"Oh, there's so much I don't know about astrophysics. I wish I'd read that book by that wheelchair guy."

-- Homer Simpson

Last Tuesday I went to a talk on numerical general relativity. Specifically, it dealt mostly with simulations of two black holes colliding to determine what kinds of gravitational waves are emitted. One of the challenges writing computer programs to do GR is that it's very hard to check them to make sure they're giving sensible answers. Usually, if you're writing a physics simulation, once you debug your code you do several "toy problems", ones you can work out by hand, in order to make sure your program gets the right answer. This gives you some confidence that you haven't made any major errors. The other check that's often done is to solve a problem that someone else has already solved with a different technique (one that the community is pretty confident is correct). The problem is that GR is just very difficult, involving in general 16 coupled, nonlinear, partial differential equations. There are very few problems you can solve by hand, and most of those are of trivial complexity when compared to the problems the code is supposed to solve. Furthermore, the field is young enough that there just aren't that many trusted results to compare against.

At the end of this talk, one of the professors asked (to the best of my recollection), "Couldn't you run the simulation and get the gravitational waves, then start run the simulation backward, starting with the end state black hole and sending the gravitational waves back in? Since GR is time reversal invariant, you should get back the two original black holes at the end of that that you had at the beginning." The speaker said something to the effect of, "Yes, people have talked about doing that off and on for a long time. To the best of my knowledge no one has done it yet," and that was it. It was an unsatisfying answer, since you'd like to know why no one has tried it or if it will be tried in the future.

After the talk, however, this exchange started to bother me. Instead of talking about two black holes colliding, think about one star undergoing gravitational collapse to form a black hole. The same logic would seem to imply that you could take whatever gravitational waves were produced by the collapse and feed them back in, reversing the collapse process and resulting with the star in the state just prior to collapse. That certainly doesn't seem right. Now certainly this ignores the fact that once you're dealing with a star it's a complex, macroscopic object that's properly treated thermodynamically, but still even if you assume your "star" is a sphere made up of an ideal fluid, such a reversal shouldn't be possible. Indeed, some checking in the black bible reminded me that there is a theorem that the surface area of the event horizon of a black hole cannot decrease in classical GR1. This fact is a vital part of the reason one may equate event horizon surface area to entropy in black hole thermodynamics. So in purely classical GR, such a reversal is not possible. That agrees with my intuition, since an event horizon wouldn't be very meaningful if you could make it go away, and this would allow you to recover the information that went inside.

The question that's been bugging me since then is, "Where is the flaw in the argument?" I can only see two possible flaws: 1) time reversal invariance in GR has an unintuitive meaning that does not imply the sort of "unformation" of the black hole I described above, or 2) GR is not time reversal invariant. The reason I considered 1) is that in GR there are many different time coordinates for the different coordinate systems of different observers, so there's a question of which time you reverse. Do you reverse the coordinate time or proper time? Upon further thought, I think the answer to that question is that it doesn't matter. Since the proper time along any timelike geodesic is monotonically increasing with coordinate time, I believe that reversing time in any valid, physical coordinates is equivalent to reversing time in any other set of coordinates. I'm not sure, though, that this is really what it means to do time reversal in GR. The difficulty is that in GR space-time itself is dynamical, so it seems one is asking a rather strange question like, "Is the behavior of time the same irrespective of time reversal?"

The question of time reversal invariance is always the question of whether you can take a solution of the theory, reverse its time dependence, and have it still be a solution to the theory. So, without getting two bogged down in what the question means, it seems like the simplest way to look at time reversal in GR would be to ask if a Einstein tensor and stress-energy tensor are solutions to Einstein's equation and you reverse the coordinate time, do they still solve Einstein's equation? The answer: a resounding "I don't know"...yet. The problem is that while the time reversal properties of the stress-energy tensor and metric seem pretty simple, deriving the time reversal properties of the Einstein tensor is more involved. In the end, I suspect the answer is that GR is not in general time reversal invariant. I'm sure someone already knows the answer to this, but I haven't run across it yet. Then again, I don't study GR; this question just piqued my interest.

I'm not sure now why I decided this would make a good daylog. Looking at what I've written, I'm not sure that it will be accessible or interesting to anyone. I guess we'll see. :-) Hopefully, I'll find the answer soon and post an update.

[Addendum: There is now an update.]

  1. This is violated by black hole evaporation, but that is due to Hawking radiation, which comes from semiclassical physics.