# Photon Spheres and Rotating Black Holes

This is another post on my current theme of black holes. Recently, I wrote a blog post about light orbiting a Schwarzchild black hole. These so called Photon spheres are sufficiently interesting to wonder whether they exist for more complicated black holes. They do, and we’re going to find out about them. This post is based on a fantastic paper [1] I read the other week. If you find this stuff interesting, and you’ve got an introductory level of GR you should definitely check out the paper here.

One of the coolest thing about black holes is you can describe any black hole with just three numbers: Mass $(M)$, Charge $(Q)$, and Angular Momentum $(J)$. Of course, if we’re being completely general we also need to describe where the black hole is and how fast it’s moving through space – but we can always perform a Lorentz transformation so that we’re in the rest-frame of the black hole and so we really only care about $M$, $Q$ and $J$. The fact that you can describe black holes with only three numbers is one of the main reasons I find them so interesting. Consider for example, two black holes, A and B. Black hole A is made entirely from matter – protons, neutrons, electrons and the like. On the other hand, black hole B is made entirely from anti-matter – anti-protons, anti-neutrons, anti-electrons and the like. Further, suppose A and B have the same charge, angular momentum and mass. How can we tell A from B? That is, what experiment can we do to tell the difference between the matter black hole or the anti-matter black hole? The answer is, we can’t tell the difference. For all intents and purposes, there is no way of telling whether a black hole is made of matter or anti-matter. Indeed, a third black hole C, with the same mass, charge and angular momentum, but made entirely from light, would be similarly indistinguishable from the first two.

As promised in the title, we’re going to be looking at rotating black holes. A rotating black hole is, like the Schwarzchild black hole, uncharged ($Q=0$). Unlike the Schwarzchild solution however, the rotating black hole has a non-zero angular momentum $J$. Rotating black holes are also called Kerr black holes, after Roy Kerr who was the first to write down an exact solution for one. We will consider an uncharged black hole which is rotating with constant angular momentum. For convenience, we define the angular momentum per unit mass $a = \frac{J}{M}$. Since the black hole is rotating, we lose the spherical symmetry we have with the Schwarzchild metric, so we expect the solution to be a little more complicated. We do, however, still have axial-symmetry, i.e. rotational symmetry around the axis of rotation.

It turns out that neither spherical polar coordinates nor the usual cartesian coordinates are particularly suited for describing a rotating black hole, so we’ll be using Boyer-Lindquist coordinates $(t, \, r, \, \theta , \, \phi)$. These are related to the usual cartesian coordinates by

$x = \sqrt{r^2 + a^2} \sin \theta \cos \phi$

$y = \sqrt{r^2 + a^2} \sin \theta \sin \phi$

$z = r \cos \theta$

In these coordinates, the metric takes the following form:

$ds^2 = -\left(1-\frac{2Mr}{\Sigma}\right)dt^2 - 2\left(\frac{2Mr}{\Sigma}\right)a \sin^2 \theta \, dt \, d\phi + \Sigma\left(\frac{dr^2}{\Delta} + d\theta^2\right) + \frac{\mathcal{A}}{\Sigma}\sin^2\theta \, d\phi^2$

where

$\Delta = r^2 - 2Mr + a^2$

$\Sigma = r^2 + a^2 \cos^2 \theta$

$\mathcal{A} = (r^2 + a^2)^2 - \Delta a^2 \sin^2 \theta$

As expected, this is more complicated than the Schwarzchild solution, but that’s ok because it’s also more interesting than the Schwarzchild solution!

Now’s probably as good a time as any to mention that I’m using natural units here, so $c = G = 1$, because the Kerr metric is even more complicated if you are using SI units. Because I want to focus on photon orbits, I’m not going to derive some of the cool consequences of having a rotating black hole, but I can’t help but mention a few things.

One of the peculiar predictions general relativity makes about rotating objects is something called frame-dragging, or the Lense-Thirring effect. Frame-dragging predicts that objects moving close to the black hole will be forced to rotate with the black hole, not because of an applied force or torque, but because of the nature of the spacetime around the black hole. Indeed, Kerr black holes have a region, called the ergosphere, where it is impossible to stand still – the spacetime around in the ergosphere is ‘pulled along’ by the black hole faster than the local speed of light. Frame dragging can be seen as the gravitational analogue of electromagnetic induction (leading to the study of Gravitomagnetism).

Because the ergosphere is outside the event horizon, it is possible to enter the ergosphere, gain energy from the rotational motion of the black hole, then exit the ergosphere, carrying that energy away from the black hole. This process of removing energy from the black hole is known as the Penrose process. It should be noted that as energy is taken away from the black hole in this way, the angular momentum of the black hole decreases and the ergosphere gets smaller. As the angular momentum decreases to zero, the ergosphere disappears as the black hole becomes a Schwarzchild black hole.

Another cool thing about rotating black holes is the existence of a ring singularity (ringularity). In a Schwarzchild black hole the center of the black hole is a point of infinite density. For a rotating black hole, it is no longer a point, but a circle of infinite density. Although we won’t go into it here, ring singularities have been proposed as candidates for traversable wormholes, and are used as toy models for studying the physics of wormholes.

Going back to photon spheres, we could, in principle, do a computation similar to how we calculated the radius of the photon sphere of a Schwarzchild black hole. I was going to go through the actual calculation but the Christoffel symbols turn out to be rather nasty, and so I will instead just say that, in the equatorial plane, there are two photon spheres. The radii of these spheres are

$r_1 = 2M\left[1 + \cos \left( \frac{2}{3} \cos^{-1} \left( - \frac{|a|}{M} \right) \right) \right]$

$r_2 = 2M\left[1 + \cos \left( \frac{2}{3} \cos^{-1} \left( \frac{|a|}{M} \right) \right) \right]$

and we have that $M \leq r_1 \leq 3M \leq r_2 \leq 4M$, recalling that (in natural units), the radius of the photon sphere of a Schwarzchild black hole is $3M$.

Why are there two photon spheres? It has to do with the rotation of the black hole, and whether the light is moving with the rotation, or against it. Due to the frame dragging of the black hole, a photon moving with the rotation of the black hole (prograde motion) orbits at a lower radius than a photon moving against the rotation (retrograde motion). Frame-dragging gives weird results for regular objects orbiting a rotating black hole as well as photons – an object orbiting the black hole in an equatorial orbit weighs more when orbiting with the rotation than when it is orbiting against the rotation.

Two photon spheres is pretty cool, but it gets way cooler. So far, the photon orbits have been in the equatorial plane – that is, staying directly above the equator the whole orbit. Can we have an orbit that’s not confined to the equatorial plane? For instance, could we have a photon orbit starting at the equator but moving south?

Two lattitudinal oscillations of a photon orbit with zero angular momentum (From [1], used with permission of author)

The answer to this is a resoundingly awesome yes! In [1], Edward Teo investigates spherical photon orbits which are not confined to a plane – they wibble and they wobble all over the place, depending on their angular momentum, or equivalently, their initial position and direction.

Teo finds a one-parameter family of solutions characterized by the parameter $\Phi = \frac{L_z}{E}$, where $L_z$ and $E$ are the constants of motion corresponding to angular momentum around the $\phi$-axis and the energy respectively, and numerically computes examples of orbits. Even for the simple case of zero angular momentum, as shown on the right, the orbit is incredibly interesting.

When $\Phi \not= 0$, the orbits attain a maximum altitude, and are confined within a band around the equator. This is shown very nicely in the diagram below. It should be noted that these computations are done for an extremal black hole, i.e. a rotating black hole which has the maximum allowed angular momentum (corresponding to $a = M$).

49 oscillations of a photon orbit with $\Phi = -6M$ (From [1], used with permission of author)

To see a complete table of example orbits visit here.

When I was reading about photon spheres for Schwarzchild black hole, I often wondered exactly what you would see if you stood with your head in the photon sphere. It’s clear that light emitted from the back of your head could orbit the black hole and enter your eye, but what else would you see? It turns out that some other people have thought of this, and there is a fantastic site which has videos and gifs which visualise exactly this. The visualisations even include a photon sphere around a neutron star.

Since I’ve been thinking about rotating black holes recently, it seems natural to ask the same question for rotating black holes or neutron stars. In this case, there isn’t a single photon sphere, rather, there is a range of radii over which photon orbits exist. If we are somewhere in this region, what would we see? Would you be able to see clearly, or would the image be jumbled because the orbits of light are non-planar? How does your height (i.e. altitude) affect what you see? How about your latitude and longitude? How does the direction you are facing affect the image? I don’t know the answer to any of these questions, so if anybody does know I’d love find out.

My next post is going to be about extremal black holes and the cosmic censorship hypothesis, so stay tuned.

Reference: [1] – E. Teo, Spherical photon orbits around a Kerr black hole, http://www.physics.nus.edu.sg/~phyteoe/kerr/paper.pdf