# Is General Relativity Time Symmetric? (1 Viewer)

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#### Phrak

Is General Relativity Time Symmetric?

#### Mentz114

Gold Member
I'm not sure what you mean. Presumably invariance of something under the substitution $t\rightarrow -t$.

I believe some solutions will work if time is reversed, an expanding dust can become a collapsing dust. The theory of the Riemannian manifold specifies that the time component of tangent space basis vector must point in the t-direction. I have no idea if this can be safely reversed.

In short, I don't know, but maybe someone else does ?

#### Demystifier

2018 Award
Is General Relativity Time Symmetric?
Yes it is.

For example, black holes and their time-inverted configurations - white holes - are both solutions of GR.

#### bcrowell

Staff Emeritus
Gold Member
Yes, relativity is symmetric under time-reversal. The laws of physics in general relativity have the same form regardless of any smooth coordinate transformation whatsoever. Time-reversal is a smooth coordinate transformation.

#### Phrak

It occurs to me that I don't know how to show that it is true. Any hints or speculation as to how to proceed?

#### JustinLevy

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.).

--

Anyway, how do we show this more rigorously?
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.

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.

Maybe a better place to look is the Lagrangian. Is the ricci curvature scalar affected by a time reversal? There's probably an easy way to see this, but I'm not sure how without the messy method of looking at how it depends on the metric components.

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#### bcrowell

Staff Emeritus
Gold Member
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.).
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.

Anyway, how do we show this more rigorously?
I claim that it is rigorous, as expressed above.

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.
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).

[EDIT] Sorry, this was not quite right. The existence of time-reversal symmetry in the SM doesn't necessarily show that the SM lacks diffeomorphism invariance, because it could be a case of spontaneous symmetry breaking. However, I don't think it's true that the SM has diffeomorphism invariance. For example, whenever you write a derivative with respect to time, that's breaking diffeomorphism invariance; time is treated asymmetrically in quantum mechanics, not as an operator like position.

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.
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.

Is the ricci curvature scalar affected by a time reversal?
It's a scalar, so it's invariant under diffomorphisms, including time reversal.

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#### Dale

Mentor
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.).
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.

#### nicksauce

Homework Helper
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.

#### bcrowell

Staff Emeritus
Gold Member
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.
Equations that are invariant under a certain transformation can have solutions that are not invariant. For example, the equation $x^2=1$ is invariant under $x\rightarrow -x$, 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.

What you can say is that if a set of differential equations has a certain symmetry, and you find a certain solution under boundary conditions that respect that symmetry, then the solution will have that symmetry as well. For example, this is how we know that the Schwarzschild spacetime is time-reversal symmetric.

[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.

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#### Parlyne

[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.
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.

#### atyy

But diff invariance is a gauge symmetry, which isn't a real symmetry, is it?

I tend to think of gauge symmetries as redundancies of description.

Certainly 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"

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.

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#### bcrowell

Staff Emeritus
Gold Member
Certainly 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"
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.

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 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.

#### atyy

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.
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?

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.
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?

#### JustinLevy

I think you are reading way too much into the ability to write an equation in tensor notation.

I'm not arguing that GR doesn't have T symmetry, I'm merely pointing out that I feel you are overstating what a tensor equation gives us.

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
This ISN'T enough.

For example, I can write the equations of electrodynamics in tensor notation. I can apply a Galilean transformation to both sides of an evolution equation. And of course both sides still equal each other.

Does this mean electrodynamics has Galilean symmetry? No.
Electrodynamics has Lorentz symmetry, not Galilean symmetry.

I think what you've proved is that the standard model can't be written in terms of tensors.
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.

So I think we are disagreeing on what is even meant by Lorentz symmetry, Galilean symmetry, time translation symmetry, spatial translation symmetry, parity symmetry, time reversal symmetry, etc. which can be worded as coordinate transformations. To me, these mean much more than just requiring equations can be written in tensor notation.

I'm not sure how to proceed. So maybe someone else can suggest a precise definition of "symmetry" in this context.

#### Dale

Mentor
But diff invariance is a gauge symmetry, which isn't a real symmetry, is it?
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".

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#### Phrak

Does it make any sense to ask if General Relativity is time symmetrical, or is this an ill posed question?

I could ask the same of the spacial manifold+time of Newtonian mechanics. It makes no sense. Its the stuff that happens on the manifold that may or may not be time symmetrical. (Who asked this ridiculous question in the first place? Oh, that was me...)

I could combine thermodynamics with a curved manifold and come to the conclusion that, no, General Relativity is Not time symmetric. All I have to do is smash two planets together. So this combination doesn't work for temporal symmetry.

But, never fear. We can still ask if the vacuum metric and all it's derivative quantities are time symmetric in Einstein's spacetime manifold.

Secondly, thermodynamics sucks. The world is not a roulette wheel where the value of a function is some obscure value determined by an unknown gambler wasting his life in Monte Carlo. God does not have to play dice to utterly confuse us. We can still leave out thermodynamics and substitute the minutia of particle interaction.

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#### atyy

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".
Charge conservation via Noether's theorem comes from U(1) as a global symmetry, not a gauge symmetry.

#### Dale

Mentor
I don't know the distinction. I thought U(1) was the gauge symmetry.

#### bcrowell

Staff Emeritus
Gold Member
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?
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.

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