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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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