# Spacetime diagram of a collapse of a rotating star

There is a well-known "standard" spacetime diagram (Kruskal and Penrose) for the collapse of a spherically symmetric star to a Schwarzschild black hole (for example here, or here in EF), which stands in nice contrast to the idealised "eternal" Schwarzschild black hole/white hole spacetime.

I've been looking without success for an analogous spacetime diagram that shows a collapsing rotating star. In particular, I'm curious to see how the star decides which of the two Kerr singularities to fall into.

Does anyone have such a diagram or reference to one, or could produce a sketch?

## 2 Answers

It's not quite what you're looking for, but the article here from Physics World shows such a diagram for a charged Reissner-Nordström black hole. They note that Reissner-Nordström black holes were used to try to model the effects of incoming radiation being infinitely blueshifted at the inner (Cauchy) horizon, including the infinite blueshift of incoming gravitational waves which should cause a singularity at the Cauchy horizon. Apparently, physicists assumed that answer for a Reissner-Nordström black hole would serve as a reasonable approximation for what would happen in the case of a rotating Kerr black hole:

This picture of an unstable Cauchy horizon, which becomes singular under exposure to radiative tails, was established through numerical integrations by Penrose and Michael Simpson at Birkbeck in 1973, and by several analytical studies in the following decade. However, this earlier work assumed a fixed (Reissner-Nordström) background, and did not attempt to estimate the effect of the blue-shifted radiation on the gravitational field.

The first attempts to take this back-reaction into account were made by William Hiscock of the University of Montana in 1981. More representative models were examined by Eric Poisson and one of the authors (WI) at the University of Alberta in 1999, and by Amos Ori, then at CalTech, in 1991. To simplify the mathematics for a first reconnaissance, these early models made several approximations. It was assumed that rotating (non-spherical) black holes could be adequately modeled by spherical charged holes, since their horizon structures are similar. Gravitational (quadrupole) wave tails were modeled by spherical scalar waves. More general studies by Ori, by Patrick Brady, Chris Chambers and John Smith at the University of Newcastle-upon-Tyne in the UK, and by our group at the University of Alberta, confirm that these simplified models capture most of the essential features of the generic case.

So with that in mind, here's the diagram:

Here the red line is the outer event horizon, the blue line is the inner Cauchy horizon. Also note that it's possible for an infalling observer to cross a black line before crossing the Cauchy horizon--some recent work suggests there should actually be an additional singularity at this black line, created by the infinite blueshift of "backscattered" radiation, see this paper (the paper also seems to indicate a different Penrose diagram for the interior, see Fig. 1) along with this section of the book The Science of Interstellar by physicist Kip Thorne which has a layman's-level discussion of this "outflying" singularity. Thorne also indicates there that the central singularity of a realistic rotating black hole is expected to be a BKL singularity which as mentioned here would be space-like (horizontal on a Penrose diagram), rather than the time-like singularity seen in the image above (not sure whether the expectation of a BKL singularity is based on any exact solution or if it's based on approximations though). Getting back to the image above, the article says:

The final state of a spherical, charged collapse is known as a "Reissner-Nordström" black hole. The contracting star passes in turn through the outer (event) and inner horizons. At this point, gravity at its surface actually becomes repulsive because of the large amount of electric field energy that is now located outside the shrunken star and no longer contributing to its effective internal mass. At event B the star rebounces at a non-zero radius and reexpands into a new infinite universe. It will continue to pulsate, each time entering a new universe.

The prospect of travel to other universes via a black hole sounds like science fiction, and before we get carried away we should check this picture for stability. If cosmic censorship holds, inclusion of spin and other non-sphericities should not prevent an event horizon from forming and the external field from converging asymptotically to a stationary condition. These stationary states are uniquely the three-parameter Kerr-Newman geometries, and they have a lattice-like structure entirely similar to figure 3.

Since the energy density goes to infinity at the Cauchy horizon in GR, due to the infinite blueshift there, any of GR's predictions about the regions beyond there could well be completely incorrect in quantum gravity (leaving aside questions about whether GR's predictions about anything beyond the outer horizon might be wrong if speculations like firewalls or fuzzballs pan out).

There is no exact solution of Einstein's equation smoothly modeling the metric of a rotating star, so a diagram like this can only be a heuristic.

• A qualitative picture would be useful, too. Is that possible, or are you saying that without a model you can't even know approximately what the spacetime looks like? Commented Dec 6, 2014 at 13:19