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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources