Is General Relativity Time Symmetric?

[bcrowell]

That is true. But I am not sure that the light cone adequately defines "time". The light cone itself is null, so that is not sufficient, and then any timelike vector can be chosen as your time coordinate. So I agree that you can show light cone reversal symmetry in a coordinate independent tensor form for the usual spacetimes of interest, but my hesitation is in identifying that with time reversal symmetry. I'm certainly not saying your argument is wrong, I am just still not comfortable about how to talk about time reversal symmetry without a coordinate system and a metric expressed in that coordinate system.

In GR, we don't even expect to have global coordinates, just an atlas of coordinate patches. It's handy to talk about t->-t when we want to talk about a global time reversal operation, but really that's just a shorthand, because in general it's not possible to cover the whole manifold with a single coordinate patch on which a single t variable is defined. Reversing the light cones is exactly the right notion if you want to talk about time-reversal.

Another way of looking at it is that once you know the null cones, you can determine the metric everywhere, up to a conformal factor. (See Hawking and Ellis, pp. 60-61.) There is no need to fix the conformal factor in order to talk about time reversal. In other words, you can define time-reversal simply in terms of flipping the Penrose diagram upside down.

 

[Phrak]

That is true. But I am not sure that the light cone adequately defines "time". The light cone itself is null, so that is not sufficient, and then any timelike vector can be chosen as your time coordinate.

It was a poor choice of language. I am not well equipped to turn concepts into words.

However, the light cones serve to separate spacetime into regions. We can take a pair of cones at each point in a model spacetime, and move it all around, so long as we don't run into a singularity such as a Schwarzschild coordinate singularity, we can populate the entire mainifold with cone-pairs at each point. Fill one of the cone pairs with blue and the other fill with red. At each point on the manifold there is a corresponding red-blue pair of cones. This, with the labeling, serves as the model to examine time reversal symmetry, as I see it, so long as spacetime is not a Klein bottle.

So I agree that you can show light cone reversal symmetry in a coordinate independent tensor form for the usual spacetimes of interest, but my hesitation is in identifying that with time reversal symmetry. I'm certainly not saying your argument is wrong, I am just still not comfortable about how to talk about time reversal symmetry without a coordinate system and a metric expressed in that coordinate system.

 

[bcrowell]

I thought you had decided that it should be defined in terms of timelike worldlines, time orientability etc, just like Demystifier's, Wald and Strominger's discussions of black holes, white holes etc, which are gauge invariant?

There are two different issues: (1) Can you define time-reversal? (2) If so, is GR form-invariant under it? You need orientability to decide on which spacetimes the answer to #1 is yes. But I would still claim that the affirmative answer to #2 (for spacetimes where the answer to #1 is yes) follows from diff-invariance.

Wrt to Maxwell's equations in Minkowski spacetime, the solutions may break the symmetries, but in contrast, because diff invariance is a gauge symmetry, it cannot be broken by any solution of GR.

I'm willing to be convinced, but so far this doesn't work for me. On a time-orientable spacetime, there is a uniquely defined time-reversal operator T, and this operator is an element of the diffeomorphism group D. Minkowski space doesn't break T symmetry, and in fact it's invariant under every element of D. A Schwarzshild metric fails to be invariant under almost every element of D, except for certain special ones like rotations and T. Realistic cosmological models break T.

Maybe we have different ideas of what it would mean to be invariant under a member of D. The way I see it, e.g., a Killing vector would generate a family of diffeomorphisms that are all members of D. The spacetime might or might not have the symmetry represented by that Killing vector.

I suppose you could take the point of view that a diffeomorphism is just a change of coordinates, and coordinates are just names for points, so a diffeomorphism is just a renaming, like calling the t coordinate \tau, in which case obviously a point in a spacetime can't "care" what it's named.

Say I tell you, "The stress-energy tensor is zero at point P." Then you apply a diffeomorphism. You could take the diffeomorphism as simply translating into a new language, in which P isn't even a word, and the new word for that point is Q. In that case obviously the same sentence is still true. On the other hand, you could take it as saying that P has been redefined to refer to some other point in space, say the center of the earth, in which case the sentence is now false.

Possibly both of these ways of looking at it are equally valid, in which case it could be valid to prove time-reversal symmetry by my argument, but one would have to specify which point of view was being adopted.

 

[atyy]

There are two different issues: (1) Can you define time-reversal? (2) If so, is GR form-invariant under it? You need orientability to decide on which spacetimes the answer to #1 is yes. But I would still claim that the affirmative answer to #2 (for spacetimes where the answer to #1 is yes) follows from diff-invariance.

But if you use orientability in #1, that is already using #2, since orientability is invariant under change of coordinates, so #1 and #2 are not distinct.

Possibly both of these ways of looking at it are equally valid, in which case it could be valid to prove time-reversal symmetry by my argument, but one would have to specify which point of view was being adopted.

I was using the second one because you indicated that you wanted to take diff invariance - general covariance.

 

[atyy]

Maybe we have different ideas of what it would mean to be invariant under a member of D. The way I see it, e.g., a Killing vector would generate a family of diffeomorphisms that are all members of D. The spacetime might or might not have the symmetry represented by that Killing vector.

Also, does the qualification "local" diffeomorphism for a Killing vector make any difference to your argument?

 

[bcrowell]

Maybe we have different ideas of what it would mean to be invariant under a member of D. The way I see it, e.g., a Killing vector would generate a family of diffeomorphisms that are all members of D. The spacetime might or might not have the symmetry represented by that Killing vector.

Also, does the qualification "local" diffeomorphism for a Killing vector make any difference to your argument?

That's why I said "e.g." All Killing vectors generate diffeomorphisms, but not all diffeomorphisms can be generated from Killing vectors. No, I don't think this has any effect on my argument. The only reason I mentioned Killing vectors was that I wanted to give some concrete examples of how a spacetime could have some proper subset of the symmetries given by the diffeomorphism group.

 

[atyy]

That's why I said "e.g." All Killing vectors generate diffeomorphisms, but not all diffeomorphisms can be generated from Killing vectors. No, I don't think this has any effect on my argument. The only reason I mentioned Killing vectors was that I wanted to give some concrete examples of how a spacetime could have some proper subset of the symmetries given by the diffeomorphism group.

OK, I agree.