Rotating black holes, causality and time travel ramble

[Elroch]

It is well-known that associated with the Kerr solution which represents a rotating black hole, there can be a region of space-time where there are loops in space time (non simply connected paths which are navigable in principle). If this is so, it breaks causality and permits time travel in these regions of space-time.

The question is whether this is really so. Can these types of region of space time even be created? It is difficult to picture how a region with linear time could join to one with cyclic time. Topologically, joining a line to a circle is clearly impossible without an anomalous point, but perhaps the other 3 dimensions of space-time make it possible. [It's not so easy to picture general 4-dimensional topology, even without adding a light cone structure. ]

If one imagines a path where time progresses normally but then crosses itself, but where space-time is otherwise perfectly normal locally, there is a problem with people (or fermions) navigating it. Suppose you plan to take this path and set off. When you reach the point where the path will cross itself, you are likely to meet yourself. And by meet yourself, this means be in exactly the same place you want to go. So suppose you try this, but see yourself coming back. Obviously the laws of physics say you have to change your path slightly to dodge around yourself, so you as a physical object are prevented from crossing the same point twice. [Physicists might say it was the fermions in you that refuse to occupy the same state]. But I see no problem with observing yourself (as I don't believe in causality as a dogma, just as something which is a fact about all the regions of space time we have empirical knowledge of). So my best guess is with regard to loops in space time, people may meet themselves from their future, but may simply interact with them physically as if they were a different physical person. Yes, they could tell you things about your future, and they would have been told those same things when they were in the same place as you were (because they are you). No, they can't kill you, because the probability that they would be where they are if they had done so is zero. And the same goes for your grandfather, if your family had settled in this region of space time a long time ago. To me it's a bit like saying what is the probability of a photon being vertically polarised if it has been measured to be horizontally polarised. Yes, being vertically polarised is a thing a photon can do, but not if you already know it isn't.

There is another superficial paradox that if you have a person or fermion in such a loop, it would expect to meet infinite numbers of copies of itself. But we know even the first copy can't take exactly the same path, so if you try to stay in this loop, you will have to make detours to avoid as many other copies of yourself as you come across. Eventually there will not be room for any more.

But now we have a problem. If the first time you arrive at the loop you see a vaste swarm of copies of yourself and decide it was a bad idea, and you should not enter the loop at all, then suddenly all those copies of yourself never entered the loop either, so why are they there? Perhaps the answer is that in the Kerr solution, you don't have a choice about entering this region - it is deep in a black hole after all. I really don't know. The breach of causality does complicate things, but perhaps no more so than solving differential equations on a non-simply connected manifold. But it is possible that such this factor might make it more difficult to extend the solution of a differential equation, which is related to what we do when inferring the future from the past.

What the truth is, and what would really happen, I am not sure. But what's for sure is that something happens if such regions exist. And if so it looks like causality does get broken in such a region. If so, we should not consider it a dogma and be very cautious about using it as a given to prove something else. For example, perhaps at very small scales, causality is broken even in ordinary space-time. The universe may be loopy and, if so, we have to accept it.

 

[Markus Hanke]

I would be very surprised indeed if we ever came across closed time-like geodesics which are causally connected to the rest of the universe. I believe one of these will happen instead :

1. A theory of quantum gravity will rule out the initial conditions needed to create such curves
2. The metrics describing such curves ( e.g. Kerr-Newman metric ) will turn out to be topologically unstable
3. Curves likes these might exist, but they always cross an event horizon, thereby preventing an observer from detecting any causality violations

I think at the moment we are missing the crucial link between GR and QM, so really there is no way to pass a final judgement on this.

 

[Elroch]

I believe the Kerr solution is unstable in some sense, but the question appears to be still open what you do get if you start with a large mass with some angular momentum and it collapses to a black hole. Clearly you get a black hole with angular momentum. From the event horizon outwards, I believe it settles into the Kerr solution, but what the heck happens inside? Is it possible that any angular momentum might even prevent the formation of a singularity? Given that the Kerr solution has a singularity that repels rather than attracts, this seems at least plausible.

It's striking that, although Schwarzschild black holes are often thought of as "normal" and are often what people discuss, there is no chance of such a black hole ever being created, since any realistic mass has angular momentum. Hence black holes with angular momentum are the only type of black hole with real world significance!

 

[Markus Hanke]

I believe the Kerr solution is unstable in some sense, but the question appears to be still open what you do get if you start with a large mass with some angular momentum and it collapses to a black hole. Clearly you get a black hole with angular momentum. From the event horizon outwards, I believe it settles into the Kerr solution, but what the heck happens inside? Is it possible that any angular momentum might even prevent the formation of a singularity? Given that the Kerr solution has a singularity that repels rather than attracts, this seems at least plausible.

It's striking that, although Schwarzschild black holes are often thought of as "normal" and are often what people discuss, there is no chance of such a black hole ever being created, since any realistic mass has angular momentum. Hence black holes with angular momentum are the only type of black hole with real world significance!

You are right, in any real-life scenario we would get a singularity with angular momentum. I should point out, however, that the momentum needs to be very large in order for the metric to differ substantially from the Schwarzschild metric. If this is the case you get a ring-shaped singularity instead of a point-like one; here of course is where it gets interesting, and you get the well-publicised phenomena like closed time-like curves and frame-dragging.
The problem with all this is of course that in a physical sense a true singularity never really forms; once a certain energy density is reached during the collapse quantum effects become significant, and the normal structure of space-time breaks down. The usual Riemann manifold is no longer smooth, and the tensor equations of GR no longer hold. What happens here is impossible to say without a theory of quantum gravity.

 

[Elroch]

You are right, in any real-life scenario we would get a singularity with angular momentum. I should point out, however, that the momentum needs to be very large in order for the metric to differ substantially from the Schwarzschild metric. If this is the case you get a ring-shaped singularity instead of a point-like one; here of course is where it gets interesting, and you get the well-publicised phenomena like closed time-like curves and frame-dragging.

I would agree with you about what the black hole would look like outside the event horizon (except to point out that frame-dragging occurs for any rotating mass, just like gravitational waves do).
But inside is a quite different matter.

Intuitively, when a large mass with any amount of angular momentum heads for a singularity, the importance of the angular momentum might be expected to grow as the region gets smaller. I don't believe that even with a small amount of angular momentum it is likely that there would be a region around a singularity anything like a Schwarzschild solution. Perhaps there is some published work on this?

The problem with all this is of course that in a physical sense a true singularity never really forms; once a certain energy density is reached during the collapse quantum effects become significant, and the normal structure of space-time breaks down. The usual Riemann manifold is no longer smooth, and the tensor equations of GR no longer hold. What happens here is impossible to say without a theory of quantum gravity.

True, we know there is a scale at which our laws of physics need to be modified. But to a macroscopic being, getting crushed into a diameter of a Planck length or so is a pretty good approximation to a singularity.

 

[Markus Hanke]

You are right, I didn't take this into account...as the radius of the ( spherical ) object decreases during the gravitational collapse the momentum is conserved, so the rotational speed increases. Elementary physics !
Yeah, and being crushed down to Planck length isn't something you are very likely to forget any time soon
But coming back to the subject at hand - the whole idea of a naked singularity just feels wrong to me; I bet you there will be something preventing such a phenomenon from occurring. And once the singularity is hidden behind an event horizon most of the other problematic issues like causality violations etc will likely take care of themselves.
Also, I have problems understanding what the notion of an unstable geometry actually means in a physical sense ??

 

[Elroch]

Singularities may not exist at all in the final theory.

String theory already avoids singularities associated with individual particles by turning them into little loops.

Adding more dimensions can reveal singularities to be an illusion. For example, the colliding brane picture of the big bang involves two smooth higher dimensional manifolds colliding, producing what appears to be a singularity at the big bang at the very first moment that they touch. A singularity can appear to exist merely because we only see part of the picture. It is quite plausible that the same could be true for a black hole (a feature somewhere on the colliding branes).