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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?

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  • $\begingroup$ The singularity isn't at the center. The singularity is a spacelike surface that is in the future for all observers. $\endgroup$ – user4552 Jan 19 at 18:25
  • $\begingroup$ @BenCrowell The Kerr singularity is timelike. $\endgroup$ – Javier Jan 19 at 19:19
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    $\begingroup$ @Javier: Interesting, I didn't know that. I don't quite see how that can be reconciled with the fact that Schwarzschild is a special case of Kerr. Maybe I should ask a separate question about that, unless you have a quick answer...? $\endgroup$ – user4552 Jan 19 at 20:59
  • $\begingroup$ @BenCrowell that would certainly be a good question to ask (and you might as well ask about the Reissner-Nordström metric, which is simpler), though if I had to come up with something, I'd say the process of constructing a Penrose diagram is not necessarily continuous, in some sense. The singularity is after all not part of the original spacetime. $\endgroup$ – Javier Jan 19 at 23:50
  • $\begingroup$ Wikipedia says "The ergosphere is a region located outside a rotating black hole's outer event horizon." $\endgroup$ – Chappo Hasn't Forgotten Monica Jan 19 at 23:50
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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 ergo sphere. (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) enter image description here

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.

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  • $\begingroup$ Very enlightening, thank you. $\endgroup$ – Quaerendo Jan 20 at 9:32

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