Is General Relativity Time Symmetric?

[JustinLevy]

... because we don't have global coordinates that have the right interpretation.

Hmm... we can both feel it; there is some kind of physical "meaning"/"interpretation" we want to associate with the transformations ... and for the Galilean transformation, we can "feel" that it doesn't work.

But there must be some mathematical way to be explicit about this physical "baggage", so we can ask about symmetries in a methodical and clear manner.

I've asked a couple friends in physics and no one has come up with an answer. Maybe our starting point of associating symmetries with coordinate transformations is a crutch from when we started physics where it was just assumed the coordinates themselves had direct physical meaning. Maybe there is a deeper / coordinate independent method that demonstrates symmetries, and which reduces to merely playing with coordinate transformations in our "special case inertial frames" or something.

Thanks for your help and discussion. I clearly need to ruminate on this more. But if you run into any further insights, please do let me know.

 

[Phrak]

I think I can supply an operational meaning behind temporal inversion that should be seen to be coordinate free despite reference to indices.

Each upper or lower index that appears in a tensor, should the tensor be indexed in any coordinate system, is multiplied by a tensor with signature (-1,1,1,1), having all other entries zero.

With this as a definition, every tensor equation is, as bcrowell has called it, form invariant.

(Is the Jacobian of the metric unchanged?)

bcrowell, I don't understand your deference to diffeomorphic invarance. In my understanding, it doesn't seem to apply. Though please correct me if I'm wrong. Diffeomorphic invariance seems to be simply a generalization of what was once called a continuous transformation, which temporal inversion is not.

A general coordinate transformation involves dilations, shears and rotations. The Lorentz transformation that involves a single spatial coordinate and the time coordinate has a parameter theta, where elements of the Lorentz tensor are cosh(theta) and sinch(theta). For any finite theta, time never changes sign. I think that if you do find a parameter that inverts t, then at some value of the parameter t-->infinity.

bcrowell and Dale, I don't know what a smooth transformation is unless you meant a smooth transformation over a parameter(s) as the above.

 

[Dale]

I have been thinking about this thread but not posting, but the more I think about it the less I am sure that it makes sense to talk about GR having or not having time reversal symmetry. A tensor doesn't have time, and you can even use coordinate systems such as light-cone coordinates where there is not even a timelike coordinate. If a theory is not expressed in terms of some variable it seems to not make sense to ask if it is symmetric wrt changes in that variable. E.g. would you ask if Newton's laws were symmetric wrt charge reversal? I don't think so, since Newton's laws don't describe charge.

The only way that you get time in GR is when you are expressing it in terms of some specific coordinate system. So, while I don't think that you can ask if GR is time reversal symmetric, you could ask if some specific metric in some specific coordinate system is time reversal symmetric. That should be pretty easy to determine based on the power of any t and dt terms in the given coordinate system and metric.

Anyway, I am not terribly confident about this position, but I thought I would go ahead and post it.

 

[bcrowell]

I have been thinking about this thread but not posting, but the more I think about it the less I am sure that it makes sense to talk about GR having or not having time reversal symmetry. A tensor doesn't have time, and you can even use coordinate systems such as light-cone coordinates where there is not even a timelike coordinate. If a theory is not expressed in terms of some variable it seems to not make sense to ask if it is symmetric wrt changes in that variable. E.g. would you ask if Newton's laws were symmetric wrt charge reversal? I don't think so, since Newton's laws don't describe charge.

The only way that you get time in GR is when you are expressing it in terms of some specific coordinate system. So, while I don't think that you can ask if GR is time reversal symmetric, you could ask if some specific metric in some specific coordinate system is time reversal symmetric. That should be pretty easy to determine based on the power of any t and dt terms in the given coordinate system and metric.

I would pretty much agree with this analysis, but maybe with a different spin on it. The role of time in GR is not quite analogous to the role of charge in Newton's laws. At least locally, GR clearly does have something that plays the role of time. It's true that the field equations don't explicitly single out a different role for time, and in fact the field equations function very happily if you feed them a metric with a ++++ signature. But in practice we always look for solutions that have a +--- metric, and then we do have light cones and a notion of time defined, at least locally. What I'm claiming is that for the vast majority of spacetimes of physical interest, this makes it possible to define a unique and sensible time-reversal operation. This is because nearly all spacetimes of physical interest are time-orientable.

The part that I still don't feel satisfied with is my understanding of is the role of background-independence. For example, consider Lorentz invariance versus Galilean invariance. Let theory T1 be Maxwell's equations in vacuum as expressed in the language of x, y, z, t, div, and curl. I'm pretty sure that T1 is form-invariant under a Lorentz transformation, but not form-invariant under a Gaillean transformation. On the other hand, let theory T2 be the full electrovac field equations of GR ( http://en.wikipedia.org/wiki/Electrovacuum_solution ), I'm pretty sure T2 is form-invariant under any diffeomorphism. In particular, say you pick some point P in your spacetime, and you write down some coordinate transformation G that, in the neighborhood of P, is a Galilean transformation. As long as G is defined globally (i.e., on every coordinate patch in your manifold), and is smooth enough to be a diffeomorphism, then I'm pretty sure T2 is form-invariant under G. That doesn't mean that G deserves to be interpreted as a global Galilean transformation -- there is no such thing in a general spacetime.

What I would like to understand better is how and where this difference in behavior between T1 and T2 creeps in. For example, we could imagine intermediate levels between T1 and T2. Maybe T1.5 could be Maxwell's equations written on a background of fixed curvature. Is T1.5 form-invariant under G, or not? How can we tell? We would have to consider things like whether the derivativatives are covariant derivatives or plain old partial derivatives, etc. It also gets confusing because in the language of T1, we don't distinguish between the covariant and contravariant forms of something like the electric field vector. Another thing I have to understand better is the implications of the fact that a theory like T2 needs a supplementary field equation on the Maxwell tensor F, rather than just the Einstein field equations.

Probably this discussion can be simplified quite a bit by taking something simpler than G. For example, we could take a point P, define local Minkowski coordinates t,x,y,z, and then define a dilation D that only acts on one coordinate, so x'=kx. If we like, we can extend this in some stupid way to a global coordinate chart, with the understanding that at points far from P it loses its geometrical interpretation as a one-dimensional dilation. I claim that T1 is not form-invariant under D, T2 is form-invariant under D, and that if we could understand exactly why this is, that understanding would also apply to G.

 

[JustinLevy]

Beautiful summary, both of you. Much better than I was able to say it.

Dalespam's view on discussing these symmetries in GR is a bit more pessimistic than the outcome I was hoping for, but it is hard to find a good way to argue with mathematically. My gut feeling still currently falls more towards bcrowell's view that we should still be able to discuss local symmetries in a meaningful way ... I just don't know how.

It also gets confusing because in the language of T1, we don't distinguish between the covariant and contravariant forms of something like the electric field vector. Another thing I have to understand better is the implications of the fact that a theory like T2 needs a supplementary field equation on the Maxwell tensor F, rather than just the Einstein field equations.

For the first one, when forced to choose I've seen people preferably choose to identify the E and B fields as components of the covariant field tensor (or the contravariant dual to the field tensor), so as to leave the "source free" maxwell's equations least "changed" as opposed to the source equation. However I think it is just purely convention choice, with no real meaning (like the sign choice of the inertial frame metric as -1,1,1,1 or 1,-1,-1,-1). So our choice shouldn't matter.

I'm not sure what exactly you are pointing out in the second one. Maybe I'm missing some implications here entirely.

For example, we could imagine intermediate levels between T1 and T2. Maybe T1.5 could be Maxwell's equations written on a background of fixed curvature. Is T1.5 form-invariant under G, or not? How can we tell? We would have to consider things like whether the derivativatives are covariant derivatives or plain old partial derivatives, etc.

Regarding partial vs covariant derivatives, I think we could just start by looking at Maxwell's equations in flat spacetime. I may be making a mistake here, but I think if we want to write Maxwell's equations in tensor notation for a fixed geometry, we'd already have to write it as covariant derivatives.

Consider a simple case of coordinate system in which the components of the metric in flat spacetime depends on the position: Rindler coordinates for an "accelerating frame". We're still in flat spacetime, but for the equations to work, we need to use covariant derivatives. So maxwell's equations as tensor equations seem to already require:
$$F^{\alpha\beta}{}_{;\beta} = \mu_0 J^{\alpha}$$
$$F_{ \alpha \beta ; \gamma } + F_{ \beta \gamma ; \alpha } + F_{ \gamma \alpha ; \beta } = 0$$
or in terms of the dual electromagnetic field tensor $G^{\gamma\delta} = \frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\alpha\beta}$
$$G^{\alpha\beta}{}_{;\beta} = 0$$
instead of partial derivatives (which it reduces to when the components of the metric don't depend on position).

Which would appear to tell us yes, T1.5 is invariant to G. So it appears merely writing it in full tensor notation, removes our ability to discuss symmetries as related to coordinate transformations ... which makes sense along DaleSpam's reasoning.

It really seems to me, we are missing a mathematical tool for asking symmetry questions for tensor equations. This has most likely been considered and solved long ago. So what is this missing tool?

 

[bcrowell]

Which would appear to tell us yes, T1.5 is invariant to G. So it appears merely writing it in full tensor notation, removes our ability to discuss symmetries as related to coordinate transformations

I disagree. I think generalizing to curved spacetime is what removes our ability to discuss a global symmetry like G. Say I ask question Q: "Does theory T have invariance under a global Galilean transformation?" Question Q is a sensible question if T is QED, but if T is GR then question Q is a nonsensical question, like asking whether love smells good.

It really seems to me, we are missing a mathematical tool for asking symmetry questions for tensor equations. This has most likely been considered and solved long ago. So what is this missing tool?

I don't think we're missing a mathematical tool. The tool of tensors just makes it easier to detect that Q is a nonsense question when T is GR. Without tensors, it would just be hard to tell whether (1) Q is an interesting question, and we need to compute the answer by laborious methods, or (2) Q is a nonsense question. Tensors make the computations trivial, so we can focus on whether Q is sensical or nonsensical.

[EDIT] Actually, maybe I would argue for the usefulness of one mathematical tool, which is the notion of orientability. Time-orientability is what makes it sensible to ask whether GR is time-reversal symmetric, because it let's us understand the conditions under which we can meaningfully define a time-reversal operator. It's possible that there are other kinds of orientability that are useful as well. E.g., it might be meaningful to ask whether GR has symmetry under parity inversion.

 

[bcrowell]

Here's a nice easy way to get at some of the issues we've been discussing. Anything that introduces a preferred scale violates diff-invariance. As a really simple example, suppose that I couple GR to classical, pointlike, identical particles of mass m. Then a Schwarzschild spacetime with mass m is a solution of the Einstein field equations. But if I subject this spacetime to a coordinate transformation where all the space and time coordinates are contracted by a factor of 2, this becomes a Schwarzschild spacetime with mass parameter m/2, which isn't a valid spacetime of this theory. IMO this is an example of how GR has a much higher degree of symmetry than other physical theories, the symmetry is expressed by GR's diff-invariance, and it's broken by coupling to specific types of matter.

 

[atyy]

Here's a nice easy way to get at some of the issues we've been discussing. Anything that introduces a preferred scale violates diff-invariance. As a really simple example, suppose that I couple GR to classical, pointlike, identical particles of mass m. Then a Schwarzschild spacetime with mass m is a solution of the Einstein field equations. But if I subject this spacetime to a coordinate transformation where all the space and time coordinates are contracted by a factor of 2, this becomes a Schwarzschild spacetime with mass parameter m/2, which isn't a valid spacetime of this theory. IMO this is an example of how GR has a much higher degree of symmetry than other physical theories, the symmetry is expressed by GR's diff-invariance, and it's broken by coupling to specific types of matter.

By diff invariance, do you mean general covariance?

As I understand it, general covariance is just gauge, and not meaningful.

 

[bcrowell]

By diff invariance, do you mean general covariance?

Yes. (Diffeomorphism invariance=general covariance.)

As I understand it, general covariance is just gauge,

Some people like to refer to it a gauge symmetry, others don't. It's a matter of taste. In any case, it's certainly at least analogous to a gauge symmetry.

and not meaningful.

...which is a whole different issue, and would depend on what you meant by "meaningful." That's pretty much what we've been debating in this thread.

 

[atyy]

...which is a whole different issue, and would depend on what you meant by "meaningful." That's pretty much what we've been debating in this thread.

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?

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.

 

[atyy]

I don't think we're missing a mathematical tool. The tool of tensors just makes it easier to detect that Q is a nonsense question when T is GR. Without tensors, it would just be hard to tell whether (1) Q is an interesting question, and we need to compute the answer by laborious methods, or (2) Q is a nonsense question. Tensors make the computations trivial, so we can focus on whether Q is sensical or nonsensical.

I think it's related to the question - how do you know your diff eq is really nonlinear, and not just a linear equation to which you have applied a nonliear change of coordinates?

Apparently, there is a method to find this out using Lie groups.
http://books.google.com/books?id=nFSJn7dIYysC&dq=stephani+lie+hans&source=gbs_navlinks_s
http://books.google.com/books?id=Yu...etries+invariants+olver&source=gbs_navlinks_s

I don't know how far it extends to other symmetries. For example, http://arxiv.org/abs/1007.5472 talks about "known or unknown 'hidden symmetries'" of a well-studied theory, so I'd guess there's no simple way to uncover all symmetries.

 

[Haelfix]

I must admit I am a little stumped by this question. Locally, surely everyone agrees that there is a good notion of time reversal symmetry.

Globally, I am not so sure and I think this is subtle.

The problem as I see it is topological. Certain spacetimes do not admit time orientability, b/c there is an obstruction to this identification (think pure De Sitter space).

Consequently, it strikes me that diffeomorphism invariance alone does not guarantee that solutions of GR are time symmetric. I suppose you could say that the diffeomorphism invariance is spontaneously broken by the choice of topology. Otoh that's a little bit backwards, hence my confusion

 

[Phrak]

I have been thinking about this thread but not posting, but the more I think about it the less I am sure that it makes sense to talk about GR having or not having time reversal symmetry. A tensor doesn't have time, and you can even use coordinate systems such as light-cone coordinates where there is not even a timelike coordinate. If a theory is not expressed in terms of some variable it seems to not make sense to ask if it is symmetric wrt changes in that variable. E.g. would you ask if Newton's laws were symmetric wrt charge reversal? I don't think so, since Newton's laws don't describe charge.

The only way that you get time in GR is when you are expressing it in terms of some specific coordinate system. So, while I don't think that you can ask if GR is time reversal symmetric, you could ask if some specific metric in some specific coordinate system is time reversal symmetric. That should be pretty easy to determine based on the power of any t and dt terms in the given coordinate system and metric.

Anyway, I am not terribly confident about this position, but I thought I would go ahead and post it.

I've vacillated over this many times in 24 hrs. However, the light cone at each point is independent of coordinates. No matter what coordinates we choose, it's invariant.

Assume at first, for simplicity, that spacetime is orientable. In addition, the light cone at any arbitrary point is equipped with labels + and - for each branch. This can be translated over the entire manifold as long as we ignore points that are singular. Doing it this way we can ask about interchanging the labels + and -. Interchanging labels has no effect on any structure that inherits its characteristics from the metric, and as long as we confine our criteria to metric structures and it's derivatives, this is good to know.

The labeled light cone pair is invariant over proper Lorentz transforms. It is not invariant over improper Lorentz transforms that include spacetime inversion, which is why I objected to arguments over diffeomorphic invariance.

The map is not the territory. Anytime we talk about an abstract manifold, implicit or not, an additional element is the direction of +time. This established the light cone labels. Implicitly, when we talk about the Schwarzschild metric, we all think it's a black hole, rather than a white hole, because the labels are implied.

 

[Phrak]

As long as we claim that the answer to the question "Is General Relativity Time Symmetric?" is confined only to the metric and it's derivatives", then the answer is yes, because equalities involving the metric and it's derivatives are invariant with respect to this labeling.

If electric charge is introduced, for instance, the question must be reconsidered.

In the end, Dale, we are seem to be in agreement, in as much as t or dt is invariably and implicitly introduced whenever a model of spacetime is argued to correspond with elements of physical reality, in so doing placing a label on each leg of the cone.

 

[Dale]

In addition, the light cone at any arbitrary point is equipped with labels + and - for each branch.

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.

 

[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.