Kerr disputes singularities in Kerr Black Holes

[.Scott]

In September 1963, Roy Kerr described the geometry of uncharged rotating black holes (ie, Kerr Black Holes). This Kerr geometry included (or was presumed to include) a feature called a ring singularity.

But after apparently staring at the blackboard for 60 years, Roy just got up and drew a large and emphatic question mark over that ring singularity thing.

In a 20 page article published on ResearchGate, Roy states:

The consensus view for sixty years has been that all black holes have singularities. There is no direct proof of this, only the papers by Penrose outlining a proof that all Einstein spaces containing a ”trapped surface” automatically contain FALL’s [finite affine length light]. This is almost certainly true, even if the proof is marginal. It was then decreed, without proof, that these must end in actual points where the metric is singular in some unspecified way. Nobody has constructed any reason, let alone proof for this. The singularity believers need to show why it is true, not just quote the Penrose assumption.

So, Roy will let you slide on that FALL thing, but if you want to claim a singularity it's time to show the proof.
Just to be clear, I have not made any serious attempt to follow the math in Kerr's new paper.
I have read the "Cliff's Notes" equivalent (am I dating myself?) here: "Big Think" article.
But I need to provide some cautions about that "Big Think" article: It provides much lighter reading, but it also includes material that goes beyond Kerr's paper. For example, the Big Think article includes this revised Penrose Diagram:

This diagram is not described or referenced in Kerr's article. Nor does Kerr's article mention worm holes.

The first thing I noticed with that diagram is the missing "You are Here" tag. And If you ever wondered how to visit the interior of a Black Hole and live to tell about it, then place that "you are here" tag in the "New Universe". Then you can wonder what will become of all this "new matter" coming into your "New Universe" once Hawking Radiation evaporates the Black Hole in the "Universe" that it was borrowed from.

 

[PeterDonis]

Roy will let you slide on that FALL thing, but if you want to claim a singularity it's time to show the proof.

The problem is that the term "singularity" is defined in the GR literature as "that FALL thing". (More precisely, as the presence of inextendible geodesics, whether or not they are lightlike.) It is not defined as "the metric becomes singular". Most GR textbooks have at least some discussion of this; Wald has a fairly detailed one. And of course the full gory details are discussed in Hawking & Ellis.

In other words, the claim Kerr claims to be rebutting, as far as I can tell, is a claim that nobody in the GR community actually makes. Kerr is correct that the presence of inextendible geodesics does not, by itself, imply that curvature invariants must increase without bound as the point of inextendibility is approached. Nor, by itself, does the presence of a trapped surface. But nobody in the GR community, as far as I know, claims that it does.

What people in the GR community do say is that, in particular cases where we have actually done the work to prove it, curvature invariants do increase without bound as points of inextendibility along geodesics are approached. For example, the $r = 0$ spacelike singularity in Schwarzschild spacetime, the $r = 0$ timelike singularity in Reissner-Nordstrom spacetime, and the $r = 0$ ring singularity in Kerr spacetime. Nobody says curvature invariants increase without bound at those singularities because of general singularity theorems by Penrose or anyone else. They say it because we can do the math explicitly for those solutions and show explicitly that the curvature invariants diverge.

Of course all of those solutions are highly idealized, and the R-N and Kerr deep interiors, at least, are considered to be unphysical because of the properties of their inner horizons; AFAIK the current belief is that in a more realistic model, those inner horizons (and everything inside them, including the singularities) are no longer there, but are replaced by something that looks more like the Schwarzschild interior, with its spacelike singularity. In numerical models of an actual gravitational collapse of a star to a black hole, I believe that is what happens. (The only closed-form models of such a collapse that I am aware of are for the uncharged, non-rotating case, and include the original 1939 Oppenheimer-Snyder model and more recent non-rotating models that, unlike O-S, are stable against small perturbations, such as the BKL singularity.) And as the spacelike singularities in these models are approached, once again, curvature invariants do increase without bound, so again any claims to that effect are not based on general theorems but on explicit computations for those particular cases.

And finally, there are models like the Bardeen black hole (which has been discussed in previous PF threads), in which the interior is de Sitter or something similar, and which, while they can look from the outside like a true black hole for a very long time (on the order of the Hawking evaporation time, i.e., $10^{67}$ years for a one solar mass hole), are not true black holes because they have no true event horizons. They do have trapped surfaces, but in their interiors they violate the energy conditions that are required to prove the singularity theorems, so they have neither any inextendible geodesics, nor any place where curvature invariants increase without bound. (If I had to put my money on one possibility at the classical level that I think will end up being close to the truth, these models are where I would put it.) Models like this completely change the game as far as accounting for our actual observations is concerned, but it doesn't look like Kerr is even aware of them.

 

[PeterDonis]

nobody in the GR community, as far as I know, claims that it does.

I do see that Kerr references claims by Penrose and Hawking. But the reference is to a 1972 paper, even before the first edition of Hawking & Ellis was published, and certainly before much of the discussion in the GR community about singularities. (In fact all but two of Kerr's references are to papers published in 1972 or earlier; the two exceptions are the 2009 updated edition of Hawking & Ellis and a 2009 book on the Kerr spacetime.) Later discussions I think are much more nuanced and much more aware of all the complexities involved.

 

[PeterDonis]

This Kerr geometry included (or was presumed to include) a feature called a ring singularity.

The fact that the maximal extension of the Kerr geometry includes a ring singularity, and that curvature invariants diverge there, is not a "presumption"; it's a mathematical fact.

In the paper Kerr talks about work that has been done to construct non-singular deep interiors (i.e., inside the inner horizon) that match to Kerr along their boundary, which would be at $r > 0$. But these would still not be viable physically, because of the issues with the inner horizon that I mentioned in an earlier post. Any physically viable matching would have to be of the other type Kerr discusses, with the boundary between the Kerr vacuum region and the object with its interior being outside the outer (event) horizon. Kerr does not appear to be claiming that such a matching would be exact (and AFAIK no such exact matching solution is known), but that it could be a very good approximation, which I would say is likely to be the case, but is a separate question from the singularity issue.

 

[PeterDonis]

the claim Kerr claims to be rebutting, as far as I can tell, is a claim that nobody in the GR community actually makes.

And in fact, at least one counterclaim Kerr makes (at least he appears to think it's a counterclaim), namely this one...

The author has no doubt, and never did, that when Relativity and Quantum Mechanics are melded it
will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong!

...is a claim that every GR physicist I'm aware of would agree with.

 

[.Scott]

Kerr:

The author has no doubt, and never did, that when Relativity and Quantum Mechanics are melded it
will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong!

...is a claim that every GR physicist I'm aware of would agree with.

I'm certainly no GR physicist, but I wonder how well even the event horizon will survive once that meld is complete.

 

[PeterDonis]

I'm certainly no GR physicist, but I wonder how well even the event horizon will survive once that meld is complete.

If a model like the Bardeen black hole turns out to be the right one, it has no event horizon. As I said in my previous post, as a matter of personal opinion, if I had to put my money somewhere, that would be where I would put it.

As for the "meld", the Bardeen black hole model is classical on its face, but the assumption it makes for the deep interior--that the geometry is de Sitter or something close to it--implicitly relies on the belief that we will end up finding that the behavior of quantum fields under such conditions looks like dark energy or something close to it--i.e., that the effective equation of state becomes very different from anything like ordinary matter because of quantum effects.

 

[PeterDonis]

This diagram is not described or referenced in Kerr's article.

Yes. What's more, Kerr dismisses maximal analytic extensions in the paper as not being useful. I think he is mistaken, and the mistake is relevant to at least one thing he discusses in the paper.

One very useful aspect of Penrose diagrams like the one you posted, which are derived from the maximal analytic extensions of various spacetimes, is not that they are generally physically realizable (for various reasons, some of which I have stated in previous posts and some of which Kerr discusses in the paper, they're not considered to be), but that they give a valuable framework for constructing models of gravitational collapse to a black hole that are physically realizable, by showing what constraints any such models have to meet.

The reason for this is that any realizable model of gravitational collapse to a black hole must have a vacuum region that is the same as some portion of the Penrose diagram of the appropriate maximal analytic extension. In other words, we must be able to draw a timelike line on the diagram that marks the boundary of the non-vacuum region (it must be timelike because the surface of the matter that is collapsing must collapse on timelike worldlines), and that line must cross the event horizon at some finite point (otherwise the entire black hole region is eliminated and we just have a model describing an ordinary object).

For example, for the 1939 Oppenheimer-Snyder model of the collapse of dust (zero pressure matter) to a Schwarzschild black hole, we would draw a boundary on the Penrose diagram of the maximal extension of the Schwarzschild geometry that would look similar to the one drawn on a Kruskal diagram in this Insights article:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-4/

Now Kerr, in the paper, discusses models of collapse to a Kerr black hole in which the deep interior does not contain any ring singularity at all, but instead contains a stable object with a surface at some radius $r > 0$. But if you try to draw the boundary of the non-vacuum region for such a collapse on the Penrose diagram you posted in the OP of this thread, you will find that it is not possible. Consider: any such timelike line will have to, first, cross the outer horizon (the event horizon) at some finite point, as noted above (otherwise we don't have a black hole at all); and second, it will have to cross the inner horizon at some finite point (because all timelike lines that cross the outer horizon must in turn cross the inner horizon; there are no timelike paths that don't do that).

But if you look at the diagram, any timelike line that crosses the inner horizon (either one--it doesn't matter whether you pick the one marked "parallel" or the other one) cannot possibly eliminate the entire ring singularity at $r = 0$ from the vacuum region. If you go up and to the left, so to speak, hoping to cross the left inner horizon as low down as possible, you can eliminate most of the left $r = 0$ line, but not all of it--but you will leave all of the right $r = 0$ line in the vacuum region (which, remember, will be the region to the right, in the diagram, of the timelike line marking the boundary of the collapsing matter). Obviously that's no good. It's better to go up and to the right, so to speak, keeping the entire left $r = 0$ line in the matter region (which means not there, because the matter region "overwrites" it)--but you still can't reach the right $r = 0$ line until some finite distance up it from the vertex where it meets the inner horizon. So some portion of the ring singularity will still be there.