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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

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 Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ and $f = 1 - s / r$ (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), and the free-fall and light-ray rules of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework (ideal freely falling massive test bodies follow future timelike geodesics with $g (U, U) = - c^{2}$ and proper time as affine parameter; ideal light rays are future null geodesics). Then:

(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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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