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

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 $f^{'} = s / r^{2}$ . Then:

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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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