Anatomy of a black hole: which are its layers?

Question

I'm trying to find out how a Kerr-type black hole is structured (that is, a black hole with non-zero angular momentum but no net electrical charge). According to the information I have found, the order in which its layers are located would be this, from outside to inside:

1. The accretion disk
2. The innermost stable circular orbit
3. The photon sphere
4. The outer event horizon
5. The inner event horizon or Cauchy horizon
6. The singularity

Is this scheme correct or is something missing? Where would then the ergosphere be placed in the list? Are the innermost stable circular orbit and the photon sphere at the same distance from the centre of the black hole?

Answer 1

I'm not sure calling the things on your list "layers" is the most accurate way of putting it. It'd rather say features. Your list is mostly complete. You could also add the innermost bound circular orbit (i.e. the unstable circular orbit with sufficient energy to reach infinity if perturbed), and as you say the ergosphere. (And I'm sure people can come up with more.)

These features are found at the following (Boyer-Lindquist) radii (restricting to the equator), using units where $c=G=M=1$.

1. Singularity $r=0$
2. Inner horizon $r=1-\sqrt{1-a^2}$
3. Outer horizon $r=1+\sqrt{1-a^2}$
4. Ergosphere $r=2$
5. Light ring $r= 2+\cos\frac{2\arccos[a]}{3}-\sqrt{3} \sin\frac{2\arcsin[a]}{3}$
6. IBCO $r = 2(1+\sqrt{1-a^2}) - a$
7. ISCO $r = \text{nasty looking cubic root}$

For convenience a plot: (negative spins correspond to retrograde orbits, for orbit related quantities)

As you can see, the special radii mostly appear in the mention order. The exception being the ergo sphere, which happens at a set radius (in Boyer-Lindquist coordinates). For low spins the ergosphere is between the outer horizon and the light ring, while for high spins all other special radii lie inside the ergosphere.