , while on components with $U V > 0$ it is less than $s$ ; both facts follow from the implicit relation once clause (c) is justified. The causal and metric meaning of the four regions is the subject of the following items.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar

Statement

Let $K = {(U, V) : U V < 1} \times S^{2}$ , $r$ , $g_{K}$ and $N_{K} = \partial_{U} + \partial_{V}$ be the Kruskal model datum of mass parameter $m_{*}$ and Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2} > 0$ (The Kruskal model datum: domain, implicit radial function, metric and time orientation). Then:

(a) The radial function. The function $φ (r) : = - (r / s - 1) e^{r / s}$ is a smooth strictly decreasing bijection of $(0, \infty)$ onto $(- \infty, 1)$ with $φ^{'} (r) = - \frac{r}{s^{2}} e^{r / s} < 0$ ; consequently the relation $φ (r (U, V)) = U V$ defines exactly one function $r : K \to (0, \infty)$ , and this function is smooth. Moreover $r (p) = s$ exactly when $U (p) V (p) = 0$ , and $r (p) \to 0$ corresponds to $U (p) V (p) \to 1$ ; the value $r = 0$ is not attained on $K$ .

(b) Lorentzian metric and time orientation. $g_{K}$ is the global smooth tensor of [L1] and is nondegenerate with inertia $(3, 1, 0)$ at every point; hence $(K, g_{K})$ is a smooth Lorentzian four-manifold of signature $(-, +, +, +)$ . The field $N_{K} = \partial_{U} + \partial_{V}$ satisfies $g_{K} (N_{K}, N_{K}) = - \frac{4 s^{3}}{r} e^{- r / s} < 0$ and is a time orientation.

(c) The four regions and the isometries. The set ${U V \neq 0}$ has four connected components. Let $r_{*}$ be the tortoise function of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart on the branch $r > s$ and define $Ψ_{+} : M \to K$ by $Ψ_{+} (x^{0}, r, ω) = (- e^{- (x^{0} - r_{*} (r)) / (2 s)}, e^{(x^{0} + r_{*} (r)) / (2 s)}, ω)$ . Then $Ψ_{+}$ is a diffeomorphism of the exterior datum onto the component ${U < 0, V > 0}$ and $Ψ_{+}^{*} g_{K} = g$ . Similarly, on the branch $0 < r < s$ with $r_{*} = r + s \log (1 - r / s)$ the map $Ψ_{-} (x^{0}, r, ω) = (e^{(r_{*} (r) - x^{0}) / (2 s)}, e^{(r_{*} (r) + x^{0}) / (2 s)}, ω)$ is a diffeomorphism of the domain $R \times (0, s) \times S^{2}$ (carrying the metric $- f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ with $f < 0$ ) onto the component ${U > 0, V > 0}$ and pulls $g_{K}$ back to that metric. The map $(U, V, ω) \mapsto (- U, - V, ω)$ is an isometry of $g_{K}$ fixing $r$ , exchanging the components ${U < 0, V > 0} \leftrightarrow {U > 0, V < 0}$ and ${U > 0, V > 0} \leftrightarrow {U < 0, V < 0}$ ; hence every component of ${U V \neq 0}$ is isometric to one of the two branches $- f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} d Ω^{2}$ with $f = 1 - s / r$ .

(d) Vacuum and Kretschmann scalar. The metric-selected connection of $g_{K}$ exists and is unique; its Ricci tensor, scalar curvature and Einstein tensor vanish identically on $K$ , and its Kretschmann scalar equals $R_{a b c d} R^{a b c d} = 12 s^{2} / r^{6}$ at every point of $K$ . In particular the curvature invariant is smooth on $K$ and grows without bound as $U V \to 1$ (that is, as $r ↓ 0$ at fixed angles), while it equals $12 / s^{4}$ on the axes $r = s$ .

(e) Scope. The statements are geometric. No maximality of the extension, no geodesic completeness, no uniqueness theorem and no global causal statement beyond those proved here are asserted. The manifold $K$ contains a copy of the exterior datum and, on the components with $U V > 0$ , points at $0 < r < s$ ; the set ${U V = 1}$ is not part of $K$ .

Facts & Assumptions

Given: The Kruskal model datum of mass parameter $m_{*}$ with $s > 0$ , the functions $U, V$ , the metric $g_{K}$ and the field $N_{K}$ .

[L1]

The domain $Ω = {U V < 1}$ is open in $R^{2}$ and $K = Ω \times S^{2}$ is a smooth four-manifold without boundary. The global tensor is $g_{K} = - 2 s^{3} e^{- r / s} r^{- 1} (d U \otimes d V + d V \otimes d U) + r^{2} ι^{*} γ_{S^{2}}$ ; in a spherical-polar chart its angular components are $r^{2} diag (1, \sin^{2} ϑ)$ , while pole charts use the smooth round metric (The Kruskal model datum: domain, implicit radial function, metric and time orientation, Euclidean spheres and closed balls as subspaces of $R^{n}$ , The round metric on the sphere as an induced metric). A smooth nondegenerate symmetric tensor with inertia $(3, 1, 0)$ is Lorentzian (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L2]

The exterior datum has global angular term $r^{2} ι^{*} γ_{S^{2}}$ and base metric $- f d (x^{0})^{2} + f^{- 1} d r^{2}$ with $f = 1 - s / r$ on $r > s$ ; on either tortoise branch, $r_{*}^{'} = 1 / f$ and $r_{*} = r + s \log ∣ r / s - 1 ∣$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart).

[L3]

The advanced extension on $M_{e x t} = R \times (0, \infty) \times S^{2}$ has metric $g_{e x t} = - f d v^{2} + 2 d v d r + r^{2} γ_{S^{2}}$ , $Ric = S = G = 0$ and Kretschmann scalar $12 s^{2} / r^{6}$ for every $r > 0$ (Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart clauses (a), (d)). On either nonhorizon branch, the smooth map $Φ_{b} (x^{0}, r, ω) = (x^{0} + r_{*} (r), r, ω)$ has inverse $(v, r, ω) \mapsto (v - r_{*} (r), r, ω)$ , and $d v = d x^{0} + f^{- 1} d r$ gives $Φ_{b}^{*} g_{e x t} = - f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ directly. Thus its domain and curvature conclusion include $0 < r < s$ , unlike the exterior-only suppliers.

[L4]

A smooth function on an open real interval with nonzero derivative at a point has a smooth local inverse there (Choice-free smooth inverse function theorem in Euclidean space, $n = 1$ ). For a bijection, these local inverses agree with its set-theoretic inverse. The mean value theorem gives strict decrease from a negative derivative on every compact subinterval (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ); a continuous strictly monotone function on an interval attains every value between its endpoint limits (A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values).

[L5]

Two continuous maps into a Hausdorff space agreeing on a dense subset agree everywhere. The Ricci tensor and the Kretschmann scalar of a smooth nondegenerate metric are smooth tensor/scalar fields, so their components in every smooth chart are continuous (Two continuous maps into a Hausdorff space that agree on a dense subset are equal, $C^{r}$ and smooth maps between smooth manifolds).

[L6]

The real exponential is smooth, has derivative $\exp$ , satisfies $\exp (a + b) = \exp (a) \exp (b)$ , and is a bijection $R \to (0, \infty)$ ; the natural logarithm is its smooth inverse with derivative $1 / x$ . Product and chain rules apply, and $φ^{'} (r) = - \frac{r}{s^{2}} e^{r / s}$ (The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ , The exponential is a continuous bijection from $R$ onto $(0, \infty)$ , The natural logarithm as the inverse of the exponential function, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, Choice-free smooth inverse function theorem in Euclidean space, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Rational powers $a^{r}$ of a positive base).

Proof

technique · establish the bijectivity and smoothness of the implicit radial function, verify the metric's matrix structure in the Kruskal chart, compute the two pullbacks explicitly, and transfer the vacuum and Kretschmann results by isometry and continuity

1.1L1L4L6algebra

(The radial function.) By [L6], $φ (r) = (1 - r / s) e^{r / s}$ is smooth with $φ^{'} (r) = - \frac{r}{s^{2}} e^{r / s} < 0$ for $r > 0$ , so $φ$ is strictly decreasing by [L4], applied on every compact subinterval of $(0, \infty)$ . Its limits are $1$ as $r ↓ 0$ and $- \infty$ as $r \to \infty$ : for $x = r / s > 1$ , the defining exponential series gives $e^{x} \geq 1 + x$ , so $(x - 1) e^{x} \geq (x - 1) (1 + x) \to \infty$ . By [L4] its image is exactly $(- \infty, 1)$ , hence it is a bijection. Its derivative never vanishes; the smooth inverse-function theorem gives a smooth local inverse at each point of the image, and bijectivity makes those local inverses agree with the global inverse. Thus $r = φ^{- 1} \circ (U V)$ is defined and smooth on all of $Ω$ , justifying clause (c) of the datum. The level relation gives $r = s$ exactly when $U V = 0$ and $r ↓ 0$ as $U V ↑ 1$ . This proves (a).

1.2L1algebra

(The metric and time orientation, including the poles.) The base block of the global tensor [L1] in $(U, V)$ is $(\begin{array}{cc} 0 & b \\ b & 0 \end{array})$ with $b = - 2 s^{3} e^{- r / s} / r < 0$ , so it is nondegenerate with one positive and one negative eigenvalue. The angular block $r^{2} γ_{S^{2}}$ is positive definite at every sphere point, including both poles. Thus the global metric has inertia $(3, 1, 0)$ . For $N_{K} = \partial_{U} + \partial_{V}$ , $g_{K} (N_{K}, N_{K}) = 2 b = - 4 s^{3} e^{- r / s} / r < 0$ everywhere, so it is a time orientation. This proves (b).

2.1L1L2L6step 1.1algebra

(The isometry with the exterior.) On $Ω_{+} = {U < 0, V > 0}$ define $U = - e^{- (x^{0} - r_{*}) / (2 s)}$ , $V = e^{(x^{0} + r_{*}) / (2 s)}$ with the branch $r > s$ of $r_{*}$ ; then $U V = - e^{r_{*} / s} = - (r / s - 1) e^{r / s}$ by [L2] (since $e^{r_{*} / s} = (r / s - 1) e^{r / s}$ on $r > s$ ), so the function defined implicitly satisfies $r (U, V) = r$ and the map $Ψ_{+}$ is a diffeomorphism onto ${U < 0, V > 0}$ with inverse $(U, V, ω) \mapsto (s \log (V / (- U)), r (U, V), ω)$ . Differentiating, $d U = - \frac{U}{2 s} (d x^{0} - r_{*}^{'} d r)$ and $d V = \frac{V}{2 s} (d x^{0} + r_{*}^{'} d r)$ , hence $d U d V + d V d U = - \frac{U V}{2 s^{2}} (d x^{0} \otimes d x^{0} - r_{*}^{' 2} d r \otimes d r)$ . Substituting into $g_{K}$ and using $\frac{s U V e^{- r / s}}{r} = - f$ and $r_{*}^{' 2} = f^{- 2}$ , the pulled-back metric is $- f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} d Ω^{2} = g$ ; this is the computation of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart restricted to the exterior. On the component ${U > 0, V > 0}$ define instead $U = e^{(r_{*} - x^{0}) / (2 s)}$ , $V = e^{(r_{*} + x^{0}) / (2 s)}$ with the branch $r_{*} = r + s \log (1 - r / s)$ of the interval $(0, s)$ ; then $U V = e^{r_{*} / s} = (1 - r / s) e^{r / s} = - (r / s - 1) e^{r / s}$ , so again $r (U, V) = r$ , and $d U d V + d V d U = \frac{U V}{2 s^{2}} (d r_{*}^{2} - d x^{0} \otimes d x^{0})$ ; the same substitution with $r_{*}^{'} = 1 / f$ (negative there) gives $- f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} d Ω^{2}$ again, now with $f < 0$ . Finally the map $(U, V, ω) \mapsto (- U, - V, ω)$ leaves both $U V$ and $d U d V + d V d U$ unchanged and fixes $r$ , $g_{ϑ ϑ}$ and $g_{φ φ}$ ; it is therefore an isometry, and it exchanges the two components with $U V < 0$ and the two with $U V > 0$ . This proves (c).

3.1step 2.1algebra

(The four connected components.) No connected subset of ${U V \neq 0}$ can change the sign of either continuous nonzero coordinate. Step 2.1 and simultaneous sign reversal identify each sign region with $R \times J \times S^{2}$ , where $J = (s, \infty)$ or $(0, s)$ . These domains are path connected: interpolate the real and radial coordinates linearly; any two sphere points can be joined by normalized straight segments, inserting an intermediate unit coordinate vector not antipodal to either endpoint when needed (among the six signed coordinate vectors at most two are excluded). All segment normalizations have nonzero denominators. Hence each sign region is connected, and the four are exactly the components asserted in (c).

3.2L3L5step 1.1step 2.1algebra

(Vacuum and Kretschmann scalar.) By step 2.1, on every component with $U V \neq 0$ the metric is isometric to a branch of the spherical metric with $f = 1 - s / r$ . By [L3] each branch is isometric to the corresponding open region of the advanced extension, whose curvature identities hold for all $r > 0$ . Uniqueness of the metric-selected connection transports it under these isometries, and therefore transports its curvature tensor and metric contractions. Thus $Ric = S = G = 0$ and the Kretschmann scalar is $12 s^{2} / r^{6}$ on both branches and their sign-reversed copies. These identities hold on the dense open subset $D = {U V \neq 0} \times (S^{2} ∖ {two poles})$ . The global metric is smooth and nondegenerate by step 1.2, so its Ricci tensor and Kretschmann scalar are smooth. In any smooth product chart their components are continuous and agree on the dense subset $D$ with zero and with $12 s^{2} / r^{6}$ , respectively; [L5] extends the identities over the axes and both poles. This proves (d), including $K = 12 / s^{4}$ at $r = s$ and $K \to \infty$ as $U V \to 1$ . Clause (e) records the scope.

4.1step 1.1step 1.2step 2.1step 3.2given∎

(Conclusion.) Steps 1.1-3.2 prove clauses (a)-(d) from the datum's definitions and the all-positive-radius advanced-extension supplier; the claims about maximality and completeness are explicitly not made, as recorded in (e).

Remarks

Why continuity suffices. The Ricci tensor and the Kretschmann scalar are defined from the metric by two derivatives. The global metric is smooth across the axes and in pole charts; the isometry calculations on the dense regular subset therefore extend the tensor identities by continuity. This establishes smoothness across $r = s$ , not real analyticity.
The implicit relation in the source. The sources write $U V = - (r / s - 1) e^{r / s}$ in the form $U V = (2 G M - r) e^{r / 2 G M} / (2 G M)$ , which is the same identity with $s = 2 G M$ ; the extraction of $r$ from $U V$ uses only that the function is a strictly decreasing bijection of $(0, \infty)$ onto $(- \infty, 1)$ , as proved in step 1.1.

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The causal cone, the escape horizon and decreasing areal radius in the Kruskal model

Statement

Let $(K, r, g_{K}, N_{K})$ be the Kruskal model datum of mass parameter $m_{*} > 0$ with Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2} > 0$ (The Kruskal model datum: domain, implicit radial function, metric and time orientation, Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar), let $(U, V, ξ^{2}, ξ^{3})$ be any Kruskal product chart, and let $γ_{i j}$ be the round-sphere components in its angular chart. In a spherical-polar Kruskal chart, $(ξ^{2}, ξ^{3}) = (ϑ, φ)$ and $γ_{i j} = diag (1, \sin^{2} ϑ)$ . Define $A : K \to (0, \infty)$ by

$A : = \frac{4 s^{3} e^{- r / s}}{r},$

so that in a product chart $g_{U V} = g_{V U} = - A / 2$ , $g_{i j} = r^{2} γ_{i j}$ and the mixed base-angular components vanish. In a spherical-polar chart this reads $g_{ϑ ϑ} = r^{2}$ , $g_{φ φ} = r^{2} \sin^{2} ϑ$ . For a $C^{1}$ curve $η : J \to K$ with $J$ a nonempty interval write $\dot{η} = (\dot{U}, \dot{V}, {\dot{ξ}}^{2}, {\dot{ξ}}^{3})$ for its coordinate components, and call $η$ future-directed causal when $g_{K} (\dot{η}, \dot{η}) \leq 0$ and $g_{K} (\dot{η}, N_{K}) < 0$ at every parameter value. Then:

(a) Causal cone criterion. At every $p \in K$ and for every $X \in T_{p} K$ with chart components $(X^{U}, X^{V}, X^{ω})$ one has: $X$ is causal and future-directed if and only if

$X^{U} \geq 0, X^{V} \geq 0, X^{U} + X^{V} > 0, A (p) X^{U} X^{V} \geq r (p)^{2} γ_{p} (X^{ω}, X^{ω}),$

where $γ_{p} (X^{ω}, X^{ω})$ is the round-sphere squared norm of the angular part. In particular every future-directed causal tangent has both $X^{U} \geq 0$ and $X^{V} \geq 0$ .

(b) Monotonicity. Along every future-directed causal $C^{1}$ curve the functions $U$ and $V$ are nondecreasing, and along every past-directed causal $C^{1}$ curve (where $g (\dot{η}, \dot{η}) \leq 0$ and $g (\dot{η}, N_{K}) > 0$ ) they are nonincreasing.

(c) Radial null geodesics and their affine parameters. For every $(U_{0}, V_{0}, ω_{0}) \in K$ there is a unique maximal solution $λ \mapsto (U (λ), V_{0}, ω_{0})$ of $\frac{d U}{d λ} = \frac{1}{A (U (λ), V_{0})}$ , $U (0) = U_{0}$ ; it is an affinely parametrized null geodesic of $g_{K}$ with $g (\dot{η}, \dot{η}) = 0$ . The same holds for $λ \mapsto (U_{0}, V (λ), ω_{0})$ with $\frac{d V}{d λ} = \frac{1}{A (U_{0}, V (λ))}$ . Both are future-directed, and $λ$ increases strictly with $U$ , respectively with $V$ , because $A > 0$ . Moreover $\partial_{U} r = - \frac{s^{2} V e^{- r / s}}{r}$ and $\partial_{V} r = - \frac{s^{2} U e^{- r / s}}{r}$ at every point of $K$ .

(d) Escape horizon. Call a future-directed causal $C^{1}$ curve $η$ escaping to the right exterior when there is a parameter value after which $η$ stays in the open quadrant ${U < 0, V > 0}$ and $r (η (t)) \to + \infty$ as $t \to + \infty$ (a point with $U \geq 0$ cannot reach the quadrant ${U < 0, V > 0}$ at all, by (b), and this is the sense in which the horizon is a one-way membrane here). Then:

no future-directed causal curve issued at a point $p$ with $U (p) \geq 0$ escapes to the right exterior;
every point $p$ with $U (p) < 0$ admits such a curve: for $t > 0$ the curve $η_{p} (t) : = (U (p), V (p) + t, ω (p))$ is a future-directed null $C^{1}$ curve, it stays in ${U < 0, V > 0}$ for $t > 1 + ∣ V (p) ∣$ and satisfies $r (η_{p} (t)) \to + \infty$ as $t \to + \infty$ .

Consequently the exact nonescaping set of the datum is ${U \geq 0}$ with boundary the null surface ${U = 0}$ , and the past-directed dual statement singles out ${V = 0}$ as the boundary of the region admitting past-directed escape to the right exterior: here escape means eventual residence in ${U < 0, V > 0}$ with $r \to + \infty$ as the past-directed parameter tends to $+ \infty$ .

(e) Decreasing areal radius. If $U (p) > 0$ and $V (p) > 0$ , then along both future directions of the radial null geodesics of (c) through $p$ the function $r$ is strictly decreasing as their affine parameter $λ$ increases: on $λ \mapsto (U_{0}, V (λ), ω_{0})$ and on $λ \mapsto (U (λ), V_{0}, ω_{0})$ .

(f) Scope. All statements are geometric properties of the smooth Kruskal datum on $K$ ; no maximality, geodesic completeness or uniqueness is claimed, the set ${U V = 1}$ (where $r = 0$ ) is not part of $K$ , and no statement is made about the physical interpretation of any curve. The escape notion of (d) is defined by the displayed conditions on $U, V$ and $r$ and does not presuppose a conformal infinity.

Facts & Assumptions

Given: The Kruskal model datum with $s > 0$ , a Kruskal chart, the function $A = 4 s^{3} e^{- r / s} / r > 0$ , and the component assignments $g_{U V} = g_{V U} = - A / 2$ , $g_{ϑ ϑ} = r^{2}$ , $g_{φ φ} = r^{2} \sin^{2} ϑ$ .

[L1]

$K = {(U, V) : U V < 1} \times S^{2}$ is a smooth four-manifold without boundary; $r : K \to (0, \infty)$ is the unique smooth function with $U V = - (r / s - 1) e^{r / s}$ , the metric $g_{K}$ is smooth Lorentzian of signature $(-, +, +, +)$ , and $N_{K} = \partial_{U} + \partial_{V}$ is a time orientation with $g_{K} (N_{K}, N_{K}) = - A < 0$ (The Kruskal model datum: domain, implicit radial function, metric and time orientation, Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar clauses (a)-(b)).

[L2]

The metric-selected connection of $g_{K}$ exists, is unique and has Christoffel symbols $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ in a product chart; a curve is a geodesic with affine parameter $λ$ precisely when its coordinate components satisfy ${\ddot{x}}^{a} + \sum_{b, c} Γ^{a}_{b c} {\dot{x}}^{b} {\dot{x}}^{c} = 0$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Metric compatibility, torsion freedom and the metric-selected connection class, Christoffel symbols of an affine connection, Geodesic of an affine connection, Coordinate geodesic equation).

[L3]

A nonzero tangent vector $X$ is causal when $g_{K} (X, X) \leq 0$ , future-directed when $g_{K} (X, N_{K}) < 0$ and past-directed when $g_{K} (X, N_{K}) > 0$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L4]

The inverse of the $2 \times 2$ block $(\begin{array}{cc} 0 & - A / 2 \\ - A / 2 & 0 \end{array})$ is $(\begin{array}{cc} 0 & - 2 / A \\ - 2 / A & 0 \end{array})$ , and the inverse metric has no mixed $U$ -angular or $V$ -angular entries; the round metric is positive definite on the angular directions (Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar, The round metric on the sphere as an induced metric).

[L5]

The function $F (r) : = (1 - r / s) e^{r / s}$ satisfies $F^{'} (r) = - (r / s^{2}) e^{r / s} < 0$ on $r > 0$ and is a smooth strictly decreasing bijection onto $(- \infty, 1)$ with smooth inverse; consequently $U V \to - \infty$ holds exactly when $r \to + \infty$ , and implicit differentiation of $U V = F (r)$ gives $\partial_{U} r = - s^{2} V e^{- r / s} / r$ , $\partial_{V} r = - s^{2} U e^{- r / s} / r$ (Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar clause (a), Choice-free smooth inverse function theorem in Euclidean space, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Rational powers $a^{r}$ of a positive base).

[L6]

A $C^{1}$ function on an interval whose derivative is $\geq 0$ everywhere is nondecreasing, and a $C^{1}$ function with derivative $> 0$ is strictly increasing; the chain, product and quotient rules hold (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, $C^{r}$ and smooth maps between smooth manifolds).

[L7]

The exponential function is smooth and satisfies $\partial_{r} e^{r / s} = s^{- 1} e^{r / s}$ ; the displayed functions involving $r$ are smooth on $K$ because $r$ is smooth and $r > 0$ (Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar, The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , Rational powers $a^{r}$ of a positive base, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · evaluate the metric on a general tangent vector to get the cone criterion, compute the Christoffel symbols along the coordinate fields, differentiate the defining relation of $r$, read monotonicity off the sign of the tangent components, and verify the geodesic, escape and trapping claims directly

1.1L1L3L4algebra

(Cone criterion.) Writing $X = X^{U} \partial_{U} + X^{V} \partial_{V} + X^{ω}$ with $X^{ω}$ angular and using the components of [L1], $g_{K} (X, X) = - A X^{U} X^{V} + r^{2} γ (X^{ω}, X^{ω})$ and $g_{K} (N_{K}, X) = g_{K} (\partial_{U} + \partial_{V}, X) = - \frac{A}{2} (X^{U} + X^{V})$ . By [L3] $X$ is causal and future-directed exactly when $- A X^{U} X^{V} + r^{2} γ (X^{ω}, X^{ω}) \leq 0$ and $X^{U} + X^{V} > 0$ . The causal inequality gives $A X^{U} X^{V} \geq r^{2} γ (X^{ω}, X^{ω}) \geq 0$ , hence $X^{U} X^{V} \geq 0$ ; if one of $X^{U}, X^{V}$ were negative the other would be $\leq 0$ and $X^{U} + X^{V} \leq 0$ , contradicting $X^{U} + X^{V} > 0$ , so both are $\geq 0$ and one is positive. Conversely, $X^{U}, X^{V} \geq 0$ with $X^{U} + X^{V} > 0$ and $A X^{U} X^{V} \geq r^{2} γ (X^{ω}, X^{ω})$ give both required inequalities. This proves (a).

1.2L2L4L5L7algebra

(Christoffel symbols along the radial coordinate fields.) The inverse $U, V$ block is $g^{U V} = g^{V U} = - 2 / A$ . Since $g_{U V} = - A / 2$ and $g_{U U} = 0$ , the Christoffel formula of [L2] gives $Γ^{U}_{U U} = \frac{1}{2} g^{U V} (\partial_{U} g_{V U} + \partial_{U} g_{U V}) = (- 2 / A) (- \frac{1}{2} \partial_{U} A) = \partial_{U} A / A$ ; similarly $Γ^{V}_{V V} = \partial_{V} A / A$ . The same substitution gives $Γ^{U}_{V V} = Γ^{V}_{U U} = 0$ , all angular components of $\nabla_{\partial_{U}} \partial_{U}$ and $\nabla_{\partial_{V}} \partial_{V}$ equal zero, and $Γ^{U}_{U V} = Γ^{V}_{U V} = 0$ . Thus $\nabla_{\partial_{U}} \partial_{U} = (\partial_{U} A / A) \partial_{U}$ and $\nabla_{\partial_{V}} \partial_{V} = (\partial_{V} A / A) \partial_{V}$ .

1.3L5algebra

(Radial derivatives of $r$ .) The relation $U V = F (r)$ with $F (r) = (1 - r / s) e^{r / s}$ of [L5] has $F^{'} (r) = - (r / s^{2}) e^{r / s} \neq 0$ on $r > 0$ , so implicit differentiation gives $V = F^{'} (r) \partial_{U} r$ , that is, $\partial_{U} r = - s^{2} V e^{- r / s} / r$ , and $U = F^{'} (r) \partial_{V} r$ , that is, $\partial_{V} r = - s^{2} U e^{- r / s} / r$ . This proves the derivative formulas of (c).

2.1step 1.1L3L6algebra

(Monotonicity.) If $η$ is future-directed causal, then by step 1.1 its tangent has $\dot{U} \geq 0$ and $\dot{V} \geq 0$ at every parameter; by [L6] applied to subintervals, $U \circ η$ and $V \circ η$ are nondecreasing. For past-directed causal curves the same argument with $N_{K}$ replaced by $- N_{K}$ gives $\dot{U} \leq 0$ , $\dot{V} \leq 0$ and hence nonincreasing functions. This proves (b).

2.2step 1.1step 1.2L2L6algebra

(Radial null geodesics and maximality.) The supplier The geodesic initial-value problem and the exponential map of the metric-selected connection gives the unique maximal geodesic through $p$ with initial tangent $A (U_{0}, V_{0})^{- 1} \partial_{U}$ . On the open scalar domain ${U : U V_{0} < 1}$ , the smooth equation $U^{'} = 1 / A (U, V_{0})$ has a unique local solution at every initial value by Picard-Lindelöf local existence and uniqueness for first-order systems. Fixing $V = V_{0}$ and the sphere point, this solution satisfies all geodesic equations: by step 1.2, $U^{''} + Γ^{U}_{U U} (U^{'})^{2} = - A^{- 3} \partial_{U} A + (\partial_{U} A / A) A^{- 2} = 0,$ and all other component equations vanish. Uniqueness makes it agree locally with the maximal geodesic. On the latter's connected parameter interval, the set where $V = V_{0}$ , the sphere point is $ω_{0}$ , $V^{'} = 0$ , the angular tangent vanishes, and $U^{'} = 1 / A (U, V_{0})$ is nonempty, closed by continuity, and open by this same local construction and geodesic uniqueness. Thus the conditions hold throughout that interval. This yields a scalar solution on the whole maximal geodesic domain. Conversely, any scalar extension would give a geodesic extension by the verified equations, contradicting maximality; hence the scalar and geodesic domains are both maximal. The curve is null because $g (\partial_{U}, \partial_{U}) = 0$ , future-directed by step 1.1, and its affine parameter $λ$ increases with $U$ because $1 / A > 0$ . Exchanging $U, V$ proves the second family, in any angular chart including pole charts.

3.1step 1.3step 2.2L5L6algebra

(Decreasing radius in the interior quadrant.) If $U (p) > 0$ and $V (p) > 0$ , then step 1.3 gives $\partial_{V} r = - s^{2} U e^{- r / s} / r < 0$ and $\partial_{U} r = - s^{2} V e^{- r / s} / r < 0$ at $p$ ; along either future branch, the varying coordinate is nondecreasing from its positive initial value and the other coordinate remains fixed and positive; hence both remain positive throughout that future branch. The derivative formulas therefore stay strictly negative, so by [L6], $r$ is strictly decreasing along the geodesics of step 2.2 through $p$ in the directions of increasing $V$ , respectively increasing $U$ , which are the future directions in the affine parameters of step 2.2. This proves (e).

3.2step 1.1step 2.1L1L5algebra

(Escape and nonescape.) Let $η$ be future-directed causal with $U (η (0)) = U (p) \geq 0$ . By step 2.1, $U \circ η \geq U (p) \geq 0$ ; hence $η$ never enters the quadrant ${U < 0, V > 0}$ and cannot escape to the right exterior, proving (d)1. Conversely let $U (p) < 0$ and consider $η_{p} (t) : = (U (p), V (p) + t, ω (p))$ for $t > 0$ . Since $U (p) < 0$ , its product $U (p) (V (p) + t)$ decreases from a value below $1$ and remains below $1$ , so this $C^{1}$ curve stays in $K$ for all $t > 0$ . Its tangent is $(0, 1, 0, 0)$ ; by step 1.1 it is future-directed null. Since $U$ is constant and negative, the curve is in ${U < 0}$ , and for $t > 1 + ∣ V (p) ∣$ it is in ${U < 0, V > 0}$ . Along it $U V = U (p) (V (p) + t) \to - \infty$ , so by [L5] $r \to + \infty$ . This proves (d)2 and identifies the nonescaping set as ${U \geq 0}$ with boundary ${U = 0}$ , a null surface because $\partial_{V}$ is null there. For the past-directed dual, $V$ is nonincreasing, so no point with $V \leq 0$ can enter the right exterior. If $V (p) > 0$ , the past-directed null curve $(U (p) - t, V (p), ω (p))$ , $t \geq 0$ , stays in $K$ , eventually enters the right exterior and has $U V \to - \infty$ , hence $r \to + \infty$ . Thus its admitting region is exactly ${V > 0}$ , with boundary ${V = 0}$ , whose null generator is $\partial_{U}$ .

4.1step 1.1step 1.3step 2.1step 2.2step 3.1step 3.2given∎

(Conclusion.) Step 1.1 proves (a), step 2.1 proves (b), steps 1.3 and 2.2 prove (c), step 3.1 proves (e), and step 3.2 proves (d). Clause (f) records that all statements are geometric, that maximality and completeness are not claimed and that the escape notion is the explicit one of the statement.

Remarks

One-way membrane, not a singularity. The nonescape statement uses only $\dot{U} \geq 0, \dot{V} \geq 0$ along future causal curves; the metric and its curvature remain smooth at $U = 0$ by the Kruskal properties lemma of this page, so the horizon's one-way character is a global causal property of the smooth model, not a defect of the metric at the surface.
Trapping without the expansion machinery. Clause (e) states the strictly decreasing areal radius along both future radial null directions at points with $U, V > 0$ , which is the geometric content of the statement that the spheres $r =$ const in that quadrant are future-trapping; the phrase of the sources is not used as a definition, since the expansion of a null congruence is not defined on this page.

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The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor

Statement

Let $V \subseteq R^{2}$ be a nonempty open set with global coordinates $y = (y^{0}, y^{1})$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, $C^{r}$ and smooth maps between smooth manifolds), let $(q_{A B})$ be a smooth symmetric $2 \times 2$ matrix of functions on $V$ with

$Δ_{q} (y) : = \det (q_{A B} (y)) < 0 (y \in V),$

so that $q$ is a smooth symmetric covariant $2$ -tensor field on $V$ whose value at every point is nondegenerate with inertia $(1, 1, 0)$ (Smooth covariant 2-tensor fields on a smooth manifold, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space), and let $r : V \to (0, \infty)$ be a smooth function. Indices $A, B, C, D$ run over ${0, 1}$ and indices $i, j, k, l$ over the sphere directions below. Define on $V$ :

$(q^{A B}) : = (q_{A B})^{- 1}, Γ^{A}_{B C} : = \frac{1}{2} \sum_{D} q^{A D} (\partial_{B} q_{D C} + \partial_{C} q_{B D} - \partial_{D} q_{B C}),$

${Hess}_{A B} r : = \partial_{A} \partial_{B} r - \sum_{C} Γ^{C}_{A B} \partial_{C} r, Δ_{q} r : = \sum_{A, B} q^{A B} {Hess}_{A B} r, z : = \sum_{A, B} q^{A B} \partial_{A} r \partial_{B} r .$

Further define

$R^{A}_{B C D} : = \partial_{C} Γ^{A}_{D B} - \partial_{D} Γ^{A}_{C B} + \sum_{E} (Γ^{A}_{C E} Γ^{E}_{D B} - Γ^{A}_{D E} Γ^{E}_{C B}), R_{A B C D} : = \sum_{E} q_{A E} R^{E}_{B C D},$

$Ric (q)_{B D} : = \sum_{A, C} q^{A C} R_{A B C D}, S_{q} : = \sum_{B, D} q^{B D} Ric (q)_{B D} .$

Let $S^{2} : = {y \in R^{3} : ∥ y ∥_{2} = 1}$ be the unit Euclidean sphere with its induced smooth structure and round metric $γ_{S^{2}}$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ , The round metric on the sphere as an induced metric), let $M : = V \times S^{2}$ carry the product smooth structure (Products of smooth manifolds have a canonical product smooth structure), and let $π : M \to V$ and $ι : M \to S^{2}$ be the projections. Define the global symmetric covariant $2$ -tensor

$g : = π^{*} q + r^{2} ι^{*} γ_{S^{2}}$

on $M$ , where $r$ in this formula is pulled back from $V$ . In any product chart $(y^{0}, y^{1}, ξ^{2}, ξ^{3})$ , let $γ_{i j}$ be the round-metric components of the angular chart. The components of $g$ are

$g_{A B} : = q_{A B}, g_{A i} : = 0, g_{i j} : = r^{2} γ_{i j} (A, B \in {0, 1}, i, j \in {2, 3}),$

Then:

(a) The metric. The global pullback formula defines a smooth symmetric covariant $2$ -tensor field $g$ on $M$ ; at every point it is nondegenerate with inertia $(3, 1, 0)$ , so $(M, g)$ is a smooth Lorentzian four-manifold of signature $(-, +, +, +)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

(b) Christoffel symbols. Let $\nabla$ be the metric-selected connection of $g$ , unique by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (b) (Metric compatibility, torsion freedom and the metric-selected connection class), and let $Γ^{μ}_{ν ρ}$ be its Christoffel symbols in a product chart. Then the only nonzero symbols, beyond the displayed lower-index symmetry $Γ^{μ}_{ν ρ} = Γ^{μ}_{ρ ν}$ , are

$Γ^{A}_{B C} of the first display, Γ^{i}_{A j} = \frac{\partial_{A} r}{r} δ^{i}_{j}, Γ^{A}_{i j} = - r q^{A B} \partial_{B} r γ_{i j}, Γ^{i}_{j k} the round-sphere symbols in the chosen chart,$

In a spherical-polar chart these are $Γ^{2}_{33} = - \sin ϑ \cos ϑ$ and $Γ^{3}_{23} = Γ^{3}_{32} = \cot ϑ$ , with all other $Γ^{i}_{j k}$ vanishing. Pole charts use their own round-metric symbols.

(c) Ricci tensor. With $Ric$ the Ricci tensor of $\nabla$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (c)), in every product chart

${Ric}_{A B} = Ric (q)_{A B} - \frac{2}{r} {Hess}_{A B} r, {Ric}_{A i} = 0, {Ric}_{i j} = (1 - z - r Δ_{q} r) γ_{i j} .$

(d) Scope. The statements are identities of the displayed component functions in product charts and use no field equation, no vacuum condition and no sign or positivity assumption on $s$ ; nothing is asserted about completeness, geodesics, extensions or the physical reading of the coordinates. The hypothesis $Δ_{q} < 0$ is used for nondegeneracy and inertia of $g$ ; it is the only signature input.

Facts & Assumptions

Given: An open set $V \subseteq R^{2}$ , a smooth symmetric matrix $q_{A B}$ with $\det q < 0$ on $V$ , a smooth $r : V \to (0, \infty)$ , the product manifold $M = V \times S^{2}$ , the assigned components $g_{A B} = q_{A B}$ , $g_{A i} = 0$ , $g_{i j} = r^{2} γ_{i j}$ in product charts, and the auxiliary functions of the statement.

[L1]

A smooth symmetric covariant $2$ -tensor field on a smooth manifold is a family of smooth bilinear forms whose chart components transform by the covariant $2$ -tensor law on overlaps (Smooth covariant 2-tensor fields on a smooth manifold, Chart independence and the transformation law of a smooth covariant 2-tensor field, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L2]

The product $M = V \times S^{2}$ with the product smooth structure is a smooth four-manifold without boundary; its projections are smooth and product charts cover $M$ , including charts about the two poles. On the dense regular angular locus the polar round-metric components are $γ_{22} = 1$ , $γ_{23} = γ_{32} = 0$ , $γ_{33} = \sin^{2} ϑ$ ; in every sphere chart the components transform as a $(0, 2)$ -tensor (Products of smooth manifolds have a canonical product smooth structure, Smooth manifolds and their smooth charts, Euclidean spheres and closed balls as subspaces of $R^{n}$ , The round metric on the sphere as an induced metric).

[L3]

The Levi-Civita theorem for a smooth four-manifold with a smooth nondegenerate symmetric covariant $2$ -tensor field supplies the unique metric-selected connection and its Christoffel symbols $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ in every chart (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clauses (a)-(b), Metric compatibility, torsion freedom and the metric-selected connection class, Christoffel symbols of an affine connection).

[L4]

Sums, products, quotients, powers and compositions of smooth functions are smooth and satisfy the usual derivative rules; $r > 0$ on $V$ and $\sin ϑ > 0$ on the polar-chart domain $0 < ϑ < π$ ( $C^{r}$ and smooth maps between smooth manifolds, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Rational powers $a^{r}$ of a positive base).

[L5]

The inverse of the block matrix $diag (q_{A B}, r^{2} γ_{i j})$ is the block matrix $diag (q^{A B}, r^{- 2} γ^{i j})$ with $γ^{i j}$ the inverse of the round metric; the components of the Ricci tensor are those of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (c) with components $R^{a}_{b c d}$ of clause (b) and the coordinate formula of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (b), applied on the locally time-oriented restrictions constructed in step 3.1 (The inverse metric, musical isomorphisms and index raising, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Tensoriality of the curvature, its coordinate formula and its point-normal metric formula).

[L6]

The coordinate derivations $\partial_{a}$ form a basis of each tangent space in a chart, the Christoffel symbols are those functions with $\nabla_{\partial_{b}} \partial_{c} = \sum_{a} Γ^{a}_{b c} \partial_{a}$ , and all index sums in the displays are finite (Coordinate derivations form a basis of the tangent space, Christoffel symbols of an affine connection, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · define the warped metric globally with the round-metric pullback, compute in local sphere charts (using polar coordinates for the explicit Ricci calculation), and extend the smooth tensor identities to the poles by continuity

1.1L1L2L4given

(Global smooth tensor.) The product projection $π : M \to V$ and sphere projection $ι : M \to S^{2}$ are smooth by [L2]. Thus $π^{*} q + r^{2} ι^{*} γ_{S^{2}}$ is a globally defined smooth symmetric covariant $2$ -tensor field; its product-chart components are exactly $g_{A B} = q_{A B}$ , $g_{A i} = 0$ , $g_{i j} = r^{2} γ_{i j}$ . This proves the global tensor assertion, including the pole charts.

1.2L2L4L5givenalgebra

(Preliminary identities.) On $V$ , differentiating $\sum_{B} q^{A B} q_{B C} = δ^{A}_{C}$ gives $\partial_{D} q^{A B} = - \sum_{E, F} q^{A E} q^{B F} \partial_{D} q_{E F}$ , and substituting the definition of $Γ^{A}_{B C}$ shows that the identity $\partial_{C} q_{A B} = \sum_{D} (Γ^{D}_{C A} q_{D B} + Γ^{D}_{C B} q_{A D})$ holds, because its right-hand side equals $\frac{1}{2} (\partial_{C} q_{A B} + \partial_{A} q_{C B} - \partial_{B} q_{C A}) + \frac{1}{2} (\partial_{C} q_{B A} + \partial_{B} q_{C A} - \partial_{A} q_{C B}) = \partial_{C} q_{A B}$ ; moreover $\partial_{C} (r^{2} γ_{i j}) = 2 r \partial_{C} r γ_{i j}$ , and the inverse-metric block $diag (q^{A B}, r^{- 2} γ^{i j})$ has no mixed entries.

2.1L2L3L5L6step 1.1givenalgebra

(Nondegeneracy and inertia.) At a point $(y, ω) \in M$ the coordinate derivations form a basis of the tangent space by [L6], and the matrix of $g$ in the product chart frame is block diagonal with the $2 \times 2$ block $q_{A B} (y)$ and the $2 \times 2$ block $r (y)^{2} γ_{i j} (ω)$ . The first block has determinant $Δ_{q} (y) < 0$ , hence two eigenvalues of opposite signs and no zero eigenvalue; the second block is positive definite because $r (y)^{2} > 0$ and the round metric is positive definite. Accordingly $g$ has one negative and three positive eigenvalues, that is, inertia $(3, 1, 0)$ , and is nondegenerate, so by [L3] its metric-selected connection exists and is unique.

2.2L3L4step 1.2algebra

(Base-type symbols.) In the Christoffel formula of [L3] with all three indices $A, B, C \in {0, 1}$ , all entries $g_{A B} = q_{A B}$ are independent of the angular coordinates and $g_{A i} = 0$ , so the formula reduces to $Γ^{A}_{B C} = \frac{1}{2} q^{A D} (\partial_{B} q_{D C} + \partial_{C} q_{B D} - \partial_{D} q_{B C})$ , the displayed base symbols. If exactly one lower index is angular, then either the differentiations produce only $g_{A i} = 0$ terms or the angular derivative annihilates the base block, and the resulting expression vanishes; this gives $Γ^{A}_{B i} = 0$ and $Γ^{i}_{A B} = 0$ .

2.3L3L4step 1.2algebra

(Mixed and sphere-type symbols.) With both lower indices angular, the formula gives $Γ^{A}_{i j} = \frac{1}{2} g^{A D} (\partial_{i} g_{D j} + \partial_{j} g_{D i} - \partial_{D} g_{i j}) = - \frac{1}{2} q^{A D} \partial_{D} (r^{2} γ_{i j}) = - r q^{A D} \partial_{D} r γ_{i j}$ , using $g_{A i} = 0$ , $\partial_{i} g_{A B} = 0$ and step 1.2. With one lower index angular and one base, $Γ^{i}_{A j} = \frac{1}{2} g^{i k} (\partial_{A} g_{k j} + \partial_{j} g_{k A} - \partial_{k} g_{A j}) = \frac{1}{2} r^{- 2} γ^{i k} \partial_{A} (r^{2} γ_{k j}) = (\partial_{A} r / r) δ^{i}_{j}$ , and $Γ^{i}_{j A}$ is the same by symmetry. Finally, with all three indices angular the formula is $\frac{1}{2} γ^{i k} (\partial_{j} γ_{k l} + \partial_{l} γ_{k j} - \partial_{k} γ_{j l})$ , the round-sphere symbols of the statement. This proves (b).

3.1L3L5step 2.1givenalgebra

(Local time orientations and intrinsic curvature.) Fix $y_{0} \in V$ . Since $q (y_{0})$ has inertia $(1, 1, 0)$ , there is a vector $v \in R^{2}$ with $q (y_{0}) (v, v) < 0$ . Continuity gives an open neighborhood $W \subset V$ of $y_{0}$ on which the constant base field $v^{A} \partial_{A}$ is timelike. Its lift $N_{W}$ to $W \times S^{2}$ is smooth and satisfies $g (N_{W}, N_{W}) = q (v, v) < 0$ , so [L5]'s time-oriented curvature suppliers apply there. The global connection supplied by [L3] restricts to the same unique metric-selected connection on each such open set. The curvature operator, lowering and contraction formulas of [L5] contain no $N_{W}$ , so their tensors agree on overlaps and define smooth global curvature and Ricci tensors even when no global time orientation exists. No simultaneous choice of local orientations is needed: this argument is at an arbitrary point.

3.2L4step 1.2step 2.2algebra

(The trace identity.) Metric compatibility in step 1.2 gives $\partial_{A} q^{A B} = - Γ^{A}_{A D} q^{D B} - Γ^{B}_{A D} q^{A D} .$ Consequently $\partial_{A} (q^{A B} \partial_{B} r) = q^{A B} \partial_{A} \partial_{B} r - Γ^{A}_{A D} q^{D B} \partial_{B} r - Γ^{B}_{A D} q^{A D} \partial_{B} r .$ Adding $Γ^{A}_{A B} q^{B D} \partial_{D} r$ cancels the first connection term after relabelling dummy indices. Relabel the remaining term and use symmetry of $q^{A B}$ and of the lower connection indices to obtain $Δ_{q} r = q^{A B} \partial_{A} \partial_{B} r - q^{A B} Γ^{C}_{A B} \partial_{C} r = \partial_{A} (q^{A B} \partial_{B} r) + Γ^{A}_{A B} q^{B D} \partial_{D} r .$ This is the trace identity used in the Ricci contraction below.

4.1L3L5step 2.2step 2.3step 3.1algebra

(Preparation of the Ricci computation.) By [L5] and step 3.1, locally at every base point the Ricci components are ${Ric}_{b d} = \sum_{a} \partial_{a} Γ^{a}_{d b} - \sum_{a} \partial_{d} Γ^{a}_{a b} + \sum_{a, e} Γ^{a}_{a e} Γ^{e}_{d b} - \sum_{a, e} Γ^{a}_{d e} Γ^{e}_{a b}$ , with all indices running over ${0, 1, 2, 3}$ . In the sums we separate the base values $a, e \in {0, 1}$ from the sphere values; by the symbol list of (b) the nonvanishing symbols are the base symbols, the sphere symbols, $Γ^{i}_{A j} = (1 / r) \partial_{A} r δ^{i}_{j}$ and $Γ^{A}_{i j} = - r q^{A D} \partial_{D} r γ_{i j}$ , and every symbol of mixed type with both lower indices base and upper index sphere, or exactly one lower index sphere and upper index base, vanishes; moreover the base symbols, $q$ , $r$ and their derivatives are independent of the angular coordinates, and $\sum_{l} Γ^{l}_{l i} = \partial_{i} \log \sqrt{\det γ} = \cot ϑ δ^{2}_{i}$ .

5.1L3step 1.2step 4.1algebra

(The components ${Ric}_{A B}$ .) Take $b = B$ , $d = A$ base. The terms with all summed indices base give $Ric (q)_{A B}$ . Among the sphere-index terms, the first Ricci sum vanishes because $Γ^{i}_{A B} = 0$ ; the second contributes $- \partial_{A} \sum_{i} Γ^{i}_{i B} = - 2 \partial_{A} (\partial_{B} r / r) = - 2 r^{- 1} \partial_{A} \partial_{B} r + 2 r^{- 2} \partial_{A} r \partial_{B} r$ . The third contributes $2 r^{- 1} \partial_{D} r Γ^{D}_{A B}$ from $\sum_{i} Γ^{i}_{i D} Γ^{D}_{A B}$ ; its other possible sphere-index term is zero because $Γ^{l}_{A B} = 0$ . The fourth contributes $- 2 r^{- 2} \partial_{A} r \partial_{B} r$ from $- \sum_{i, l} Γ^{i}_{A l} Γ^{l}_{i B}$ ; its other mixed terms vanish because $Γ^{i}_{A D} = Γ^{D}_{i B} = 0$ . Adding these contributions gives $- 2 r^{- 1} (\partial_{A} \partial_{B} r - Γ^{D}_{A B} \partial_{D} r) = - 2 r^{- 1} {Hess}_{A B} r$ . Thus ${Ric}_{A B} = Ric (q)_{A B} - 2 r^{- 1} {Hess}_{A B} r$ .

5.2L5step 2.3step 4.1algebra

(The mixed components vanish.) The four sums below are those of the component formula of [L5]. Take $b = i$ , $d = A$ . The first sum $\sum_{a} \partial_{a} Γ^{a}_{i A} = \sum_{l} \partial_{l} ((1 / r) \partial_{A} r δ^{l}_{i})$ vanishes because $r$ is independent of the angular coordinates and the only angular dependence sits in the constant coefficient $δ^{l}_{i}$ ; the second sum $\sum_{a} \partial_{A} Γ^{a}_{a i} = \partial_{A} (\sum_{C} Γ^{C}_{C i} + \sum_{l} Γ^{l}_{l i})$ vanishes because $Γ^{C}_{C i} = 0$ and $\sum_{l} Γ^{l}_{l i} = \cot ϑ δ^{2}_{i}$ is angular. In the third sum the nonvanishing terms are $\sum_{l} Γ^{l}_{l i} ((1 / r) \partial_{A} r) = \cot ϑ δ^{2}_{i} \partial_{A} \log r$ ; in the fourth sum the nonvanishing terms are $- \sum_{l} Γ^{l}_{i l} ((1 / r) \partial_{A} r) = - \cot ϑ δ^{2}_{i} \partial_{A} \log r$ , since the remaining possibilities have at least one symbol of a vanishing mixed type. The two contributions cancel and ${Ric}_{A i} = 0$ .

5.3L6step 2.3step 4.1step 3.2algebra

(The components ${Ric}_{i j}$ .) Take $b = j$ , $d = i$ sphere indices. The terms of the four sums of step 4.1 whose indices are all sphere are exactly the curvature and Ricci computation of the round unit sphere; with the symbols of step 2.3 the direct components are $Ric (γ_{S^{2}})_{22} = 0 + \frac{1}{\sin^{2} ϑ} - \cot^{2} ϑ = 1$ , $Ric (γ_{S^{2}})_{33} = \sin^{2} ϑ - \cos^{2} ϑ - \cos^{2} ϑ + 2 \cos^{2} ϑ = \sin^{2} ϑ$ and $Ric (γ_{S^{2}})_{23} = 0$ , so these terms contribute $Ric (γ_{S^{2}})_{i j} = γ_{i j}$ . The remaining, base-involving terms are: the value $a = A$ of the first sum, $- γ_{i j} \partial_{A} (r q^{A B} \partial_{B} r)$ ; the terms $Γ^{k}_{k A} Γ^{A}_{j i} = - 2 z γ_{i j}$ and $Γ^{A}_{A B} Γ^{B}_{j i} = - r γ_{i j} (\sum_{A} Γ^{A}_{A B}) q^{B D} \partial_{D} r$ of the third sum, where $\sum_{k} Γ^{k}_{k A} = 2 \partial_{A} \log r$ ; and the terms $- Γ^{A}_{j l} Γ^{l}_{A i} - Γ^{l}_{j A} Γ^{A}_{l i} = 2 z γ_{i j}$ of the fourth sum, each of the two equal to $γ_{i j} \sum_{A, B} q^{A B} \partial_{A} r \partial_{B} r = z γ_{i j}$ . Since $\partial_{A} (r q^{A B} \partial_{B} r) = z + r \partial_{A} (q^{A B} \partial_{B} r)$ , these base-involving terms sum to $- γ_{i j} (z + r \partial_{A} (q^{A B} \partial_{B} r)) - 2 z γ_{i j} - r γ_{i j} (\sum_{A} Γ^{A}_{A B}) q^{B D} \partial_{D} r + 2 z γ_{i j} = - γ_{i j} (z + r Δ_{q} r)$ by the trace identity of step 3.1. Adding the sphere contribution gives ${Ric}_{i j} = (1 - z - r Δ_{q} r) γ_{i j}$ .

6.1step 1.1step 2.1step 2.2step 2.3step 5.1step 5.2step 5.3L1L2given∎

(Extension to pole charts and conclusion.) The explicit polar-chart calculations in steps 2.2-5.3 establish the Christoffel and Ricci formulas on the dense regular angular locus. The formulas in clause (b) are the general product-coordinate formulas with the sphere connection of the chosen chart. Each Ricci side in clause (c) is a smooth tensor field: the base block is smooth, the mixed block is zero, and the sphere block is a smooth scalar multiple of $γ_{S^{2}}$ . Since both sides agree on a dense subset, their components agree in every pole chart by continuity. Thus (a)-(c) hold globally; (d) records that no equation on $g$ and no sign of $s$ was used, and $\det q < 0$ entered only through the inertia computation of step 2.1.

Remarks

Where the hypothesis $\det q < 0$ is used. It gives the two-dimensional base the Lorentzian inertia $(1, 1, 0)$ and hence $g$ the inertia $(3, 1, 0)$ of step 2.1; all other computations are algebraic and would hold verbatim for any nondegenerate symmetric $q$ .
The sign convention. The signs of $R^{a}_{b c d}$ and of the Ricci tensor are those of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, which fixes $R (\partial_{c}, \partial_{d}) \partial_{b} = R^{a}_{b c d} \partial_{a}$ and ${Ric}_{b d} = \sum_{a, c} g^{a c} R_{a b c d}$ . With the opposite curvature convention, every curvature and Ricci component changes sign; the right-hand sides of clause (c), evaluated with the convention used here, must therefore be negated. The Christoffel symbols, Hessian, $z$ and $Δ_{q} r$ do not change.
Why $q$ is not assumed to come from a global two-manifold. The statement is local in the base coordinates; it applies to every warped-form expression of a smooth Lorentzian metric in a product chart of the form $V \times S^{2}$ , which is the form in which the vacuum computation of the spherical model is performed.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Two-dimensional base identities for the warped spherical metric: pure-trace Ricci, the vacuum reduction and the extra Killing field

Statement

Let $V \subseteq R^{2}$ be a connected nonempty open set, let $(q_{A B})$ be a smooth symmetric matrix of functions on $V$ with $D_{q} : = \det (q_{A B}) < 0$ everywhere, and let $r : V \to (0, \infty)$ be smooth; let $Γ^{A}_{B C}$ , ${Hess}_{A B} r$ , $Δ_{q} r$ , $z$ , $R^{A}_{B C D}$ , $R_{A B C D}$ , $Ric (q)_{A B}$ and $S_{q}$ be the functions defined in the statement of The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor, so that all index sums run over ${0, 1}$ . Assume in addition that $z \neq 0$ everywhere on $V$ and that the warped vacuum equations

$Ric (q)_{A B} = \frac{2}{r} {Hess}_{A B} r, 1 - z - r Δ_{q} r = 0 on V$

hold, where the left-hand equation is the vanishing of the components ${Ric}_{A B}$ and the right-hand equation the vanishing of the components ${Ric}_{i j}$ of the warped metric of The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor clause (c) (the mixed components vanish there identically). Define the base volume form by

$ε_{A B} : = \sqrt{- D_{q}} η_{A B}, η_{01} : = 1, η_{10} : = - 1, η_{00} : = η_{11} : = 0,$

and the volume-form field by

$K_{A} : = \sum_{B} ε_{A B} \partial^{B} r, \partial^{B} r : = \sum_{C} q^{B C} \partial_{C} r, K^{A} : = \sum_{B} q^{A B} K_{B} .$

For a smooth base vector field $ξ = ξ^{A} \partial_{A}$ , define a Killing field of $q$ to mean $L_{ξ} q = 0$ , with Lie derivative defined by $L_{ξ} q = (d / d t) ∣_{t = 0} Φ_{t}^{*} q$ for the local flow $Φ_{t}$ (The Lie derivative of a tensor field). Its coordinate components are $(L_{ξ} q)_{A B} = ξ^{C} \partial_{C} q_{A B} + q_{C B} \partial_{A} ξ^{C} + q_{A C} \partial_{B} ξ^{C}$ by The coordinate formula for the Lie derivative of a covariant tensor. Here $\nabla_{A} ξ_{B} : = \partial_{A} (q_{B C} ξ^{C}) - Γ^{C}_{A B} ξ_{C}$ and the base gradient is $\nabla r = (q^{A B} \partial_{B} r) \partial_{A}$ .

Then:

(a) Pure trace in two dimensions. $Ric (q)_{A B} = \frac{1}{2} S_{q} q_{A B}$ and $S_{q} = \sum_{A, B} q^{A B} Ric (q)_{A B}$ on all of $V$ .

(b) Vacuum reduction of the Hessian. ${Hess}_{A B} r = \frac{1}{2} Δ_{q} r q_{A B}$ , $Δ_{q} r = \frac{1 - z}{r}$ and $S_{q} = \frac{2 Δ_{q} r}{r} = \frac{2 (1 - z)}{r^{2}}$ on $V$ .

(c) Constancy of the mass parameter. $\partial_{C} z = \frac{1 - z}{r} \partial_{C} r$ for $C = 0, 1$ , hence $\partial_{C} (r (1 - z)) = 0$ ; consequently there is a constant $s \in R$ with

$r (1 - z) = s, that is, z = 1 - \frac{s}{r} on V .$

(d) The extra Killing field. $K (r) = 0$ , $\sum_{A} K^{A} \partial_{A} r = 0$ , $\sum_{A, B} K^{A} q_{A B} K^{B} = - z$ , the field is nonzero at every point, $\nabla_{A} K_{B} + \nabla_{B} K_{A} = 0$ for the base connection $Γ^{A}_{B C}$ (so $K$ is a Killing field of $q$ in the local definition above), and the fields $K$ and $\nabla r$ are pointwise linearly independent: the equalities $\sum_{A} K^{A} \partial_{A} r = 0$ , $\sum_{A, B} K^{A} q_{A B} K^{B} = - z$ and $\sum_{A, B} \partial_{A} r q^{A B} \partial_{B} r = z \neq 0$ show that neither field vanishes and that the two are not proportional.

(e) Explicit components of the Killing field. With $λ : = \sqrt{- D_{q}} > 0$ one has $K^{0} = - \frac{\partial_{1} r}{λ}$ and $K^{1} = \frac{\partial_{0} r}{λ}$ .

(f) Scope. The statements are identities of the displayed functions and use only the two warped vacuum equations, the strict positivity of $r$ and the hypothesis $z \neq 0$ ; no field equation of the four-dimensional metric beyond those two components, no global hypothesis on $V$ and no sign or physical reading of $s$ is assumed. The constant $s$ is determined on the connected set $V$ ; that it is the same constant for a larger region requires that region to be connected as well.

Facts & Assumptions

Given: A connected open set $V \subseteq R^{2}$ , a smooth symmetric matrix $q_{A B}$ with $\det q < 0$ on $V$ , a smooth $r : V \to (0, \infty)$ with $z \neq 0$ everywhere, the auxiliary functions of The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor, and the two warped vacuum equations.

[L1]

The functions $Γ^{A}_{B C}$ , ${Hess}_{A B} r$ , $Δ_{q} r$ , $z$ , $R^{A}_{B C D}$ , $R_{A B C D}$ , $Ric (q)_{A B}$ and $S_{q}$ are the explicit functions of The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor, in particular $R^{A}_{B C D} = \partial_{C} Γ^{A}_{D B} - \partial_{D} Γ^{A}_{C B} + \sum_{E} (Γ^{A}_{C E} Γ^{E}_{D B} - Γ^{A}_{D E} Γ^{E}_{C B})$ , $R_{A B C D} = \sum_{E} q_{A E} R^{E}_{B C D}$ , $Ric (q)_{B D} = \sum_{A, C} q^{A C} R_{A B C D}$ and $S_{q} = \sum_{B, D} q^{B D} Ric (q)_{B D}$ ; the compatibility identity $\partial_{C} q_{A B} = \sum_{D} (Γ^{D}_{C A} q_{D B} + Γ^{D}_{C B} q_{A D})$ and the trace identity $Δ_{q} r = \sum_{A} \partial_{A} (q^{A B} \partial_{B} r) + \sum_{B} (\sum_{A} Γ^{A}_{A B}) q^{B D} \partial_{D} r$ hold.

[L2]

For an invertible $2 \times 2$ matrix $q$ with inverse $q^{- 1}$ , Cramer's rule gives $q^{00} = q_{11} / D_{q}$ , $q^{11} = q_{00} / D_{q}$ and $q^{01} = q^{10} = - q_{01} / D_{q}$ , and $q^{00} q^{11} - (q^{01})^{2} = 1 / D_{q}$ and $D_{q} < 0$ (Invertible matrices and the general linear group ${GL}_{n} (F)$ , algebra).

[L3]

The exterior derivative of a smooth function is the covector of its partial derivatives; the chain, product and quotient rules hold, and a smooth function on a connected nonempty open Euclidean set whose differential is zero is constant (A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant, with $m = 2$ , $q = 1$ ; The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{r}$ and smooth maps between smooth manifolds, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L4]

The base Killing definition is $L_{ξ} q = 0$ ; its coordinate Lie-derivative formula is the one in the Statement, valid in every dimension (The Lie derivative of a tensor field, The coordinate formula for the Lie derivative of a covariant tensor). Step 4.2 proves its equivalence with the symmetrized base derivative using [L1].

[L5]

The square root $\sqrt{- D_{q}} > 0$ is smooth because $D_{q} < 0$ and $D_{q}$ is smooth, and $\partial_{C} \log (- D_{q}) = \sum_{A, B} q^{A B} \partial_{C} q_{A B} = 2 \sum_{D} Γ^{D}_{C D}$ (Rational powers $a^{r}$ of a positive base, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Continuity and derivatives of positive-base real powers, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · compute the lowered curvature formula and its pair antisymmetries, derive the two-dimensional pure-trace identity from them, reduce the vacuum equations to the Hessian identity, differentiate $z$, and verify the Killing and component claims for the volume-form field

1.1L1algebra

(The two curvature antisymmetries.) Let $X, Y, Z, W$ be coordinate vector fields on $V$ , so $[X, Y] = 0$ by Coordinate vector fields commute. The compatibility identity in [L1] says $X q (Z, W) = q (\nabla_{X} Z, W) + q (Z, \nabla_{X} W)$ for the base connection defined by the displayed Christoffel symbols. Apply $X$ to the compatibility identity for $Y q (Z, W)$ , apply $Y$ to the one for $X q (Z, W)$ , and subtract. The mixed terms cancel in pairs, and $X Y q (Z, W) - Y X q (Z, W) = 0$ ; using the curvature definition in [L1] gives [L1, algebra] $q (R (X, Y) Z, W) + q (Z, R (X, Y) W) = 0.$ With $X = \partial_{C}$ , $Y = \partial_{D}$ , $Z = \partial_{B}$ , $W = \partial_{A}$ , this is $R_{A B C D} = - R_{B A C D}$ . The defining curvature formula also changes sign when $C, D$ are exchanged, so $R_{A B C D} = - R_{A B D C}$ . In dimension two, a component can therefore be nonzero only if each of $(A, B)$ and $(C, D)$ is $(0, 1)$ or $(1, 0)$ ; the only possible components are $R_{0101}$ and its three skew variants. No pair-exchange identity is used.

1.2L1L5algebra

(The base volume form is parallel.) Set $ε_{01} = λ$ , $ε_{10} = - λ$ , $ε_{00} = ε_{11} = 0$ , where $λ : = \sqrt{- D_{q}} > 0$ . By [L5], $\partial_{C} λ = λ (Γ^{0}_{C 0} + Γ^{1}_{C 1})$ . The coordinate formula for the covariant derivative of a covariant two-tensor gives $\nabla_{C} ε_{A B} = \partial_{C} ε_{A B} - Γ^{D}_{C A} ε_{D B} - Γ^{D}_{C B} ε_{A D}$ . For $(A, B) = (0, 0)$ and $(1, 1)$ the two connection terms cancel because the only nonzero entries of $ε$ are opposite; for $(A, B) = (0, 1)$ the value is $\partial_{C} λ - Γ^{0}_{C 0} λ - Γ^{1}_{C 1} λ = 0$ , and for $(A, B) = (1, 0)$ it is the negative of this. Thus $\nabla_{C} ε_{A B} = 0$ for every choice of indices.

2.1step 1.1L2algebra

(Pure trace.) By step 1.1 the only possibly nonzero lowered components are $R_{0101}$ , $R_{0110} = - R_{0101}$ , $R_{1001} = - R_{0101}$ and $R_{1010} = R_{0101}$ ; write $ρ : = R_{0101}$ . In the sums $Ric (q)_{B D} = \sum_{A, C} q^{A C} R_{A B C D}$ every term has an index repeated inside a pair except those with ${A, B} = {C, D} = {0, 1}$ , so $Ric (q)_{00} = q^{11} R_{1010} = q^{11} ρ$ , $Ric (q)_{11} = q^{00} R_{0101} = q^{00} ρ$ and $Ric (q)_{01} = Ric (q)_{10} = q^{10} R_{1001} = - q^{01} ρ$ . Cramer's rule [L2] gives $q^{00} q^{11} - (q^{01})^{2} = 1 / D_{q}$ , hence $S_{q} = q^{00} Ric (q)_{00} + 2 q^{01} Ric (q)_{01} + q^{11} Ric (q)_{11} = ρ (2 q^{00} q^{11} - 2 (q^{01})^{2}) = 2 ρ / D_{q}$ . Therefore $\frac{1}{2} S_{q} q_{00} = ρ q_{00} / D_{q} = ρ q^{11} = Ric (q)_{00}$ by [L2], $\frac{1}{2} S_{q} q_{11} = ρ q_{11} / D_{q} = ρ q^{00} = Ric (q)_{11}$ , and $\frac{1}{2} S_{q} q_{01} = ρ q_{01} / D_{q} = - ρ q^{01} = Ric (q)_{01}$ ; this proves (a).

3.1step 2.1givenalgebra

(Hessian reduction.) Substituting $Ric (q) = \frac{1}{2} S_{q} q$ of step 2.1 into the first warped vacuum equation gives ${Hess}_{A B} r = \frac{r S_{q}}{4} q_{A B}$ ; tracing with $q^{A B}$ and using symmetry gives $Δ_{q} r = \frac{r S_{q}}{4} \cdot 2 = \frac{r S_{q}}{2}$ , hence $S_{q} = 2 Δ_{q} r / r$ and ${Hess}_{A B} r = \frac{1}{2} Δ_{q} r q_{A B}$ . The second warped vacuum equation gives $r Δ_{q} r = 1 - z$ and hence $Δ_{q} r = (1 - z) / r$ and $S_{q} = 2 (1 - z) / r^{2}$ . This proves (b).

4.1step 3.1L1L2L3algebra

(Derivative of $z$ .) Differentiating $q^{A E} q_{E B} = δ^{A}_{B}$ and using the compatibility identity of [L1] gives [step 3.1, L1, L2, L3, algebra] $\partial_{C} q^{A B} = - Γ^{B}_{C D} q^{A D} - Γ^{A}_{C D} q^{D B} .$ Differentiate $z = \sum_{A, B} q^{A B} \partial_{A} r \partial_{B} r$ and substitute this identity. The two connection sums are equal after exchanging the dummy indices $A, B$ , so $\partial_{C} z = 2 \sum_{A, B} q^{A B} \partial_{A} r (\partial_{C} \partial_{B} r - \sum_{D} Γ^{D}_{C B} \partial_{D} r) = 2 \sum_{A, B} q^{A B} \partial_{A} r {Hess}_{C B} r .$ By step 3.1, ${Hess}_{C B} r = \frac{1}{2} Δ_{q} r q_{C B}$ , so $\partial_{C} z = Δ_{q} r \partial_{C} r = \frac{1 - z}{r} \partial_{C} r$ . Hence $\partial_{C} (r (1 - z)) = (1 - z) \partial_{C} r - r \partial_{C} z = 0$ for $C = 0, 1$ ; since $V$ is connected and open and $r (1 - z)$ is smooth, [L3] makes it constant, and writing this constant as $s$ gives $z = 1 - s / r$ because $r > 0$ . This proves (c).

4.2step 1.2step 3.1L2L4algebra

(The Killing field.) By step 1.2, $\nabla_{C} ε_{A B} = 0$ , so $\nabla_{C} K_{A} = \sum_{B} ε_{A B} q^{B D} {Hess}_{C D} r$ ; with $Hess = \frac{1}{2} Δ_{q} r q$ of step 3.1 this is $\frac{1}{2} Δ_{q} r ε_{A C}$ , and adding $\nabla_{A} K_{C} = \frac{1}{2} Δ_{q} r ε_{C A}$ gives zero by antisymmetry of $ε$ . For any smooth $ξ$ , expanding the locally defined covariant derivatives gives $\nabla_{A} ξ_{B} + \nabla_{B} ξ_{A} = q_{B C} \partial_{A} ξ^{C} + q_{A C} \partial_{B} ξ^{C} + ξ^{C} (\partial_{A} q_{B C} + \partial_{B} q_{A C} - 2 Γ^{D}_{A B} q_{D C})$ . The compatibility identity in [L1] and lower-index symmetry of $Γ$ reduce the parenthesis to $\partial_{C} q_{A B}$ . This is exactly $(L_{ξ} q)_{A B}$ in [L4]. Applying it to $K$ proves $L_{K} q = 0$ , so $K$ is a Killing field under the stated two-dimensional definition. Put $w^{A} : = \partial^{A} r$ . From $K^{A} = q^{A C} ε_{C B} w^{B}$ we have $K (r) = K^{A} \partial_{A} r = w^{C} ε_{C B} w^{B} = 0$ , since $w^{C} w^{B}$ is symmetric and $ε_{C B}$ antisymmetric. Also $K \neq 0$ because $q (K, K) = - z \neq 0$ . The components $K^{0} = - \partial_{1} r / λ$ , $K^{1} = \partial_{0} r / λ$ follow from the definition of $K_{A}$ , Cramer's rule [L2] and $λ^{2} = - D_{q}$ ; direct substitution gives $q (K, K) = λ^{- 2} (q_{00} (\partial_{1} r)^{2} - 2 q_{01} \partial_{0} r \partial_{1} r + q_{11} (\partial_{0} r)^{2}) = λ^{- 2} D_{q} z = - z$ . Since $z \neq 0$ , $\nabla r \neq 0$ ; a linear relation $a K + b \nabla r = 0$ would give $0 = q (K, a K + b \nabla r) = a (- z)$ and $0 = q (\nabla r, a K + b \nabla r) = b z$ , hence $a = b = 0$ . This proves (d) and (e).

5.1step 1.1step 2.1step 3.1step 4.1step 4.2given∎

(Conclusion.) Step 1.1 supplies the pair antisymmetries, step 2.1 the pure trace (a), step 3.1 the reduction (b), step 4.1 the constancy (c), and step 4.2 the Killing field with its properties (d)-(e); clause (f) records the exact hypotheses, in particular that only the two displayed components of the four-dimensional vacuum equations, the positivity of $r$ and $z \neq 0$ were used, and that the constant $s$ is produced on the connected set $V$ .

Remarks

The role of $z \neq 0$ . The hypothesis makes $\nabla r$ non-null and nowhere zero, so that $K$ and $\nabla r$ are independent by step 4.2 and the coordinate construction of the Birkhoff theorem of this page can be carried out. The vacuum equations force $z = 1 - s / r$ , so $z = 0$ precisely at points with $r = s$ . In a horizon-crossing chart this can occur along a level set while $r$ varies; the hypothesis excludes such points from the open set under consideration, not only a degenerate chart with constant $r$ .
Sign of $s$ . Nothing in this item fixes the sign of the constant $s$ : $s = 0$ gives $z = 1$ and the flat warped form, and the proof uses $1 - z = s / r$ only through the smoothness of $r (1 - z)$ . The positivity of a mass parameter requires separate physical hypotheses or regular-center conditions.

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Local Birkhoff theorem: a vacuum warped product with a round sphere factor is locally Schwarzschild with an extra Killing field

Statement

Let $V \subseteq R^{2}$ be a connected nonempty open set with global coordinates $y$ , let $(q_{A B})$ be a smooth symmetric matrix of functions on $V$ with $D_{q} : = \det (q_{A B}) < 0$ everywhere, let $r : V \to (0, \infty)$ be smooth and let $z = \sum_{A, B} q^{A B} \partial_{A} r \partial_{B} r$ be nonzero everywhere; let $M = V \times S^{2}$ and let $g$ be the warped metric of The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor, whose Ricci tensor is given there in clause (c) by ${Ric}_{A B} = Ric (q)_{A B} - (2 / r) {Hess}_{A B} r$ , ${Ric}_{A i} = 0$ , ${Ric}_{i j} = (1 - z - r Δ_{q} r) γ_{i j}$ . Assume the vacuum condition

$Ric (g) = 0 on M,$

that is, the vanishing of all components of the Ricci tensor of $g$ in every product chart. Then:

(a) Mass parameter. There is a constant $s \in R$ such that $z = 1 - \frac{s}{r}$ on $V$ ; equivalently $r (1 - z) = s$ .

(b) Local Schwarzschild coordinates. For every $p \in V$ there are an open neighborhood $W \subseteq V$ of $p$ and a smooth function $T : W \to R$ such that $(T, r)$ is a smooth coordinate system on $W$ , the vector field $\partial_{T}$ coincides on $W$ with the volume-form field $K$ of Two-dimensional base identities for the warped spherical metric: pure-trace Ricci, the vacuum reduction and the extra Killing field, and in these coordinates

$q = - (1 - \frac{s}{r}) d T \otimes d T + (1 - \frac{s}{r})^{- 1} d r \otimes d r on W,$

where $(1 - s / r)^{- 1}$ is the smooth function $1 / z$ on $W$ .

(c) Local Schwarzschild form of the metric. In the product chart $(T, r, ϑ, φ)$ over $W \times S^{2}$ the components of $g$ are

$g_{T T} = - (1 - \frac{s}{r}), g_{r r} = (1 - \frac{s}{r})^{- 1}, g_{ϑ ϑ} = r^{2}, g_{φ φ} = r^{2} \sin^{2} ϑ,$

with all mixed components zero; that is, $g = - (1 - \frac{s}{r}) d T \otimes d T + (1 - \frac{s}{r})^{- 1} d r \otimes d r + r^{2} γ_{S^{2}}$ on the chart domain.

(d) Extra Killing field. $\partial_{T} = K$ is a Killing field of $g$ on the chart domain in the sense of Killing fields of a Lorentzian metric; it satisfies $\partial_{T} (r) = 0$ and $g (\partial_{T}, \partial_{T}) = - z = - (1 - s / r)$ . On a subregion where $z > 0$ , this field is timelike and orthogonal to the hypersurfaces $T = c o n s t a n t$ ; this is the local staticity asserted here. Where $z < 0$ , it is spacelike and no staticity is asserted. In either case it is an additional symmetry not assumed among the hypotheses.

(e) Scope. The statement is local in $V$ : a neighborhood of each point is produced, and $s$ is a constant on the connected set $V$ but need not be positive, need not be the same for a different connected component of a larger region, and no asymptotic flatness, regular-center, completeness, maximality or global uniqueness statement is made. The hypotheses $D_{q} < 0$ and $z \neq 0$ are exactly those of the two quoted lemmas, and $r > 0$ is used to divide by $r$ .

Facts & Assumptions

Given: A connected open set $V \subseteq R^{2}$ , a smooth symmetric matrix $q_{A B}$ with $\det q < 0$ , a smooth $r : V \to (0, \infty)$ with $z \neq 0$ everywhere, the warped metric $g$ on $M = V \times S^{2}$ in product charts, and the vacuum condition $Ric (g) = 0$ .

[L1]

The functions $Γ^{A}_{B C}$ , ${Hess}_{A B} r$ , $Δ_{q} r$ , $z$ , $R^{A}_{B C D}$ , $Ric (q)_{A B}$ are those of The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor, where clause (c) gives the Ricci components of $g$ in every product chart; the components of a smooth tensor field vanish identically if they vanish in every chart of an atlas (Smooth covariant 2-tensor fields on a smooth manifold, Chart independence and the transformation law of a smooth covariant 2-tensor field).

[L2]

Under the hypotheses of the statement, Two-dimensional base identities for the warped spherical metric: pure-trace Ricci, the vacuum reduction and the extra Killing field applies to the two equations ${Ric}_{A B} = 0$ and ${Ric}_{i j} = 0$ , and its clauses (c)-(e) give the constant $s$ with $z = 1 - s / r$ , the volume-form field $K$ with $K \neq 0$ , $K (r) = 0$ , $K$ pointwise independent of $\nabla r$ , the Killing property $\nabla_{A} K_{B} + \nabla_{B} K_{A} = 0$ and the component formulas $K^{0} = - \partial_{1} r / λ$ , $K^{1} = \partial_{0} r / λ$ with $λ = \sqrt{- D_{q}}$ .

[L3]

The flow-box theorem applies to the smooth vector field $K$ on the smooth surface $V$ : at every point $p$ with $K_{p} \neq 0$ there are coordinates $(u^{1}, u^{2})$ on a neighborhood of $p$ in which $K = \partial_{u^{1}}$ (The flow-box theorem, A smooth vector field is a smooth section of the tangent bundle).

[L4]

In a chart, coordinate derivations form a basis of each tangent space, and a list of $n$ smooth functions on an $n$ -manifold is a coordinate system near a point when its Jacobian determinant is nonzero there (Coordinate derivations form a basis of the tangent space, $C^{r}$ and smooth maps between smooth manifolds, Choice-free smooth inverse function theorem in Euclidean space, with $n = 2$ in the chart domain).

[L5]

Smoothness, the chain and product rules, the inverse of an invertible $2 \times 2$ matrix and the fact that a smooth function of one variable with nonzero derivative is a local coordinate are available (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Invertible matrices and the general linear group ${GL}_{n} (F)$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Choice-free smooth inverse function theorem in Euclidean space). On a connected open real interval, a smooth function with zero derivative is constant (A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant, $m = q = 1$ ).

[L6]

A smooth function on a real interval has the primitive $H (r) = \int_{r_{0}}^{r} h (u) d u$ with derivative $H^{'} = h$ (The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ , The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive).

Proof

technique · read the two vacuum equations off the warped Ricci formula, invoke the base reduction for the constant $s$ and the Killing field $K$, straighten $K$ by the flow-box theorem, use $K(r)=0$ to make $(T,r)$ coordinates, remove the cross term, and read off the metric and the Killing property

1.1L1givenalgebra

(Vacuum equations in warped form.) By [L1] the components of $Ric (g)$ in a product chart are those displayed in the statement; the vacuum condition therefore gives $Ric (q)_{A B} = (2 / r) {Hess}_{A B} r$ and $1 - z - r Δ_{q} r = 0$ on $V$ , since $γ_{i j}$ is a nondegenerate matrix and $r > 0$ .

2.1step 1.1L2given

(The constant $s$ .) The hypotheses of Two-dimensional base identities for the warped spherical metric: pure-trace Ricci, the vacuum reduction and the extra Killing field are exactly the connectedness of $V$ , $D_{q} < 0$ , $r > 0$ , $z \neq 0$ and the two equations of step 1.1, so its clause (c) gives a constant $s \in R$ with $r (1 - z) = s$ , that is, $z = 1 - s / r$ ; this is (a).

3.1step 2.1L2L3L4L5algebra

(Coordinates adapted to $K$ .) By [L2], $K$ is a nonvanishing smooth field, so [L3] supplies a flow-box chart $(u^{1}, u^{2})$ near $p$ with $K = \partial_{u^{1}}$ . In this chart the smooth map $F (u^{1}, u^{2}) = (u^{1}, r (u^{1}, u^{2}))$ has derivative matrix $(\begin{array}{cc} 1 & 0 \\ 0 & \partial_{u^{2}} r \end{array})$ , since $\partial_{u^{1}} r = K (r) = 0$ . Its determinant is nonzero: otherwise $d r$ would vanish, contrary to $z \neq 0$ . Apply the smooth Euclidean inverse-function theorem [L4] at $p$ to obtain an open neighborhood $W$ with coordinates $(T, r) = (u^{1}, r)$ . The identities $K (T) = 1$ and $K (r) = 0$ show $K = \partial_{T}$ there.

4.1step 2.1step 3.1L2L5L6algebra

(The base metric in coordinates adapted to $K$ .) In the coordinates $(T, r)$ from step 3.1, $K = \partial_{T}$ and $q_{T T} = q (K, K) = - z$ by [L2]. Since $K$ is Killing, $L_{\partial_{T}} q = 0$ , so the coordinate components $q_{T T}, q_{T r}, q_{r r}$ are independent of $T$ . After shrinking $W$ to a coordinate rectangle, write $E (r) = q_{T T}$ and $F (r) = q_{T r}$ . The assumption $z \neq 0$ gives $E = - z \neq 0$ . Define locally $H (r) : = \int_{r_{0}}^{r} \frac{F (u)}{E (u)} d u .$ The integrand is smooth, and [L6] gives $H^{'} = F / E$ . Replace the time coordinate by $\hat{T} = T + H (r)$ , leaving $r$ fixed; because the old coordinate is $T = \hat{T} - H (r)$ , the new cross coefficient is $F - E H^{'} = 0$ , while $\partial_{\hat{T}} = \partial_{T} = K$ and $q_{\hat{T} \hat{T}} = E = - z$ . In these diagonal coordinates $z = q^{r r} = 1 / q_{r r}$ by its definition, so $q_{r r} = 1 / z$ . Substituting $z = 1 - s / r$ from step 2.1 gives $q = - (1 - \frac{s}{r}) d \hat{T} \otimes d \hat{T} + (1 - \frac{s}{r})^{- 1} d r \otimes d r$ . Renaming $\hat{T}$ as $T$ proves (b).

5.1step 3.1step 4.1L1L2algebra

(The metric and the Killing field.) In the product chart $(T, r, ϑ, φ)$ the components of $g$ are $g_{T T} = q_{T T}$ , $g_{r r} = q_{r r}$ , $g_{ϑ ϑ} = r^{2}$ , $g_{φ φ} = r^{2} \sin^{2} ϑ$ and all mixed components zero, by the definition of $g$ in The warped product of a two-dimensional Lorentzian base with a round sphere: Christoffel symbols and Ricci tensor; substituting step 4.1 gives (c). After shrinking to a connected coordinate rectangle, $z$ has a constant nonzero sign. On this product domain choose $N = \partial_{T}$ when $z > 0$ and $N = \partial_{r}$ when $z < 0$ ; its squared norm is respectively $- z < 0$ or $1 / z < 0$ , so the time-oriented domain required by Killing fields of a Lorentzian metric is available. Every displayed component is independent of $T$ , and $\partial_{T}$ has components $(1, 0, 0, 0)$ in this chart, so the Lie derivative $L_{\partial_{T}} g$ has all components $\partial_{T} g_{a b} = 0$ and vanishes; by [L2] we also know $K$ is a Killing field of $q$ with $K (r) = 0$ and $q (K, K) = - z$ , and $\partial_{T} = K$ by step 3.1, so $\partial_{T} (r) = 0$ and $g (\partial_{T}, \partial_{T}) = - z = - (1 - s / r)$ . This proves (d).

6.1step 2.1step 4.1step 5.1given∎

(Conclusion.) Step 2.1 proves (a), step 4.1 proves (b), step 5.1 proves (c) and (d); clause (e) records the local scope, the role of the hypotheses and the fact that $s$ is constant only on the connected set $V$ .

Remarks

What the theorem does and does not say. It converts the two-component vacuum condition on a warped chart into the local Schwarzschild form and an extra local Killing symmetry; it does not assert that every vacuum solution is spherically symmetric, that the region is an exterior region, that $s > 0$ , or that the coordinates extend beyond $W$ . The hypothesis $z \neq 0$ excludes every point with $r = s$ ; because $V$ is connected and $r - s$ is continuous, a region satisfying it lies wholly on one side of that level. It does not merely exclude a chart on which $r$ is constantly $s$ .
Relation to the global sources. The cited treatments derive the same conclusion under stronger global or static hypotheses and in coordinates adapted to the spherical symmetry; the version proved here is the local warped-product statement, with all hypotheses displayed, and it is the form used by the spherical vacuum model of this page.

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The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter

Physical scope

Conditional model deduction inside the classical gravitational framework with the field law in its direct-equation presentation: the Schwarzschild exterior datum of mass parameter m_* > 0 with the vacuum stress-energy prescription T = 0 and zero cosmological parameter is a model of the framework satisfying the adopted field law; its ideal static clocks read the proper times of the static congruence, and the parameter enters the model through the Schwarzschild radius, the round-sphere angular speed of circular geodesics and the leading far-field coefficient. Excludes: any claim that the model or its vacuum content is realized in nature, any empirical comparison (no observation, apparatus, calibration or uncertainty), non-vacuum matter, nonzero Lambda, the interior and horizon region (which belongs to the extension and Kruskal items of the page), distributional sources, and quantum effects. All conclusions are conditional on the adopted postulates.

Statement

Adopt the postulates Adopted spacetime model and classical domain of the gravitational framework, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, Adopted Einstein field equation in the direct-equation presentation and Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework of the classical gravitational framework, with the field law in its direct-equation presentation and the unit conventions of SI unit and coordinate conventions of the classical gravitational framework. Let $(M, g, N)$ be the Schwarzschild exterior datum of mass parameter $m_{*} > 0$ with Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ and static congruence $n = f^{- 1 / 2} \partial_{0}$ , $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), and let the model's stress-energy tensor be the vacuum prescription

$T : = 0 on M, Λ : = 0.$

Then all statements below are conditional deductions inside the model so specified; they assert nothing about nature.

(a) The Schwarzschild vacuum model. $(M, g, N)$ satisfies the spacetime postulate, $T = 0$ is a smooth symmetric stress-energy tensor field on the matter domain $M$ in the sense of Stress-energy tensor field of a model, the prescription is admissible, and the adopted field law $G_{a b} + Λ g_{a b} = κ T_{a b}$ holds identically on $M$ . A vacuum solution with zero cosmological parameter here means precisely a datum satisfying that equation with $T = 0$ and $Λ = 0$ ; this datum is such a solution, read as a model of the framework. The model is not locally flat: its curvature tensor is nowhere zero, while its Ricci tensor and Einstein tensor vanish identically (Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric clause (d)).

(b) Static clock readings. The integral curves of $n$ are the static observer worldlines at constant radius $r_{0}$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clause (b)); adopting the clock rule of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (i), the reading difference of an ideal clock carried along such a worldline between two events whose coordinate times differ by $Δ t$ is

$Δ τ = \sqrt{f (r_{0})} Δ t = \sqrt{1 - \frac{s}{r_{0}}} Δ t,$

so two static clocks at radii $r_{1}, r_{2}$ compared over the same coordinate interval with $Δ t \neq 0$ satisfy $Δ τ_{1} / Δ τ_{2} = \sqrt{f (r_{1}) / f (r_{2})}$ . The static worldlines are not geodesics: their acceleration is $a = \frac{s}{2 r^{2}} \partial_{r} \neq 0$ , so the model assigns them to accelerated (constrained) bodies and not to freely falling ones.

(c) The mass parameter. The parameter $m_{*}$ enters the model only through $s = 2 G_{N} m_{*} / c^{2}$ and through the quantities below; the framework reads $m_{*}$ as the mass parameter of the model in kilograms (SI unit and coordinate conventions of the classical gravitational framework clause (a), The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clause (a)):

Let $γ : I \to M$ be a smooth future timelike geodesic, parametrized by proper time $τ$ on an interval with nonempty interior, with $r \circ γ \equiv r_{0} > \frac{3 s}{2}$ . Put $σ : = ι \circ γ : I \to S^{2}$ and $t : = (x^{0} \circ γ) / c$ . Its angular speed with respect to $t$ is $Ω : = \sqrt{γ_{S^{2}} (d σ / d t, d σ / d t)}$ , where $d σ / d t : = σ^{'} (τ) / t^{'} (τ)$ and $t^{'} (τ) > 0$ by future timelikeness. It has units $s^{- 1}$ and satisfies $Ω^{2} = G_{N} m_{*} / r_{0}^{3}$ in every angular orientation. For an equatorial curve, $ϑ = π / 2$ and $Ω^{2} = (d φ / d t)^{2}$ in local polar charts (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (a) and proof step 2.2). This is the Kepler-like law in which $m_{*}$ plays the role of the central mass;
in the isotropic chart of The isotropic coordinate form of the Schwarzschild exterior metric and the isothermal radius shift, with $a = s / (4 ρ)$ , the exact identity $g_{00} + 1 = 1 - ((1 - a) / (1 + a))^{2} = 4 a / (1 + a)^{2}$ gives $g_{00} + 1 = + \frac{s}{ρ} + O (ρ^{- 2}) = + \frac{2 G_{N} m_{*}}{c^{2} ρ} + O (ρ^{- 2}) .$ Under the weak-field convention $g_{00} = - (1 + 2 Φ / c^{2})$ to leading order (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition), this is the coefficient $- 2 Φ_{N} / c^{2}$ for the explicit exterior Newtonian potential $Φ_{N} (ρ) : = - G_{N} m_{*} / ρ$ in the comparison reading of The Newtonian comparison model and the weak-field correspondence identification.

The reading is a statement about the model's own identities and the adopted unit and weak-field conventions; no numerical value of $G_{N}$ or $m_{*}$ , no Newtonian source model realized by matter and no comparison with any reported mass is asserted.

(d) Scope. The model is an idealized vacuum model of the framework, not a claim about the actual universe: no observation, apparatus, calibration or uncertainty is used, and no comparison with data is made. The datum solves the field law with the chosen parameters $(m_{*}, Λ, T) = (m_{*}, 0, 0)$ ; other solutions of the same field law exist and are not excluded, and the exterior domain $r > s$ contains no horizon or interior region, whose treatment is the subject of the extension and Kruskal items of this page. The physical interpretation of coordinates, clocks and bodies remains the adopted one of the cited postulates.

Facts & Assumptions

Given: The adopted framework postulates, the Schwarzschild exterior datum with $m_{*} > 0$ , $s = 2 G_{N} m_{*} / c^{2}$ , its metric $g$ , its static congruence $n = f^{- 1 / 2} \partial_{0}$ , the parameters $Λ = 0$ and $T = 0$ , and the coupling $κ = 8 π G_{N} / c^{4}$ .

[A1]

The model spacetime is a smooth time-oriented Lorentzian four-manifold without boundary, and the framework adopts the model's geometry and its matter content separately; a model that prescribes $T$ directly is admissible only if the prescription meets the consistency conditions imposed by the adopted field law (Adopted spacetime model and classical domain of the gravitational framework clauses (i) and (iv), Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions clauses (ii)-(iii)).

[A2]

The field law of the direct-equation presentation is $G_{a b} + Λ g_{a b} = κ T_{a b}$ with $κ = 8 π G_{N} / c^{4}$ , and it is the model's adopted gravitational dynamics (Adopted Einstein field equation in the direct-equation presentation clause (i), SI unit and coordinate conventions of the classical gravitational framework clause (a)).

[A3]

An ideal clock is an idealized device carried along a future timelike worldline whose readings reproduce the proper time of that worldline; ideal freely falling massive test bodies follow future timelike geodesics with $g (U, U) = - c^{2}$ ; ideal light rays are future null geodesics (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clauses (i)-(iii), Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).

[L1]

The Schwarzschild exterior datum $(M, g, N)$ is a smooth time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , with $f = 1 - s / r > 0$ on $r > s$ ; its Ricci tensor, scalar curvature and Einstein tensor vanish identically, and its curvature tensor is nowhere zero (The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric clauses (c)-(d)).

[L2]

The static congruence $n = f^{- 1 / 2} \partial_{0}$ is a future-directed unit timelike smooth field whose integral curves are the curves of constant $(r, ϑ, φ)$ ; along such a curve the proper-time increment over a coordinate increment $Δ x^{0}$ is $Δ τ = \sqrt{f (r_{0})} Δ x^{0} / c = \sqrt{f (r_{0})} Δ t$ , its acceleration is $a = \frac{s}{2 r^{2}} \partial_{r} \neq 0$ , and the ratio of proper-time increments of two static observers over a common nonzero coordinate interval is $\sqrt{f (r_{1}) / f (r_{2})}$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clauses (a)-(e), Observer congruences, their observer curves and adapted charts, Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold).

[L3]

For a constant-radius future timelike geodesic, the radial equation gives $γ_{S^{2}} (σ^{'}, σ^{'}) = \frac{f^{'} (r_{0})}{2 r_{0}} ((x^{0} \circ γ)^{'})^{2}$ in every sphere chart, including pole charts (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 proof step 2.2); its equatorial longitude law is $(d φ / d t)^{2} = G_{N} m_{*} / r_{0}^{3}$ (that supplier's clause (a)). In the isotropic chart, $g_{00} = - (1 - a)^{2} / (1 + a)^{2}$ with $a = s / (4 ρ)$ , so direct algebra gives $g_{00} + 1 = 4 a / (1 + a)^{2} = + s / ρ + O (ρ^{- 2})$ . The weak-field metric convention is $g_{00} = - (1 + 2 Φ / c^{2})$ to first order, and the comparison dictionary identifies its potential with the Newtonian comparison potential; choosing $Φ_{N} (ρ) = - G_{N} m_{*} / ρ$ gives $- 2 Φ_{N} / c^{2} = + s / ρ$ (The isotropic coordinate form of the Schwarzschild exterior metric and the isothermal radius shift clause (b), Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition, The Newtonian comparison model and the weak-field correspondence identification, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Rational powers $a^{r}$ of a positive base).

[L4]

By direct substitution in [A2], any datum with $G = 0$ , $T = 0$ and $Λ = 0$ satisfies $G + Λ g = 0 = κ T$ . The covariant derivative of the zero tensor is zero, so this prescription also satisfies $\nabla^{a} T_{a b} = 0$ . This is a local algebraic deduction from Adopted Einstein field equation in the direct-equation presentation clause (i), not an application of its flat example.

Proof

technique · verify the model postulates and the field law from the vanishing Einstein tensor, then read the clock and mass-parameter statements off the cited geometric lemmas through the adopted postulates

1.1A1A2L1L4givenalgebra

(The model and its field law.) By [L1] and The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation the datum is a smooth time-oriented Lorentzian four-manifold without boundary, so it satisfies the spacetime postulate by [A1]; $T = 0$ is a smooth symmetric covariant $2$ -tensor field on the matter domain $M$ . By [L1], $G_{a b} \equiv 0$ ; with $Λ = 0$ and $T = 0$ , $G_{a b} + Λ g_{a b} = 0 = κ T_{a b}$ holds identically and $\nabla^{a} T_{a b} = 0$ trivially, so the prescription is admissible by [A1] and satisfies the field law by [A2], which is the vacuum solution of [L4]. That the model is not locally flat is [L1]. This proves (a).

1.2A3L2algebra

(Static clock readings.) By [L2] the integral curves of $n$ are the constant-radius worldlines and are smooth future timelike curves; by [A3] the model assigns to an ideal clock carried along each of them the reading equal to the proper time, and [L2] gives the increment $Δ τ = \sqrt{f (r_{0})} Δ t$ and, for a common $Δ t \neq 0$ , the ratio $\sqrt{f (r_{1}) / f (r_{2})}$ for two radii; the denominator is nonzero since $f (r_{2}) > 0$ . Since [L2] also gives $a = \frac{s}{2 r^{2}} \partial_{r} \neq 0$ , the static worldlines are not geodesics, so by [A3] the model does not assign them to freely falling test bodies. This proves (b).

1.3L3algebra

(The mass parameter.) By The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clauses (a) and (c), the parameter $m_{*}$ enters the associated geometry through $s = 2 G_{N} m_{*} / c^{2}$ . For a circular geodesic $γ$ as in (c)(1), future timelikeness gives $g (γ^{'}, N) = - f (r_{0}) (x^{0} \circ γ)^{'} < 0$ , hence $t^{'} = (x^{0} \circ γ)^{'} / c > 0$ . Dividing the radial identity of [L3] by $(t^{'})^{2}$ gives $Ω^{2} = c^{2} f^{'} (r_{0}) / (2 r_{0}) = c^{2} s / (2 r_{0}^{3}) = G_{N} m_{*} / r_{0}^{3}$ . This identity uses the round metric and is independent of the angular chart; at the equator its polar expression is $(d φ / d t)^{2}$ . For the isotropic chart, the exact coefficient in [L3] gives $g_{00} + 1 = \frac{4 a}{(1 + a)^{2}} = 4 a + O (a^{2}) = + \frac{s}{ρ} + O (ρ^{- 2}) = + \frac{2 G_{N} m_{*}}{c^{2} ρ} + O (ρ^{- 2}), a = \frac{s}{4 ρ},$ as $ρ \to \infty$ with fixed $s$ , by the exact identity $(1 + a)^{- 2} = 1 + O (a)$ . The weak-field convention and comparison dictionary in [L3] identify the leading coefficient as $- 2 Φ_{N} / c^{2}$ for $Φ_{N} (ρ) = - G_{N} m_{*} / ρ$ . The framework's unit conventions of SI unit and coordinate conventions of the classical gravitational framework read $m_{*}$ in kilograms, and these are the model's own occurrences of the parameter; no numerical value, source model or observation enters. This proves (c).

2.1step 1.1step 1.2step 1.3given∎

(Status and scope.) Steps 1.1-1.3 establish (a)-(c) as consequences of the adopted postulates and the cited mathematics, using no observation, apparatus or measured input; clause (d) records that the model is one solution among many, that the exterior domain excludes the horizon and interior and that no empirical claim is made.

Remarks

One model among many. Nothing in the deduction selects the Schwarzschild exterior uniquely: the vacuum field law admits many other solutions, and the local Birkhoff theorem of this page shows only that a vacuum warped product with spherical orbits is locally of this form under the stated hypotheses. The model is used on this page as the exact spherical comparison model, not as a claim about the world.
Where the staticity entered. Clause (b) is a statement about the model's static congruence and its ideal clocks; it does not assert that any real clock is ideal, and the acceleration statement is the geometric reason why holding a clock at fixed radius is not free fall in this model.

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Restricted redshift prediction of the Schwarzschild vacuum model: the observer-energy ratio between static observers

Physical scope

Conditional prediction of the adopted Schwarzschild exterior vacuum model: for a specified ideal light ray (a future null geodesic) joining two events on static observer worldlines at radii r_E and r_R, the ratio of the observer energies measured with the static congruence is sqrt(f(r_E)/f(r_R)) with f = 1 - 2G_N m_*/(c^2 r), and the redshift parameter defined from that ratio satisfies 1 + z = sqrt(f(r_R)/f(r_E)). The statement is an exact consequence of the model metric, the null-geodesic light-ray rule and the observer-energy definition; it asserts no observed wavelength, spectral line, calibration or measurement, and it excludes moving observers, non-geometric-optics propagation and dispersion. Any comparison with a reported redshift requires a separate apparatus, calibration and statistical model.

Statement

Adopt the Schwarzschild exterior vacuum model of The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter: the exterior datum $(M, g, N)$ of mass parameter $m_{*} > 0$ with $s = 2 G_{N} m_{*} / c^{2}$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), its metric-selected connection $\nabla$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), its static congruence $n = f^{- 1 / 2} \partial_{0}$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios), and the light-ray rule that ideal light rays are future null geodesics of $\nabla$ (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (iii)). Let $η : J \to M$ be the specified ray, a smooth affinely parametrized geodesic of $\nabla$ on an interval $J$ , with tangent $W = η^{'}$ , $g (W, W) = 0$ and $W$ future-directed (Geodesic of an affine connection, Timelike, null, spacelike and causal vectors of a Lorentz vector space). Let $λ_{E}, λ_{R} \in J$ with $λ_{E} < λ_{R}$ and suppose $η (λ_{E})$ and $η (λ_{R})$ lie on static observer worldlines of radii $r_{E} = r (η (λ_{E}))$ and $r_{R} = r (η (λ_{R}))$ with $r_{E}, r_{R} \in (s, \infty)$ . For each parameter value define the observer energy of the ray relative to the static observers by

$E (λ) : = - c g_{η (λ)} (W (λ), n (η (λ))),$

the value of the observer-energy definition of Observer-relative energy, spatial momentum and rest mass of a tangent vector clause (a) for the unit observer $n$ . Then:

(a) Constancy of the Killing pairing. $K = \partial_{0}$ is a Killing field of $g$ and the function $C (λ) : = - g_{η (λ)} (W (λ), K (η (λ)))$ is constant along $J$ ; moreover $C > 0$ .

(b) The observer-energy ratio. $E (λ) > 0$ along the ray and

$\frac{E (λ_{R})}{E (λ_{E})} = \sqrt{\frac{f (r_{E})}{f (r_{R})}} = \sqrt{\frac{1 - s / r_{E}}{1 - s / r_{R}}} .$

(c) Redshift parameter. If the redshift parameter of the pair of events is defined by $z : = \frac{E (λ_{E})}{E (λ_{R})} - 1$ , equivalently by $1 + z = E (λ_{E}) / E (λ_{R})$ , then

$1 + z = \sqrt{\frac{f (r_{R})}{f (r_{E})}} .$

The ratio in (b) and (c) is unchanged if $W$ is replaced by $μ W$ with a positive constant $μ$ : a future null ray determines its future tangent only up to positive scale, and only the ratio of energies is asserted. No observed wavelength, spectral line, calibration or measurement enters the statement; the identification of the ratio with a frequency ratio of an ideal ray is part of the framework's geometric-optics reading, and moving observers require additional factors.

(d) Scope. The statement is a model prediction for the specified ideal ray and the specified static observers; it asserts no redshift measurement and no property of any real source, and it excludes non-geometric-optics propagation, dispersion and any apparatus or analysis model.

Facts & Assumptions

Given: The Schwarzschild exterior datum with $m_{*} > 0$ and $s = 2 G_{N} m_{*} / c^{2}$ , its metric $g$ , its connection $\nabla$ , its static congruence $n = f^{- 1 / 2} \partial_{0}$ , a future null geodesic $η : J \to M$ with tangent $W$ , parameters $λ_{E} < λ_{R}$ in $J$ with radii $r_{E}, r_{R} > s$ , and the energies $E (λ)$ of the statement.

[L1]

In a spherical-polar product chart the metric components are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ with $f = 1 - s / r > 0$ on $r > s$ , and the static congruence is $n = f^{- 1 / 2} \partial_{0}$ with $g (n, n) = - 1$ , $g (n, \partial_{0}) = - f^{1 / 2}$ (The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clause (a)).

[L2]

The product factor $R$ defines a global time-translation field $K = \partial_{0}$ on the exterior datum. The global metric is invariant under these translations, so $K$ is Killing; on every product chart its coordinate expression is $\partial_{0}$ . For any geodesic contained in one such chart, $g (W, K)$ is constant by the Killing-field conservation identity (Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model clause (a), Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clause (b), Killing fields of a Lorentzian metric).

[L3]

The observer energy of a tangent vector $P$ relative to an observer $n$ is $E_{n} (P) = - c g (P, n)$ ; for a nonzero future causal $P$ it is positive (Observer-relative energy, spatial momentum and rest mass of a tangent vector clause (a), Pairing of an observer with future causal vectors claim 1, Observers, rest spaces and the spatial projection at an event).

[L4]

The ideal light ray of the model is a future null geodesic; when its future direction is retained, its tangent is determined only up to a positive constant affine rescaling, since a negative factor reverses time orientation (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (iii), Geodesic of an affine connection, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L5]

The square root $\sqrt{f (r)} > 0$ is smooth on $r > s$ , and $\sqrt{f (r_{R})} / \sqrt{f (r_{E})} = \sqrt{f (r_{R}) / f (r_{E})}$ (Rational powers $a^{r}$ of a positive base, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

Proof

technique · contract the null tangent with the Killing field, express the observer energy through the static congruence, and take the ratio at the two parameter values

1.1L2L3L1algebra

(Constancy and positivity of the pairing.) The globally defined field $K = \partial_{0}$ is Killing by [L2]. Each parameter value has a neighborhood in $J$ whose ray segment lies in a product chart; on that segment the Killing conservation identity gives constancy of the scalar function $C (λ) : = - g_{η (λ)} (W (λ), K (η (λ)))$ . These local constants agree on overlaps because the pairing is a globally defined scalar, so $C$ is locally constant on the connected interval $J$ and therefore constant on $J$ . Since $W$ is nonzero future causal and $K$ is future timelike, [L3] with the unit observer $\tilde{n} : = K / \sqrt{f}$ at each point gives $g (W, \tilde{n}) < 0$ , hence $C = - g (W, K) > 0$ .

2.1step 1.1L1L3L4L5algebra

(The observer energy along the ray.) By [L3] and [L1], writing $n = f^{- 1 / 2} K$ at each point and using the constancy of step 1.1, $E (λ) = - c g (W, n) = - c f (r (η (λ)))^{- 1 / 2} g (W, K) = c C f (r (η (λ)))^{- 1 / 2} > 0$ . Evaluating at $λ_{E}$ and $λ_{R}$ and dividing gives $E (λ_{R}) / E (λ_{E}) = \sqrt{f (r_{E}) / f (r_{R})}$ , which is claim (b); the definition $1 + z = E (λ_{E}) / E (λ_{R})$ gives $1 + z = \sqrt{f (r_{R}) / f (r_{E})}$ , which is (c). Replacing $W$ by $μ W$ with $μ > 0$ multiplies $C$ and every $E (λ)$ by $μ$ , leaving both ratios unchanged, and by [L4] only this positive scale ambiguity is present for future tangents.

3.1step 1.1step 2.1given∎

(Conclusion and status.) Steps 1.1-2.1 establish (a)-(c) as consequences of the model, the light-ray rule and the observer-energy definition, using no observed quantity; clause (d) records that the statement is a conditional model prediction for the specified ideal ray and static observers, with moving observers and apparatus models excluded.

Remarks

Model prediction, not an observation. The theorem compares two ideal observer energies of one ideal null geodesic in the adopted model. A reported redshift is a measured quantity with an emission line, a spectrograph, calibration, resolution and uncertainty; identifying the two requires an apparatus and analysis model, and this is a subject for experiment items and for the experimental-interface page, not a consequence of the mathematics proved here.
Where each assumption entered. The metric form gave the pairing $g (W, \partial_{0})$ and the factor $f^{- 1 / 2}$ ; the Killing property of $\partial_{0}$ gave constancy; the light-ray rule gave the identification of the ideal ray with a future null geodesic; the observer-energy definition entered through $E = - c g (W, n)$ . No field equation and no matter model was used: the result holds for every Schwarzschild exterior datum with $s > 0$ irrespective of the field law, and it is the physical reading of the model that attaches the frequency interpretation.

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Restricted geodesic predictions of the Schwarzschild vacuum model: circular free-fall orbits, the stability threshold and the photon sphere

Physical scope

Conditional model predictions of the adopted Schwarzschild exterior vacuum model for ideal test bodies and ideal light rays: for every radius r_0 > 3s/2 the model assigns a circular free-fall worldline to an ideal freely falling massive test body with coordinate angular velocity (dphi/dt)^2 = G_N m_*/r_0^3; there is no circular free-fall worldline with s < r_0 <= 3s/2; the effective radial potential has a strict local minimum at the circular-orbit value exactly for r > 3s and a strict local maximum for 3s/2 < r < 3s; and the model admits an ideal light ray on the null circular geodesic at r = 3s/2. The statements are exact consequences of the model metric and the free-fall and light-ray postulates in the test-field idealization; they exclude nonlinear stability, radiation reaction, self-force, finite-size effects, orbit evolution and any observation or measurement, and the stability claim is the sign of the second derivative of the stated potential only.

Statement

(a) Circular free-fall orbits. For every $r_{0} > \frac{3 s}{2}$ the model assigns to an ideal freely falling massive test body with the corresponding initial event and four-velocity a physical history that is a restriction of the circular timelike geodesic $γ : R \to M$ of radius $r_{0}$ in Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (a), with $e, ℓ$ as there. This all-real mathematical geodesic is the unique maximal continuation of that history; the physical domain need not be maximal. On the physical history, the tangent is future timelike with $g (γ^{'}, γ^{'}) = - c^{2}$ and proper time affine parameter, and its coordinate angular velocity satisfies

$(\frac{d φ}{d t})^{2} = \frac{G_{N} m_{*}}{r_{0}^{3}} .$

(b) No circular free-fall orbit below the threshold. If a future timelike geodesic of the datum, parametrized by proper time, has $r \equiv r_{0} > s$ constant on an interval of positive length, then $r_{0} > \frac{3 s}{2}$ , regardless of angular orientation; in particular no ideal freely falling massive test body is assigned a circular worldline of radius $s < r_{0} \leq \frac{3 s}{2}$ .

(c) Stability threshold of the restricted radial potential. For each $r > \frac{3 s}{2}$ set $ℓ_{r}^{2} = c^{2} s r^{2} / (2 r - 3 s)$ . Differentiate $V_{ℓ_{r}} (x) = \frac{1}{2} f (x) (c^{2} + ℓ_{r}^{2} / x^{2})$ with respect to $x$ while holding $ℓ_{r}$ fixed, then evaluate at $x = r$ . The result is

$V_{ℓ_{r}}^{''} (r) = \frac{c^{2} s (r - 3 s)}{r^{3} (2 r - 3 s)}, r > \frac{3 s}{2},$

so the potential has a strict local minimum at the circular-orbit value exactly for $r > 3 s$ and a strict local maximum exactly for $\frac{3 s}{2} < r < 3 s$ ; the radius $r = 3 s$ is the innermost stable circular orbit radius of the restricted model in this potential sense.

(d) Photon sphere as an ideal light orbit. The null circular geodesic of Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (d) at $r = \frac{3 s}{2}$ is the all-real maximal mathematical continuation for an ideal light ray with the corresponding initial event and future tangent. The ray's physical history is a restriction to its actual affine-parameter domain; along it

$(\frac{d φ}{d t})^{2} = \frac{4 c^{2}}{27 s^{2}} .$

(e) Scope. All statements are conditional model predictions for ideal test bodies and ideal rays in the exterior domain: no self-force, radiation reaction, finite-size coupling, nonlinear orbital stability, orbit decay or evolution, no interior or horizon region and no observation enters. The stability claim of (c) is the sign of the second derivative of the stated potential; the circular worldlines of (a) are geodesics of the model, and the model assigns their restrictions to freely falling bodies through the adopted free-fall rule.

Facts & Assumptions

Given: The Schwarzschild exterior datum with $m_{*} > 0$ , $s = 2 G_{N} m_{*} / c^{2}$ and $f = 1 - s / r$ , its connection $\nabla$ , the adopted free-fall and light-ray rules, and the geodesic lemmas of the page.

[A1]

An ideal freely falling massive test body has as worldline a future timelike geodesic with $g (U, U) = - c^{2}$ , proper time an affine parameter, and the worldline is determined by its release event and four-velocity; an ideal light ray is a future null geodesic (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clauses (ii)-(iii), Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).

[A2]

The model satisfies the field law with $T = 0$ and $Λ = 0$ and its spacetime is the smooth time-oriented Lorentzian four-manifold of the exterior datum (The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter clause (a), The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L1]

For every $r_{0} > \frac{3 s}{2}$ the curve of clause (a) of Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 is a future timelike geodesic with $g (γ^{'}, γ^{'}) = - c^{2}$ , conserved quantities $e, ℓ$ of Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model, and coordinate angular velocity $(d φ / d t)^{2} = G_{N} m_{*} / r_{0}^{3}$ ; conversely any future timelike geodesic parametrized by proper time with constant radius $r_{0} > s$ on an interval of positive length satisfies $r_{0} > \frac{3 s}{2}$ , and in the equatorial case it also satisfies $ℓ^{2} = c^{2} s r_{0}^{2} / (2 r_{0} - 3 s)$ ; and, after differentiating at fixed $ℓ$ and evaluating at $r > \frac{3 s}{2}$ with $ℓ^{2} = c^{2} s r^{2} / (2 r - 3 s)$ , one has $V_{ℓ}^{''} (r) = c^{2} s (r - 3 s) / (r^{3} (2 r - 3 s))$ (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clauses (a)-(c)).

[L2]

For $ℓ \neq 0$ the curve with $r = \frac{3 s}{2}$ and $x_{0}^{0}, φ_{0}$ as in clause (d) is a future null geodesic with conserved quantities $e_{0}, ℓ$ and $(d φ / d t)^{2} = 4 c^{2} / (27 s^{2})$ (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (d), Geodesic of an affine connection, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L3]

The static congruence of the model is $n = f^{- 1 / 2} \partial_{0}$ with $f = 1 - s / r > 0$ , and it is not geodesic; the coordinate $t$ is the coordinate time $x^{0} / c$ of the datum (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clauses (a) and (e), The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clause (d), Observer congruences, their observer curves and adapted charts).

[L4]

The statements of the geodesic lemmas are identities of the smooth exterior datum and involve no matter model, no field equation and no observation (Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model clause (d), Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (e), Rational powers $a^{r}$ of a positive base, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

Proof

technique · apply the free-fall and light-ray rules to the exact geodesic lemmas of the page and read off the angular-velocity and potential statements

1.1A1A2L1L3algebra

(Circular free-fall orbits.) By [A2] the adopted model satisfies the field law with $T = 0$ and $Λ = 0$ and its spacetime is the exterior datum. Let $r_{0} > \frac{3 s}{2}$ and let $γ$ be the curve of [L1]; by [L1] it is a future timelike geodesic with $g (γ^{'}, γ^{'}) = - c^{2}$ , so by [A1] its restriction to the actual domain of an ideal freely falling massive test body with the same initial event and tangent is the body's physical history. The all-real curve is its unique maximal mathematical continuation by The free-fall worldline of an ideal test body is determined by its initial event and four-velocity; proper time is an affine parameter, and [L1] gives $(d φ / d t)^{2} = G_{N} m_{*} / r_{0}^{3}$ with $t = x^{0} / c$ as in [L3]. This proves (a).

1.2A1L1algebra

(No circular orbit below the threshold.) Let $γ$ be any future timelike geodesic parametrized by proper time with $r \equiv r_{0} > s$ on an interval of positive length; by [L1] it satisfies $r_{0} > \frac{3 s}{2}$ without an angular-orientation restriction. Hence no such geodesic exists for $s < r_{0} \leq \frac{3 s}{2}$ , and by [A1] the model assigns no circular worldline of those radii to an ideal freely falling massive test body. This proves (b).

1.3L1algebra

(Stability threshold.) Fix $r > 3 s / 2$ and $ℓ^{2} = c^{2} s r^{2} / (2 r - 3 s)$ , held constant during differentiation. By [L1] the second derivative of $V_{ℓ}$ at this circular-orbit value is $V_{ℓ}^{''} (r) = c^{2} s (r - 3 s) / (r^{3} (2 r - 3 s))$ for $r > \frac{3 s}{2}$ ; its sign is the sign of $r - 3 s$ , so $V_{ℓ}^{''} (r) > 0$ exactly for $r > 3 s$ and $V_{ℓ}^{''} (r) < 0$ exactly for $\frac{3 s}{2} < r < 3 s$ , with $V_{ℓ}^{''} (3 s) = 0$ . The strict-extremum assertions, including the absence of either extremum at $3 s$ , follow from clause (c) of Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 in [L1], whose endpoint proof factors $V_{ℓ}^{'} (x) = c^{2} s (x - 3 s)^{2} / (2 x^{4})$ at $ℓ^{2} = 3 c^{2} s^{2}$ . This proves (c) in its stated radial-potential sense.

1.4A1L2algebra

(Photon sphere.) The curve of [L2] is an all-real future null geodesic. By [A1] a physical ideal ray with its initial event and tangent is a geodesic on its actual affine-parameter interval. The geodesic uniqueness theorem The geodesic initial-value problem and the exponential map of the metric-selected connection identifies it with a restriction of the displayed all-real curve, which is therefore its unique maximal mathematical continuation. Along that restriction, [L2] gives $(d φ / d t)^{2} = 4 c^{2} / (27 s^{2})$ ; this is (d).

2.1step 1.1step 1.2step 1.3step 1.4L4given∎

(Conclusion and status.) Steps 1.1-1.4 establish (a)-(d) as consequences of the model, the adopted coupling rules and the exact geodesic lemmas; clause (e) records the test-field idealization, the purely radial-potential sense of the stability claim and the exclusion of observations, nonlinear stability and radiation effects.

Remarks

Geodesics versus observed orbits. A circular geodesic is a property of the smooth model; whether any astronomical body follows one depends on the model's applicability, on matter couplings and on observational modelling, none of which is asserted here. The Kepler-like law (a) uses the model's parameter $m_{*}$ and is not a statement about a measured mass.
What stability means here. Claim (c) is the sign of $V_{ℓ}^{''}$ at the circular-orbit value, the standard linear potential test; it is not a nonlinear or Lyapunov stability theorem, and the potential itself is a function of the reduced one-dimensional problem, not a physical energy of the model.

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Horizon versus singularity in the Schwarzschild model: a chart breakdown, a smooth null surface and a curvature obstruction

Physical scope

Scope note for the spherical vacuum model page: it distinguishes the coordinate breakdown of the Schwarzschild chart at r = s from the curvature obstruction at r = 0 in the smooth model, records that the advanced-extension horizon is a smooth null hypersurface, that the Kruskal level r = s is the union of two smooth null hypersurfaces intersecting transversely at the bifurcation sphere (their union is not smooth there), and that the right-exterior future escape boundary is U = 0 with the stated one-way property, and states what is not claimed (maximality, geodesic completeness, singularity theorems, collapse and any observation). It asserts no physical fact about nature beyond the model-conditional statements of the cited items and reports no measurement.

What the Schwarzschild chart does at $r=s$

The exterior datum of The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation is defined on the open domain $r > s$ only. At every point of this domain a product-chart frame remains a basis. The component $g_{11} = f (r)^{- 1} = (1 - s / r)^{- 1}$ diverges as $r ↓ s$ , and the chart contains no point with $r = s$ , so there is no frame limit there in this chart. This is a statement about one component, not an invariant: the metric tensor is smooth and nondegenerate on $r > s$ and has the separate regular extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart.

The advanced horizon and the two Kruskal null hypersurfaces

Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart constructs on $M_{e x t} = R \times (0, \infty) \times S^{2}$ a smooth nondegenerate Lorentzian metric $g_{e x t}$ whose restriction to $r > s$ is the exterior datum; the level set $r = s$ is a smooth null hypersurface with generator direction $\partial_{v}$ , and the Ricci tensor and Kretschmann scalar of $g_{e x t}$ extend smoothly across it. In the Kruskal model (The Kruskal model datum: domain, implicit radial function, metric and time orientation, Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar) the full level set ${r = s}$ is ${U V = 0} = {U = 0} \cup {V = 0}$ . Each axis times $S^{2}$ is a smooth hypersurface: $d U$ and $d V$ never vanish, and the inverse base matrix has $g^{U U} = g^{V V} = 0$ , so each has null normal covector. They intersect transversely at the bifurcation sphere ${U = V = 0} \times S^{2}$ . Their tangent spaces there together span the full four-dimensional tangent space, so their union cannot be a smooth three-dimensional hypersurface at that sphere. The function $r$ and metric remain smooth there; the curvature invariant is $12 / s^{4}$ on both axes. For future escape to the right exterior, the boundary is specifically ${U = 0}$ , while ${V = 0}$ is the past escape boundary. The one-way character in the model is the exact statement of The causal cone, the escape horizon and decreasing areal radius in the Kruskal model clause (d): no future-directed causal curve issued at a point with $U \geq 0$ reaches the quadrant ${U < 0, V > 0}$ with $r \to + \infty$ , while every point with $U < 0$ admits such a curve along the future radial null geodesic at constant $U$ . This is a causal property of the smooth model; it is not a divergence of the metric at the surface, and the model's time orientation is part of the datum.

The obstruction at $r=0$ is curvature, not coordinates

The Kretschmann scalar of the exterior datum is $K = R_{a b c d} R^{a b c d} = 12 s^{2} / r^{6} = \frac{48 G_{N}^{2} m_{*}^{2}}{c^{4} r^{6}}$ (Orthonormal-frame curvature components and the Kretschmann scalar 12s^2/r^6 of the Schwarzschild exterior metric clause (c)): on $r > s$ it takes values in $(0, 12 / s^{4})$ and has the finite boundary value $12 / s^{4}$ as $r ↓ s$ ; on the smooth extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart, whose domain has $r > 0$ and on which the same expression $12 s^{2} / r^{6}$ is the Kretschmann scalar, it grows without bound as $r ↓ 0$ . In the Kruskal model it is smooth on $K$ and grows without bound as $U V \to 1$ , that is, as $r ↓ 0$ (Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar clause (d)). The value $r = 0$ is not attained on $K$ ; the set ${U V = 1}$ is not part of the model. The invariant is a scalar built from the metric, so it is chart-independent, and the following argument shows that $r = 0$ cannot be appended as a regular $C^{2}$ metric boundary point: if a $C^{2}$ Lorentzian metric on a manifold containing $K$ together with a boundary point $q$ approached with $r \to 0$ restricted to $g_{K}$ on $K$ , then its curvature tensor would be continuous near $q$ and its Kretschmann scalar bounded there, contradicting the unboundedness of the same invariant along sequences in $K$ . The distinction between the two values of $r$ is therefore a statement about the smooth curvature of the model, not about the chosen coordinates.

What is not claimed

No singularity theorem and no collapse. Nothing above proves that any solution of the field law develops the region $0 < r < s$ , that it is realized by collapsing matter, or that any observed object is described by the model; the interior region and the second exterior region of the Kruskal model are geometric regions of that model only.
No maximality or completeness. The extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart is not claimed maximal, and no geodesic completeness or incompleteness statement for $K$ is made here; the maximal-development and boundary theorems are deferred to their own modules of the track.
No asymptotic or global mass. The parameter $m_{*}$ is the model's parameter (The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter clause (c)); no ADM-type mass at infinity and no boundary-asymptotic comparison is asserted on this page.
No observation. The words horizon and singularity here refer to properties of the model; whether the model applies to any observed system is an empirical question belonging to experiment items and to the experimental-interface page, with apparatus, calibration, sampling and uncertainty qualifications.

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Scope and status of the spherical vacuum model page

Physical scope

Scope and status note for the spherical vacuum model page: it records that the page's items are conditional results of the adopted smooth classical model, that no observation or measured value is asserted on the page, that the empirical interpretation of any redshift, orbit or horizon statement requires separate apparatus and analysis models, and that maximality, global mass, collapse and non-vacuum interior topics are deferred. It asserts no physical fact about nature and reports no measurement.

What the page defines and proves

The model. The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation fixes the datum: the manifold $R \times (s, \infty) \times S^{2}$ , the global smooth metric $g = - f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ with diagonal components $- f, f^{- 1}, r^{2}, r^{2} \sin^{2} ϑ$ in spherical-polar product charts, the time orientation $\partial_{0}$ and the parameters $c, G_{N}, m_{*} > 0$ with $s = 2 G_{N} m_{*} / c^{2}$ . The components carry the unit readings of SI unit and coordinate conventions of the classical gravitational framework; no numerical value of any constant is asserted.
The vacuum calculation. Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric computes the Christoffel symbols, the curvature, the vanishing Ricci and Einstein tensors and the nonvanishing curvature of the datum, and The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter reads the result as a model of the framework's field law with $T = 0$ and $Λ = 0$ , together with the model's static clock readings and the reading of the mass parameter.
Local uniqueness. Local Birkhoff theorem: a vacuum warped product with a round sphere factor is locally Schwarzschild with an extra Killing field shows that a vacuum warped product with spherical orbits is locally of this form with an extra local Killing field, under the explicit hypotheses of that theorem; the conclusion is local and does not assert that every vacuum solution or every spherical vacuum region is an exterior region with positive mass.
Restricted predictions. Restricted redshift prediction of the Schwarzschild vacuum model: the observer-energy ratio between static observers and Restricted geodesic predictions of the Schwarzschild vacuum model: circular free-fall orbits, the stability threshold and the photon sphere are conditional model predictions for ideal clocks, test bodies and light rays, using the adopted coupling rules of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework; they are not measurements and involve no apparatus model. The horizon and singularity distinctions are recorded in Horizon versus singularity in the Schwarzschild model: a chart breakdown, a smooth null surface and a curvature obstruction.

What is not asserted

No observation and no empirical comparison. The page contains no experiment item, no reported observation, no calibration or uncertainty record and no statistical inference. A comparison of the redshift ratio, an orbital frequency or a horizon statement with data would test the model together with an apparatus, preparation and analysis model and would be evidence about the combination, not a proof of the model (Adopted Einstein field equation in the direct-equation presentation and the postulates fix only the model side).
No numerical values. Neither $G_{N}$ , $c$ , $m_{*}$ nor any derived number is evaluated; the statements are identities in the parameters.
Deferred topics. Maximality and global structure of the Kruskal model, causal-boundary and maximal-development theorems, boundary-asymptotic mass, rotating and charged models, collapse and singularity theorems, and the empirical interface are separate modules of the track; nothing on this page supplies or presupposes them. Where an item uses the extension or Kruskal models it says so explicitly, and where it uses the weak-field reading of the mass parameter it carries the comparison warning of the isotropic coordinate lemma.

5 · Examples, experiments and counterexamples

None yet.