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Explicit circular-orbit data of the Schwarzschild exterior at the ISCO radius 3s and the photon-sphere radius 3s/2

Example

Let $(M, g, N)$ be the Schwarzschild exterior datum with Schwarzschild radius $s > 0$ , $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), with the circular-orbit data of Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 and the radial potential $W_{ℓ} (r) = f (r) (c^{2} + ℓ^{2} / r^{2})$ of Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model clause (c). Then the following exact evaluations hold.

In every $V_{ℓ}^{''}$ evaluation below, differentiate with $ℓ$ fixed as a constant and substitute the circular-orbit value only after differentiation.

The innermost stable circular orbit at $r_{0} = 3 s$ . With $ℓ^{2} = \frac{c^{2} s r_{0}^{2}}{2 r_{0} - 3 s}$ one has $ℓ^{2} = 3 c^{2} s^{2}$ , $e^{2} = \frac{8}{9} c^{2}$ and $e = \frac{2 \sqrt{2}}{3} c$ , and the second derivative of $V_{ℓ} = \frac{1}{2} W_{ℓ}$ vanishes: $V_{ℓ}^{''} (3 s) = \frac{c^{2} s (3 s - 3 s)}{(3 s)^{3} (6 s - 3 s)} = 0$ . The circular angular velocity at this radius satisfies $(\frac{d φ}{d t})^{2} = \frac{G_{N} m_{*}}{(3 s)^{3}} = \frac{c^{2}}{54 s^{2}}$ .
The photon sphere at $r_{0} = \frac{3 s}{2}$ . For $ℓ \neq 0$ , with $e_{0} = \frac{∣ ℓ ∣ \sqrt{f (r_{0})}}{r_{0}} = \frac{2 ∣ ℓ ∣}{3 \sqrt{3} s}$ one has $f (\frac{3 s}{2}) = \frac{1}{3}$ and $(\frac{d φ}{d t})^{2} = \frac{c^{2} f (r_{0})}{r_{0}^{2}} = \frac{4 c^{2}}{27 s^{2}}$ ; the null potential $W_{ℓ}^{n} (r) = f (r) ℓ^{2} / r^{2}$ has $(W_{ℓ}^{n})^{'} (\frac{3 s}{2}) = 0$ and $(W_{ℓ}^{n})^{''} (\frac{3 s}{2}) = - \frac{32 ℓ^{2}}{81 s^{4}} < 0$ .
A stable comparison radius. At $r_{0} = 4 s$ one has $ℓ^{2} = \frac{c^{2} s \cdot 16 s^{2}}{8 s - 3 s} = \frac{16}{5} c^{2} s^{2}$ and $V_{ℓ}^{''} (4 s) = \frac{c^{2} s (4 s - 3 s)}{(4 s)^{3} (8 s - 3 s)} = \frac{c^{2}}{320 s^{2}} > 0$ , a strict local minimum: the two threshold values $3 s / 2$ (existence) and $3 s$ (stability) separate the unstable circular orbits $\frac{3 s}{2} < r_{0} < 3 s$ from the stable ones $r_{0} > 3 s$ .
Blow-up at the existence threshold. As $r_{0} ↓ \frac{3 s}{2}$ the required angular momentum diverges, $ℓ^{2} = \frac{c^{2} s r_{0}^{2}}{2 r_{0} - 3 s} \to + \infty$ , so there is no finite angular momentum at the limiting radius; for $r_{0} \leq \frac{3 s}{2}$ no circular timelike geodesic exists (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (b)).

Facts & Assumptions

Given: The exterior datum with $s > 0$ , and the function $f = 1 - s / r$ , the potential $W_{ℓ} (r) = f (r) (c^{2} + ℓ^{2} / r^{2})$ , its half $V_{ℓ}$ , and the null potential $W_{ℓ}^{n} (r) = f (r) ℓ^{2} / r^{2}$ , with $ℓ \neq 0$ in the photon-sphere clause.

[L1]

For $r_{0} > \frac{3 s}{2}$ the equatorial circular timelike geodesic constructed in Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (a) has $ℓ^{2} = \frac{c^{2} s r_{0}^{2}}{2 r_{0} - 3 s}$ , $e = \frac{c (r_{0} - s) \sqrt{2}}{\sqrt{r_{0} (2 r_{0} - 3 s)}}$ , and its angular velocity satisfies $(\frac{d φ}{d t})^{2} = \frac{G_{N} m_{*}}{r_{0}^{3}}$ ; conversely every future timelike geodesic with $r \equiv r_{0}$ constant on an interval of positive length has $r_{0} > \frac{3 s}{2}$ , and if equatorial it has this $ℓ^{2}$ (that supplier's clause (b), The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation).

[L2]

Differentiating at fixed $ℓ$ and then imposing $ℓ^{2} = c^{2} s r^{2} / (2 r - 3 s)$ gives the circular stability value $V_{ℓ}^{''} (r) = \frac{c^{2} s (r - 3 s)}{r^{3} (2 r - 3 s)}$ for $r > \frac{3 s}{2}$ (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (c), Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model clause (c)).

[L3]

For $ℓ \neq 0$ , the null circular orbit at $r_{0} = \frac{3 s}{2}$ has $e_{0} = \frac{∣ ℓ ∣ \sqrt{f (r_{0})}}{r_{0}}$ , angular velocity $(\frac{d φ}{d x^{0}})^{2} = \frac{f (r_{0})}{r_{0}^{2}}$ , and the null potential satisfies $(W_{ℓ}^{n})^{'} (\frac{3 s}{2}) = 0$ , $(W_{ℓ}^{n})^{''} (\frac{3 s}{2}) = - \frac{32 ℓ^{2}}{81 s^{4}}$ (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (d)).

[L4]

For $s > 0$ the identities $f (3 s) = \frac{2}{3}$ , $f (\frac{3 s}{2}) = \frac{1}{3}$ , $(3 s)^{3} = 27 s^{3}$ , $(\frac{3 s}{2})^{2} = \frac{9 s^{2}}{4}$ , $G_{N} m_{*} = \frac{s c^{2}}{2}$ hold, and $6 s - 3 s = 3 s$ , $8 s - 3 s = 5 s$ ; all divisions below have nonzero denominators (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clause (a), 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).

Proof

technique · direct; substitute the two threshold radii into the conserved-quantity and potential formulas of the cited lemma, and evaluate with the exact identities $f(3s)=2/3$, $f(3s/2)=1/3$

1.1L1L2L4algebra

(ISCO data at $3 s$ .) By [L4] and [L1], at $r_{0} = 3 s$ the denominator is $2 r_{0} - 3 s = 3 s$ , so $ℓ^{2} = \frac{c^{2} s \cdot 9 s^{2}}{3 s} = 3 c^{2} s^{2}$ . Then $c^{2} + ℓ^{2} / r_{0}^{2} = c^{2} + \frac{3 c^{2} s^{2}}{9 s^{2}} = c^{2} + \frac{c^{2}}{3} = \frac{4}{3} c^{2}$ , so $e^{2} = f (3 s) \cdot \frac{4}{3} c^{2} = \frac{2}{3} \cdot \frac{4}{3} c^{2} = \frac{8}{9} c^{2}$ and $e = \frac{2 \sqrt{2}}{3} c > 0$ . The angular velocity is $(\frac{d φ}{d t})^{2} = \frac{G_{N} m_{*}}{(3 s)^{3}} = \frac{s c^{2} / 2}{27 s^{3}} = \frac{c^{2}}{54 s^{2}}$ . By [L2], $V_{ℓ}^{''} (3 s) = \frac{c^{2} s \cdot 0}{(3 s)^{3} \cdot 3 s} = 0$ . This is clause 1.

1.2L3L4algebra

(Photon-sphere data at $3 s / 2$ .) By [L4], $f (\frac{3 s}{2}) = \frac{1}{3}$ , so $e_{0} = \frac{∣ ℓ ∣ \sqrt{1 / 3}}{3 s / 2} = \frac{2 ∣ ℓ ∣}{3 \sqrt{3} s}$ ; the angular velocity satisfies $(\frac{d φ}{d x^{0}})^{2} = \frac{f (3 s / 2)}{(3 s / 2)^{2}} = \frac{1 / 3}{9 s^{2} / 4} = \frac{4}{27 s^{2}}$ , hence $(\frac{d φ}{d t})^{2} = \frac{c^{2} \cdot 4}{27 s^{2}}$ . The two potential identities are those of [L3] evaluated at the stated radius. This is clause 2.

1.3L1L2L4algebra

(Stable comparison radius $4 s$ .) At $r_{0} = 4 s$ the denominator is $2 r_{0} - 3 s = 5 s$ , so $ℓ^{2} = \frac{c^{2} s \cdot 16 s^{2}}{5 s} = \frac{16}{5} c^{2} s^{2}$ ; by [L2], $V_{ℓ}^{''} (4 s) = \frac{c^{2} s (4 s - 3 s)}{(4 s)^{3} \cdot 5 s} = \frac{c^{2} s^{2}}{64 s^{3} \cdot 5 s} = \frac{c^{2}}{320 s^{2}} > 0$ , a strict local minimum of $V_{ℓ}$ at the circular-orbit radius. Combined with clause 1, the sign of $V_{ℓ}^{''}$ changes at $r = 3 s$ , separating the strict local maxima on $\frac{3 s}{2} < r_{0} < 3 s$ from the strict local minima on $r_{0} > 3 s$ ; this is clause 3.

2.1L1L4algebra∎

(Threshold divergence.) For $r_{0} > \frac{3 s}{2}$ the denominator $2 r_{0} - 3 s$ is positive with $2 r_{0} - 3 s \to 0^{+}$ as $r_{0} ↓ \frac{3 s}{2}$ , while the numerator $c^{2} s r_{0}^{2} \to c^{2} s \cdot \frac{9 s^{2}}{4} > 0$ ; hence $ℓ^{2} \to + \infty$ . By the converse clause of [L1], no future timelike geodesic in any angular orientation has $r \equiv r_{0}$ constant on an interval of positive length for $s < r_{0} \leq \frac{3 s}{2}$ , since that would force $r_{0} > \frac{3 s}{2}$ . This is clause 4.

Remarks

What the example computes. The quantities $3 c^{2} s^{2}$ , $2 \sqrt{2} c / 3$ , $c^{2} / (54 s^{2})$ , $4 c^{2} / (27 s^{2})$ and $c^{2} / (320 s^{2})$ are exact evaluations of the cited formulas at the stated radii; the factor $2 \sqrt{2} / 3$ in $e / c$ is irrational. The longitude velocities and angular momenta refer to the constructed equatorial curves. The example reports no orbit of any observed body and no measurement, and the word "stable" refers only to the sign of the second derivative of the stated effective potential of the restricted model.

Explicit circular-orbit data of the Schwarzschild exterior at the ISCO radius 3s and the photon-sphere radius 3s/2

Example

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources