# 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 , Charge , and Angular Momentum . 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 , and . 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 (). Unlike the Schwarzchild solution however, the rotating black hole has a non-zero angular momentum . 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 . 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.