Is General Relativity Time Symmetric?
To put it in more formal terminology, the field equations of GR are form-invariant under diffeomorphisms, and time-reversal is a diffeomorphism. Therefore any solution is also a solution under time-reversal.bcrowell: "Time-reversal is a smooth coordinate transformation."
Hmm... I think I see what you mean here by "smooth" : the mapping of old coordinates to new coordinates is a smooth function.
But I have also heard of parity or time-reversal to not be considered smooth transformations because one cannot smoothly transition from the initial to final coordinate maps by a series of infinitesimal steps (like rotations, or stretching/scaling, etc.).
I claim that it is rigorous, as expressed above.Anyway, how do we show this more rigorously?
I think what you've proved is that the standard model can't be written in terms of tensors. That is, I don't think the standard model is diffeomorphism invariant. (It doesn't have general covariance.) If it were diffeomorphism invariant, it would be expressed without a background spacetime; but the standard model is expressed with a certain assumed background spacetime (flat spacetime).Merely being able to write the equations in tensor notation doesn't seem sufficient, as we can do that for the standard model, but it isn't time reversal invariant.
You don't have to crank through the complete rederivation of the curvature tensors. The curvature tensors change under a change of coordinates according to the tensor transformation laws. Since both sides of the Einstein field equations are tensors, they transform identically, and a time-reversed solution is still a solution.Under time reversal, the metric g_uv -> g'_uv will change sign on the dx dt, dy dt, dz dt, pieces. This will effect some terms of the curvature, maybe? This looks like it will get complicated. I'm not sure how it will affect the field equations.
It's a scalar, so it's invariant under diffomorphisms, including time reversal.Is the ricci curvature scalar affected by a time reversal?
You are probably thinking in terms of the differentiable symmetries of Noether's theorem. Parity is not an infinitesimal transform of the sort that would indicate a differential symmetry, but it is a smooth transform that indicates another kind of symmetry.But I have also heard of parity or time-reversal to not be considered smooth transformations because one cannot smoothly transition from the initial to final coordinate maps by a series of infinitesimal steps (like rotations, or stretching/scaling, etc.).
Equations that are invariant under a certain transformation can have solutions that are not invariant. For example, the equation [itex]x^2=1[/itex] is invariant under [itex]x\rightarrow -x[/itex], but the solution x=1 is not invariant under that transformation. Or consider that Newton's laws are invariant under rotations, but a particle moving along the x axis in the positive x direction is a solution that is not invariant under rotations.So I'm not so sure about this, but maybe I'm missing something. I figure it would be easy to imagine a metric, which is not invariant under t-> -t, (just put some term in the metric proportional to t, for example). As long as the stress tensor is not completely ridiculous, you'd end up with a solution that is not time symmetric.
Of course, in such cases the distinct solution you get by applying the symmetry operation to the symmetry-broken state will also be a valid solution if the underlying dynamics respect the symmetry.[EDIT] Actually this is not quite right, as I realized by asking some questions over in Beyond the Standard Model and browsing WP. You can have spontaneous symmetry breaking. In cosmological solutions, Lorentz invariance is spontaneously broken by the occurrence of a preferred frame (the CMB's frame, or the frame of the Hubble flow). I guess T symmetry is also broken in perpetually expanding cosmologies.
This paper might be relevant: http://arxiv.org/abs/gr-qc/0603087Certainly at the level of the equations of motion every theory can be made diff invariant, and apparently even at the level of the action http://arxiv.org/abs/1010.2535: "The point here is that any local theory (e.g., a single free scalar field) can be written in diffeomorphism-invariant form through a process known as parametrization"
I think diff invariance implies exactly the same thing as what you're saying about Newtonian mechanics.In Newtonian mechanics, time reversal symmetry means if you reverse time and momenta, the different physical situation obtained is also a solution of the equations of motion. I think Demystifier was giving a comment along these lines, which seems to be different from the diff invariance perspective.
I agree GR has no fixed background, which is different from diff invariance (or in the language of that paper, diff covariance). But what has no fixed background got to do with time reversal?This paper might be relevant: http://arxiv.org/abs/gr-qc/0603087
Take a look at p. 7. They show an example of a heat diffusion equation, which isn't diff-invariant, and definitely not time-reversal invariant. They turn it into a diff-invariant equation, but in order to do that they have to introduce a background field that points in the preferred direction of time. I think the key is that this background field is written as if it were a tensor, but it doesn't transform like a tensor; it's postulated to be constant in its upper-index form.
The field equations of GR aren't like this. They're written in terms of things that really transform like tensors -- no fixed background.
Time is absolute in Newtonian mechanics. The time you are referring to seems to be coordinate time, which is not gauge invariant. Is there any formulation using the absolute notion of timelike?I think diff invariance implies exactly the same thing as what you're saying about Newtonian mechanics.
In the standard formulation of Newtonian mechanics, the laws of physics are form-invariant under time reversal. Therefore any solution can be time-reversed, and it's still a solution.
In the standard formulation of GR, the laws of physics are form-invariant under diffeomorphisms, which include time reversal. Therefore any solution can be time-reversed, and it's still a solution.
I don't think there's anything mysterious or deep going on here. If you write down the tensor transformation law as applied to the rank-2 tensors in the Einstein field equations, they operate in the same way on both sides of the equation, and the only requirement is that the transformation be a diffeomorphism (so that the derivatives appearing in the transformation law don't blow up or fail to exist). The transformation can be any diffeomorphism you like. Time reversal is simply one special case.
This argument only depends on the assumption that everything in the field equations is a tensor that really transforms like a tensor -- not something that's written so as to look like a tensor, but doesn't transform like one.
This ISN'T enough.I don't think there's anything mysterious or deep going on here. If you write down the tensor transformation law as applied to the rank-2 tensors in the Einstein field equations, they operate in the same way on both sides of the equation
I believe the standard model can be written in terms of tensors. Using your limited requirement for "time reversal symmetry", we can apply any coordinate transformation to both sides of a tensor equation, so yes, in that sense (which I feel lacks meaning) it has time reversal symmetry. But the "form" of the equations changed. The weak force instead of dealing only with left handed neutrinos, now only deals with right handed neutrinos.I think what you've proved is that the standard model can't be written in terms of tensors.
Gauge symmetries certainly are real symmetries. They are the most fundamental symmetries of the standard model or any field theory. Also, per Noether's theorem they lead to conservation laws, e.g. the U(1) gauge symmetry leads to the conservation of charge.But diff invariance is a gauge symmetry, which isn't a real symmetry, is it?
Charge conservation via Noether's theorem comes from U(1) as a global symmetry, not a gauge symmetry.Gauge symmetries certainly are real symmetries. They are the most fundamental symmetries of the standard model or any field theory. Also, per Noether's theorem they lead to conservation laws, e.g. the U(1) gauge symmetry leads to the conservation of charge.
EDIT: I just realize that I have violated my usual policy of not getting involved in discussions about whether or not something is "real". So I would like to revise my above statement and simply say that a gauge symmetry is a differential symmetry that leads to a conservation law as per Noether's theorem. As to whether or not that qualifies it for status of a "real" symmetry depends on the definition of "real".
If you look at that example in the paper, the only way they were able to recast the heat diffusion equation into a diff-invariant form was by writing the field equations with a background "tensor" that doesn't really transform like a tensor. This background points in the forward time direction, and it's the only thing in the field equations that defines a direction in time.I agree GR has no fixed background, which is different from diff invariance (or in the language of that paper, diff covariance). But what has no fixed background got to do with time reversal?