Roy Kerr vs. the singularities

This preprint by Roy Kerr should be a hit (but I bet it will be ignored!) .

Kerr (yes, who found the well-known Kerr black hole solutions) disagrees with Penrose's singularity theorem and its variations. Namely these theorems prove the existence of geodesics that can't be extended beyond a finite affine length, but Kerr finds numerous examples of inextensible light rays that don't contain singularities. These geodesics go all the way to the null infinity, and yet the affine parameter remains finite. And there are such light rays through every point of the Kerr spacetime. Only some geodesics hit the ring singularity, but this region can be replaced by a nonsingular one, perhaps matter can do this. Kerr thinks that his perfectly symmetric vacuum solution doesn't happen in reality (despite the "no-hair theorem", which is in fact a conjecture improperly called "theorem", stating that all black holes evolve into a Kerr solution), even though he thinks that black holes exist.

Now, how is this possible? I mean the singularity theorems, now sealed forever by a Nobel prize, prove that there are conditions that necessarily lead to singularities. That if there's a black hole, there must be a singularity beyond its horizon. Or do they?

This is a bit of a word play. There are more meanings of the word "singularity". Normally singularity means a place where the metric blows up. Or its inverse. Or the curvature, or any field that we think it's physical. But then we can think of excluding these points from spacetime. If these points are "in the way" of the physical fields, if the evolution equations can't go beyond such a place but they should, this would be a problem even if we exclude them from spacetime. But if these singularities are somewhere at the "edge" of spacetime, and the spacetime admits a nice foliation so that the evolution equations work fine across the entire spacetime, why would this be a problem? And yet, the other definition of singularity, the one that is actually the object of the singularity theorems, includes such cases as well. That is, as a diagnostic method, it gives numerous false positives.

Here's what happened. And I don't say it's a plot against General Relativity, rather an accident, perhaps welcomed by many. If your spacetime contains singularities, we can think of excluding them from spacetime. But this, as I said, doesn't solve the problem. So maybe there is a way to detect this pathology even with the singularities removed, and talk about such a spacetime as being singular anyway. And here comes into play the redefinition of singular spacetime in terms of geodesic incompleteness. And it is said in the Hawking & Ellis bible, on page 258: 

I don't want to single out this great book, it explains well the adoption of this diagnosis, and others said similar things. But here I think lies the problem. Because this definition can be misunderstood (unintentionally I think) in a way that makes the singularity theorems seem about singularities even if there are no singularities in the interior of spacetime, even if the spacetime can be nicely foliated, offering a nice home to the evolution equations.

The singularity theorems prove (and they indeed prove this) that there are incomplete geodesics, where incomplete means they can't be extended beyond a finite affine length. Whether all of them deserve to be called "incomplete" is also questionable. If the affine length (which is not the same as geometric length anyway) of a timelike or null geodesic is finite, but it goes to the "real edge" of spacetime, as in Kerr's paper, why should it be called incomplete? This already seeds in our minds the idea that there's something wrong with them.

So, one on top of another, the meaning of words shifted so that now it's widely believed that General Relativity breaks down, due to the singularities. And Kerr gives nice rich counterexamples, all in the same spacetime of a Kerr black hole. I mean, his spacetime has a singularity, but the singularity theorem doesn't even predict that singularity. It predicts some singularities, but they are false positives, they don't occur on the geodesics up to the boundary of spacetime. It doesn't predict the ring singularity, because, as Kerr says, there is no trapped surface inside the inner horizon of the Kerr black hole. So, if we cut out the spacetime around that ring, and replace that region (and the "other universe" beyond the ring) with one without singularities, we get a spacetime without singularities (and from what we know matter may do this), and yet the singularity theorems as usually cited say it has singularities (outside that region)! 

I'd like to add that I was convinced as well, for a long time, that the singularity theorems imply the kind of metric singularities that are problematic. They were the reason why I worked to save General Relativity by reformulating it in a way that doesn't have infinities at the singularities. And I repeated numerous times the claim that the singularity theorems prove that the metric tensor has singularities, assuming that they are of this kind. And I might have regarded people who didn't believe in singularities as, let's say, not very serious. Despite being aware that there was a step in the proof of the singularity theorems that I never understood, namely exactly the step where from inextensibility we conclude the existence of such singularities. Despite never being able to find a place where this step is proved for a limited person like me. And that while knowing that I didn't understand that step, and being limited, I considered that I should trust the experts about it, or maybe just my limited understanding of what experts say. And now, after seeing Roy Kerr's counterexamples, I think I was wrong.

So yes, Kerr is right, to be able to say that General Relativity breaks down because of singularities we need a proof for exactly such singularities, and the singularity theorems alone don't do the job, and there are counterexamples showing this. But of course counterexamples are a no go mainly for the more mathematically inclined (and some of them noticed this at some times, but somehow the most spread interpretation of the singularity theorems remained unaffected). Many physicists may still use the confusion between the two notions of singular spacetimes (assuming they're aware of them) to reject classical General Relativity, and at the same time they would claim that quantum gravity doesn't have this problem, again without proof, without even a theory of quantum gravity! (The only argument is that quantum fields may violate a condition in the singularity theorems, but this doesn't prove that this avoids the alleged singularity)

But what if somebody takes notice now of Kerr's paper, and of the disambiguation of the term "singular spacetime", and finds a singularity theorem, with different conditions evidently, that is actually about such singularities? Even so, General Relativity can be formulated in terms of finite geometric objects, which can evolve beyond the singularities, as I showed some time ago https://arxiv.org/abs/1301.2231. This formulation is equivalent with the usual one outside the singularities, but it extends at the singularities too, at least in the usual cases.

So I see no reason why General Relativity is so often pronounced dead. I mean, sure, we need a quantum theory of gravity, but let's stop throwing the baby with the bathwater. There's no reason to treat like a stepchild one of the two babies, General Relativity and Quantum Theory, and favor the other one. The really naughty one ;)

 

The negative way to sentience (comments welcome!)

I wrote an essay about sentience and its relations to physics. For the moment, I  keep it on ResearchGate, and I am welcoming comments.

The negative way to sentience (comments welcome!)

Abstract. While the materialist paradigm is credited for the incredible success of science in describing the world, to some scientists and philosophers there seems to be something about subjective experience that is left out, in an apparently irreconcilable way. I show that indeed a scientific description of reality faces a serious limitation, which explains this position. On the other hand, to remain in the realm of science, I explore the problem of sentient experience in an indirect way, through its possible physical correlates. This can only be done in a negative way, which consists in the falsification of various hypotheses and the derivation of no-go results. The general approach I use here is based on simple mathematical proofs about dynamical systems, which I then particularize to several types of physical theories and interpretations of quantum mechanics. Despite choosing this scientifically-prudent approach, it turns out that various possibilities to consider sentience as fundamental make empirical predictions, ranging from some that can only be verified on a subjective basis to some about the physical correlates of sentience, which are independently falsifiable by objective means.

For the moment, you can read it and comment here.

A debate inside another one

Tim Maudlin debated Gerard 't Hooft about his cellular automaton interpretation of quantum mechanics in a series of Facebook posts, the fourth one being here https://www.facebook.com/tim.maudlin/posts/10155699914028398. Somewhere in the forest of comments I was engaged in a sort of sub-debate, with Tim, Hans, and others. Sabine was there too. The discussion was completely surrealistic, Tim and Hans completely misunderstood my point. This started by me intervening with a theoretical counterexample to a claim that all so called superdeterministic theories (in particular 't Hooft's) are not falsifiable, and of course it led to different topics. It is not known, but not a secret that the wavefunction collapse leads to violations of conservation laws, and that it is possible at least in principle to remove the collapse while remaining with a single world. But removing the collapse can be seen as superdeterministic (although I wouldn't call it like this, because it is based on spacetime, not on initial conditions), and I even proposed a principle to explain this, and experiments to test it. I paste here most of this *debate*, because there are some parts I am interested to keep. I skipped some parts in which I was not involved.

Maudlin's "(Information) Paradox Lost" paper

Tim Maudlin has an interesting paper in which he criticizes the importance given to the black hole information paradox, and even brings arguments that it is not even a problem: (Information) Paradox Lost. I agree that the importance of the problem is perhaps exaggerated, but at the same time many consider it to be a useful benchmark to test quantum gravity solutions. This led to decades of research made by many physicists, and to many controversies. I wrote a bit about some of the proposed solutions to the problem in some older posts, for example [1,2,3]. Maudlin's paper is discussed by Sabine here.

One of the central arguments in Maudlin's paper is that the well-known spacetime illustrating the information loss can be foliated into some 3D spaces (which are Cauchy hypersurfaces that are discontinuous at the singularity). These hypersurfaces have a part outside the black hole, and another one inside it, which are not connected to one another. Cauchy hypersurfaces contain the Cauchy data necessary to solve the partial differential equations, so the information should be preserved if we consider both their part inside and their part outside the black hole.

I illustrate this with this animated gif:

 

I made this gif back in 2010, when I independently had the same idea and wanted to write about it, but I don't think I made it public. Probably the idea is older. The reason I didn't write about it was that I was more attracted* to another solution I found, which led to an analytic extension of the black hole spacetime, and has Cauchy hypersurfaces but no discontinuities. I reproduce a picture of the Penrose diagram from an older post in which I say more about this:

 

A. The standard Penrose diagram of an evaporating black hole.

B The diagram from the analytic solution  I proposed.

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* The reason I preferred to work at the second solution is that it allows the information to become available after the evaporation to an external observer. The solution which relies on completing the Cauchy hypersurface with a part inside the black hole doesn't restore information and unitarity for an external observer. I don't know if this is a problem, but many physicists believe that information should be restored for an external observer, because otherwise we would observe violations of unitarity even in the most mundane cases, considering that micro black holes form and evaporate at very high energies. I don't think this argument, also given by Sabine, is very good, because there is no reason to believe that micro black  holes form at high energy under normal conditions. People arrive at high energies for normal situations because they use perturbative expansions, but this is just a method of approximation. And even so, I doubt anyone who sums over Feynman diagrams includes black holes. But nevertheless, I wouldn't like information to be lost for an outside observer after evaporation, but this is just personal taste, I don't claim that there is some experiment that proved this. And the solution I preferred to research allows recovery of information and unitarity for an external observer, and other things which I explained in the mentioned posts and my PhD thesis.