# What is a photon sphere

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

A photon sphere is a spherical surface round a non-rotating black hole (or other extremely compact spherically symmetric body) containing all the possible closed orbits of a photon.

All such orbits are circular and unstable.

The radius of the photon sphere is 3M, where M=Gm/c2 is the mass-equivalent radius of the body, and m is its mass.

Obviously, there is no photon sphere if the radius of the body is greater than 3M.

For a rotating black hole, there is an outer radius at which the only photon orbit is equatorial circular and retrograde (counter-rotating), and an inner radius at which the only orbit is equatorial circular and prograde (co-rotating).

Bound orbits for a photon exist on each sphere between these two extreme radii. Each such orbit has the approximate shape of a circle which precesses round the sphere between two fixed "latitudes" and with a characteristic angular momentum. This angular momentum increases (becomes more prograde) with decreasing radius, while the fixed "latitude" increases to polar and then decreases again.

Equations

NON-ROTATING BLACK HOLE (Schwarzschild coordinates):

radius of event horizon: $2M$

radius of photon sphere: $3M$

line element:
$$ds^2\ =\ -\frac{dr^2}{\left(1 - \frac{2M}{r}\right)}\ -\ r^2d\theta^2\ -\ r^2 sin^2\theta d\phi^2\ +\ \left(1 - \frac{2M}{r}\right)dt^2$$

free-fall equations for mass m, energy parameter E and angular momentum parameter L:

$$\frac{dt}{d\tau}\ =\ E/\left(1 - \frac{2M}{r}\right)$$
$$\frac{d\phi}{d\tau}\ =\ L/r^2$$
$$\frac{dr}{d\tau}\ =\ \pm\sqrt{E^2\ -\ \left(1 - \frac{2M}{r}\right)\left(m^2\ +\ \frac{L^2}{r^2}\right)}$$

ROTATING BLACK HOLE (with angular momentum $aM$):

radii of event horizons: $$R_\pm\ =\ M\ \pm\ \sqrt{M^2\ -\ a^2}$$

$$r_-\ =\ 2M\left(1\ +\ cos\frac{2}{3}cos^{-1}\frac{a}{M}\right)$$

$$r_+\ =\ 2M\left(1\ +\ cos\frac{2}{3}cos^{-1}\frac{-a}{M}\right)\ =\ 2M\left(1\ +\ cos\left(120^o\ -\ \frac{2}{3}cos^{-1}\frac{a}{M}\right)\right)$$

Extended explanation

The maths (outline):

A body free-falling near a non-rotating black hole follows a trajectory with three constant parameters, m E and L, which may be thought of as its mass energy and angular momentum.

For a photon, m is zero.

The usual Schwarzschild coordinates, are related to the "age", $\tau$, of a photon (measured as number of wavelengths, since of course the "proper time" of a photon does not change) by the equations:
$$\frac{dt}{d\tau}\ =\ E/(1\ -\ 2M/r)$$
$$\frac{d\phi}{d\tau}\ =\ L/r^2$$
$$\frac{dr}{d\tau}\ =\ \pm E\sqrt{1\ -\ (1\ -\ 2M/r)L^2/E^2r^2}$$

when L/E = 3√3M, the last equation is:
$$\frac{dr}{d\tau}\ =\ \pm E\sqrt{1\ +\ 6M/r}(1\ -\ 3M/r)$$
from which obviously one solution is the circular motion:
$$r\ =\ 3M\text{ and }d\phi /dt\ =\ 1/3\sqrt{3}M$$

Accordingly, a photon with L/E = 3√3M can orbit on the photon sphere ($r\ =\ 3M$) with period $6\pi\sqrt{3}M$, or can approach the photon sphere, circling ever closer either just outside or just inside it with approximately the same period, but never quite reaching it.

Lens and mirror effects:

Similarly, a photon with L/E slightly greater than 3√3M may circle the photon sphere a number of times before returning to distant space.

So a black hole can act as a lens giving rise to n ring-shaped images of a background star, each ring corresponding to light which has circled 1,2,3,..n times around, for some positive integer n (which depends on the distance beyond the black hole).

And it can act as a mirror giving rise to n ring-shaped images of a foreground star, in the same way.

These effects are too faint to be observed, but the "zeroth ring", in which light is focussed without circling the black hole at all, has been observed, and is known as gravitational lensing

By comparison, a massive particle can orbit at any distance greater than or equal to the marginally stable radius, $6M$

Rotating black hole:

For details, see "Spherical photon orbits around a Kerr black hole" by Edward Teo at http://www.physics.nus.edu.sg/~phyteoe/kerr/paper.pdf

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