Your question basically boils down to a recognition of the following fact:
The Schwarzschild metric, with spacelike $r=0$, admits an "eternal" BH to form by stellar collapse, like the one you've drawn above. It forms, then maintains a permanent static state forever, but doesn't evaporate or absorb anything else.
BH metrics with a timelike $r=0$, like Kerr (rotating), Reissner-Nordstrom (charged), and Hayward (nonsingular) do not admit a reasonable "eternal" solution, since the region beyond the inner horizon leads to strange and unwanted parts of spacetime.
This means that for rotating BHs (as well as charged and nonsingular BHs), if we want a full Penrose diagram, we need to confront a difficult question:
- What happens to the BH after if forms?
Most people agree that the BH evaporates by emitting Hawking radiation, but no one agrees on the correct semi-classical spacetime to model this process (google "evaporating black hole spacetime", e.g.). Moreover, not everyone even agrees that Hawking evaporation is the dominant process (popular alternatives include, e.g., remnants, quantum bounce models, mass inflation instability).
The good news about this is that we should have had to confront this anyway: if BHs evaporate the eternal diagram you drew above is wrong (at the top) anyway.
The bad news is that we can't say what the correct diagram is, we can only postulate some ideas about what a reasonable diagram might be.
Here is one toy-model possibility, assuming that the BH forms from a collapsing star, then evaporates by emitting an outgoing burst of Hawking radiation from just outside the trapping horizon, while absorbing a burst of negative mass. An evaporation model with ingoing/outgoing fluxes of negative/positive energy like this is motivated by the DFU stress tensor for Hawking radiation.
If this was for a charged nonrotating BH, which has a similar causal structure, this would be pretty satisfying. Then we could say:
- The collapsing star surface cuts off unwanted past regions.
- The evaporation process cuts off unwanted future regions.
Unfortunately, for the rotating BH, this only represents the $\theta=0$ axis, so there are problems:
- Not clear what other $\theta$ values look like in diagram.
- By going to other $\theta$ values, you can still go through the ring singularity and get to unwanted regions.
It is not obvious how these issues associated with rotation should be resolved, or even that the interior metric has to be exactly Kerr, since:
- We don't have a simple exact metric for a rotating star collapsing to a BH. Only numerical studies.
- The metric is only asymptotically Kerr around a rotating body, so the metric near $r=0$ might not be known.
- See this Kerr Spacetime Introduction and the Living Reviews in Relativity for discussion of these issues.
What's more, regardless of whether this BH is spinning or charged, it has some weird bad properties:
- There's a naked singularity.
- You can fall in past all the horizons and escape without anything terrible happening.
That certainly doesn't seem right.
One way to try to resolve all of these issues at once is by assuming that instead of a singularity, BHs have an extremely tiny, extremely dense core. The assumption is that classical GR holds until densities and curvatures reach the Planck scale, at which point Quantum Gravity takes over the dynamics. It might sound like this violates the singularity theorems, but it doesn't: the energy conditions required for singularity theorems to hold are already violated by the Hawking radiation, and definitely can't be assumed to hold a priori in quantum gravity. This point of view is not widely accepted, however, although personally I think it should be.
That assumption results in the theory of nonsingular (or "regular") BHs, see e.g. nonrotating and rotating cases. The rotating variety still have some technical issues, but if some good rotating nonsingular metric does exist, then the diagram would:
- Look basically like the diagram above, except that near $r=0$ in the Kerr metric would not be a vacuum, but would rather be an extremely dense core of matter, extending into the region between $r_{\pm}$.
- The ring singularity is replaced by a dense matter blob (if rotating fast probably a pancake shape), no more issue of going through the ring. No more unwanted regions.
- Anyone who falls through outer horizon falls into core and gets turned into quantum gravity soup before being emitted in the Hawking radiation.
- No more singularity = no more naked singularity.
Like I said, this is conjecture, since the right metric for this hasn't been discovered, as far as I know.
So that's my point of view of what this diagram probably should look like, but, as I've pointed out, there may be many others. My assumptions were that Hawking evaporation dominates the late time dynamics, and that the ingoing/outgoing radiation is a reasonable semiclassical model for the evaporation process.
If anyone has an alternative reasonably self-consistent diagram for this process, I think it would be very interesting to compare. Trying to sort out these "astrophysically relevant" BH scenarios seems like a good way to weed out some of the BH nonsense...
