Worldline congruence and general covariance

[TrickyDicky]

Ok, let's forget about picking a direction, what the W. P. gives is a way for anybody to have a reference, this reference being the comoving observers, without the hypersurface orthogonal condition, there is no way to choose a comoving observer time coordinate reference; is this a clearer way to put it?
Now back to general covariance, this was the whole point of the OP, it seems like there are physical laws, those that are not time translation invariant (as I said only a few but usually considered fundamental) that fail to behave in a generally covariant way in the FRW geometry.
How does this happen? Maybe it is best understood with an example that can be agreed by anyone, we all accept that spatial homogeneity is a feature of a certain spacetime slicing, it should be like this to comply with the cosmological principle, any coordinate system of the FRW metric that is not time hypersurface orthogonal should not find homogeneity in the matter distribution. This doesn't affect general covariance because the distribution of matter is not considered a physical law.
However, it is easy to check that those same coordinates that make the FRW spacetime lose its spatial homogeneity produce an observer disagreement about those physical laws that are not time invariant.

 

[PeterDonis]

We seem to be going around in circles.

Ok, let's forget about picking a direction,

But, as I've said for many, many posts now, the physical laws you are concerned about, such as the second law, *only* depend on agreement on the direction of time (meaning which half of the light cones is the future half). Given that agreement, these laws are generally covariant: all observers in whatever state of motion will agree on them. See next comment.

However, it is easy to check that those same coordinates that make the FRW spacetime lose its spatial homogeneity produce an observer disagreement about those physical laws that are not time invariant.

Really? I have said a number of times during this thread that this is not the case: the physical laws you speak of (e.g., the second law) *are* generally covariant, and will be observed to be true by any observers who agree with "comoving" observers on the direction of time. I even gave an example: we, here on Earth, observe the second law to hold, even though we are not at rest relative to "comoving" observers, and our proper time is *not* the same as "cosmic time"; the slicing of spacetime implied by our sense of simultaneity is *not* the same as the "comoving" slicing, in which spacetime appears homogeneous. Spacetime does *not* appear homogeneous to us, even "on average"; as I said before, we see a large dipole anisotropy in the CMBR. (There are other effects as well, such as anisotropy in redshifts of galaxies due to our "proper motion" relative to the comoving frame.) You have not addressed any of these arguments.

Also, the way you state your assertion above invites confusion: you say "coordinates that make the FRW spacetime lose its homogeneity", but we agreed many posts ago that homogeneity is a property of the spacetime, not of a coordinate system; the FRW spacetime is homogeneous and isotropic even if we adopt coordinates that do not make those properties manifest. I think what you meant to say is "coordinates that make the spatial slices no longer *appear* homogeneous". See further comments below on "matter distribution".

what the W. P. gives is a way for anybody to have a reference, this reference being the comoving observers, without the hypersurface orthogonal condition, there is no way to choose a comoving observer time coordinate reference; is this a clearer way to put it?

Yes, but I only agree if it is understood that the "cosmic time" is not the same as proper time for any observer not at rest in the comoving frame, and that this does not in any way contradict general covariance. See my comments above on that.

we all accept that spatial homogeneity is a feature of a certain spacetime slicing, it should be like this to comply with the cosmological principle, any coordinate system of the FRW metric that is not time hypersurface orthogonal should not find homogeneity in the matter distribution. This doesn't affect general covariance because the distribution of matter is not considered a physical law.

Again, I think this way of putting it invites confusion. First of all, as I noted above, we agreed many posts ago that homogeneity is a property of the *spacetime*, not of a particular slicing; a better way to state what I think you meant to say above is that spatial homogeneity is only *manifest* in a particular spacetime slicing. Second, a better term for what I think you meant by "distribution of matter" is "stress-energy tensor" (SET), but the SET is a generally covariant geometric object that appears on the right-hand side of the Einstein Field Equation, so it's not quite correct to say that it is not considered a physical law. A better way to say it would be that the way we typically arrive at a solution to the EFE in cosmology is to make a certain *assumption* about the stress-energy tensor (for example, that it is spatially homogeneous and isotropic, so there will be some coordinate system in which it takes a particular simple form), and then plug into the EFE and solve for the dynamics of the spacetime. Once we have such a solution, both the spacetime curvature and the "matter distribution" throughout the entire spacetime are fixed, and all physical predictions based on them are also fixed, and can't be changed by changing coordinate systems. I think you agree on this, but do you realize that it implies that there *cannot* be observer disagreement on physical laws, once the overall spacetime solution is determined? The only way to change the physical predictions is to change the starting assumptions about the SET, and thereby change the solution of the EFE you are using; that changes which spacetime you are working with, and of course in a different spacetime, with different properties, you will get different physical predictions. (But they will still be generally covariant *with respect to that spacetime*.)

 

[TrickyDicky]

I got distracted by the naughty neutrinos, let's finish this discussion properly.

Really? I have said a number of times during this thread that this is not the case: the physical laws you speak of (e.g., the second law) *are* generally covariant,

Ok, would you explain to me geometrically how can physical laws that involve time asymmetry (like the second law) be generally covariant in a manifold like the FRW universe that is isotropic but not spherically symmetric?, you see spherical symmetry is demanded for any spacetime manifold that is isotropic and that is supposed to be generally covariant for laws that involve time and that are not themselves time symmetric, since in such a spacetime all time derivatives of the metric tensor are set to zero.
As you probably know this spherical symmetry is only seen (in the context of valid solutions of the EFE) in vacuum solutions like Schwarzschild's where it implies a static spacetime by the Birkhoff theorem.

 

[PeterDonis]

I got distracted by the naughty neutrinos, let's finish this discussion properly.

So did I.

Ok, would you explain to me geometrically how can physical laws that involve time asymmetry (like the second law) be generally covariant in a manifold like the FRW universe that is isotropic but not spherically symmetric?

The FRW spacetime *is* spherically symmetric. See below.

you see spherical symmetry is demanded for any spacetime manifold that is isotropic and that is supposed to be generally covariant for laws that involve time and that are not themselves time symmetric, since in such a spacetime all time derivatives of the metric tensor are set to zero.
As you probably know this spherical symmetry is only seen (in the context of valid solutions of the EFE) in vacuum solutions like Schwarzschild's where it implies a static spacetime by the Birkhoff theorem.

Birkhoff's theorem only applies to vacuum solutions. The FRW spacetime is not a vacuum solution. For a non-vacuum solution, you can have a non-stationary metric and still have spherical symmetry and isotropy.

 

[TrickyDicky]

The FRW spacetime *is* spherically symmetric.

It is spatially isotropic but not spherically symmetric in 4-spacetime. Please show a reference where it is stated the FRW metric is spherically symmetric in spacetime.

If you think about symmetries a moment you'll realize that a time asymmetric spacetime like FRW can't be both spherically symmetric in 3-space and 4-spacetime.

For a non-vacuum solution, you can have a non-stationary metric and still have spherical symmetry and isotropy.

Are you sure? see above.

 

[PeterDonis]

It is spatially isotropic but not spherically symmetric in 4-spacetime. Please show a reference where it is stated the FRW metric is spherically symmetric in spacetime.

What does "spherically symmetric in 4-spacetime" mean? The only definition of "spherically symmetric" that I'm aware of involves isometry with respect to the spatial rotation group. See, for example, the Wiki page:

http://en.wikipedia.org/wiki/Spherically_symmetric_spacetime

 

[TrickyDicky]

What does "spherically symmetric in 4-spacetime" mean? The only definition of "spherically symmetric" that I'm aware of involves isometry with respect to the spatial rotation group. See, for example, the Wiki page:

http://en.wikipedia.org/wiki/Spherically_symmetric_spacetime

Last time I checked spacetime was 4-dimensional, has that changed?
The wiki definition is fine with me.

 

[TrickyDicky]

Actually there's some terminology confusion in the wiki article. SO(3) is a spatial symmetry, not a spacetime symmetry.

 

[TrickyDicky]

I guess spherical symmetry is usually referring to the spatial part of the manifold, but since we are dealing with general covariance of the 4-manifold, I am referring to a 4-dimensional symmetry, that wouldn't be able to accommodate asymmetrical laws of physics wrt time.

 

[PeterDonis]

Actually there's some terminology confusion in the wiki article. SO(3) is a spatial symmetry, not a spacetime symmetry.

I guess spherical symmetry is usually referring to the spatial part of the manifold, but since we are dealing with general covariance of the 4-manifold, I am referring to a 4-dimensional symmetry, that wouldn't be able to accommodate asymmetrical laws of physics wrt time.

I don't understand the distinction you're making here. The spatial part of the manifold is part of the manifold, whether the manifold is 3-D space or 4-D spacetime. A spatial symmetry *is* a spacetime symmetry; it's just not a symmetry that includes the time portion of the metric. Birkhoff's theorem says that, for a vacuum solution, the spatial symmetry under SO(3) *requires* the spacetime to be static, i.e., it implies something about the time portion of the metric; but that only applies to a vacuum solution. I'm not aware of any requirement that a non-vacuum solution have any time symmetry in order to be considered spherically symmetric, and I don't see why it would have to, since the key difference with a non-vacuum solution is that you can fill spacetime with a fluid whose density can be uniform in space, so it can still be spherically symmetric (isotropic), but can vary with time. The "density" of a vacuum can't vary with time, either because it's zero, or (if you include a cosmological constant) because it's a constant multiple of the metric.

 

[TrickyDicky]

I don't understand the distinction you're making here. The spatial part of the manifold is part of the manifold, whether the manifold is 3-D space or 4-D spacetime. A spatial symmetry *is* a spacetime symmetry; it's just not a symmetry that includes the time portion of the metric. Birkhoff's theorem says that, for a vacuum solution, the spatial symmetry under SO(3) *requires* the spacetime to be static, i.e., it implies something about the time portion of the metric; but that only applies to a vacuum solution. I'm not aware of any requirement that a non-vacuum solution have any time symmetry in order to be considered spherically symmetric, and I don't see why it would have to, since the key difference with a non-vacuum solution is that you can fill spacetime with a fluid whose density can be uniform in space, so it can still be spherically symmetric (isotropic), but can vary with time. The "density" of a vacuum can't vary with time, either because it's zero, or (if you include a cosmological constant) because it's a constant multiple of the metric.

This is fine but general covariance involves 4-D so it should imply something about the time portion of the metric, don't you think? So how can a manifold that is spherically symmetric in the standard terminology meaning that is isotropic, be also invariant for the form of physical laws under arbitrary coordinate transformations (this includes coordinate transformations that involve the time coordinate) without being also time symmetric?

 

[PeterDonis]

This is fine but general covariance involves 4-D so it should imply something about the time portion of the metric, don't you think? So how can a manifold that is spherically symmetric in the standard terminology meaning that is isotropic, be also invariant for the form of physical laws under arbitrary coordinate transformations (this includes coordinate transformations that involve the time coordinate) without being also time symmetric?

Sure, coordinate transformations include the time coordinate, but why should that have anything to do with time symmetry? General covariance doesn't say that the metric has to look identical in any coordinate system; it just says the laws of physics have to be the same in any coordinate system. Nor does general covariance say that the metric must have the same symmetry in every coordinate system; obviously a metric that looks isotropic in one coordinate system, will not look isotropic in a coordinate system that's in relative motion to the first. That's true even for a static metric; the metric of Schwarzschild spacetime won't look isotropic in a coordinate system that's moving relative to the black hole (nor will it look time-independent). But the Einstein Field Equation will still hold.

 

[TrickyDicky]

Peter, I guess we've reached a blind spot you are not able to get rid of.


Can some actual physicist look at what I'm saying in my previous post and give me an answer?

 

[TrickyDicky]

Is it not demanded in GR that the metric tensor and thus the line element must be generally covariant?

 

[PAllen]

Is it not demanded in GR that the metric tensor and thus the line element must be generally covariant?

That is basically achieved by definition. Given a completely arbitrary metric in one coordinates, its expression in all other coordinates are specified by the transformation rule for covariance. The definition of the this transform quite trivially guarantees that any computation of an invariant based on the metric tensor comes out the same. All observable physics is supposed to be defined in GR as some flavor of invariant or coordinate independent geometric quantity (generally involving the world line of the measuring instrument, so there is observer dependence but not coordinate dependence). Conversely, anything that can only be expressed in a coordinate dependent way cannot be a legitimate observable in GR.

 

[Ben Niehoff]

"Generally covariant" just means that the physics is independent of what coordinate system we use to describe it. It's another way to say that "coordinates have no intrinsic meaning". It does NOT mean that "the geometry of spacetime must have no distinguishing features".

The Schwarzschild geometry is static and has spherical symmetry, but these are coordinate-independent concepts. "Static" means that there is a timelike Killing vector which can be globally written as the gradient of some scalar function. "Spherically symmetric" means that there are three spacelike Killing vectors whose Lie algebra is that of SO(3).

In standard Schwarzschild coordinates, both of these symmetries are manifest, because

1. There are no metric functions depending on t,
2. There are no cross terms between dt and any other basis 1-form,
3. There are no metric functions depending on the angular coordinates.

In other coordinate systems, the symmetries might not be manifest. For example, in Kruskal coordinates, the time-translation symmetry is not manifest. And in boosted coordinates, neither time-translation nor spherical symmetry are manifest.

But even if the symmetries are not manifest, they are still there. This is what is implied by general covariance. In Kruskal coordinates, one can still find a timelike Killing vector that is the gradient of some scalar function. In boosted coordinates, one can still find a timelike Killing vector, and three spacelike Killing vectors that generate SO(3). That is because these notions are geometric properties that do not depend on the coordinate system.

The Schwarzschild geometry, independently of any coordinate system, DOES have some distinguishing features:

1. Every point has a preferred frame, given by the timelike Killing vector. An observer in this preferred frame is called "static",
2. There is a preferred point in space, the "center of the universe", where the singularity is. This point is picked out because it is the one point left invariant by SO(3) rotations generated by the spacelike Killing vectors,
3. There is a trapped null surface, the event horizon. This is where the timelike Kiling vector momentarily becomes null (as it transitions to being spacelike on the interior). (Note that the interior portion of Schwarzschild is not static, because it doesn't have a timelike Killing vector.)

None of these features has anything to do with general covariance. These are geometrical properties of the spacetime itself, and they will show up in any coordinate description. General covariance is precisely the idea that any real, physical feature of the spacetime must have exactly this property: that it can be defined and exists independently of any coordinate system.

So a preferred frame may exist and be generally-covariant, if it is an actual geometrical feature and can be defined independently of coordinates.

 

[TrickyDicky]

"Generally covariant" just means that the physics is independent of what coordinate system we use to describe it. It's another way to say that "coordinates have no intrinsic meaning". It does NOT mean that "the geometry of spacetime must have no distinguishing features".

The Schwarzschild geometry is static and has spherical symmetry, but these are coordinate-independent concepts. "Static" means that there is a timelike Killing vector which can be globally written as the gradient of some scalar function. "Spherically symmetric" means that there are three spacelike Killing vectors whose Lie algebra is that of SO(3).

In standard Schwarzschild coordinates, both of these symmetries are manifest, because

1. There are no metric functions depending on t,
2. There are no cross terms between dt and any other basis 1-form,
3. There are no metric functions depending on the angular coordinates.

In other coordinate systems, the symmetries might not be manifest. For example, in Kruskal coordinates, the time-translation symmetry is not manifest. And in boosted coordinates, neither time-translation nor spherical symmetry are manifest.

But even if the symmetries are not manifest, they are still there. This is what is implied by general covariance. In Kruskal coordinates, one can still find a timelike Killing vector that is the gradient of some scalar function. In boosted coordinates, one can still find a timelike Killing vector, and three spacelike Killing vectors that generate SO(3). That is because these notions are geometric properties that do not depend on the coordinate system.

The Schwarzschild geometry, independently of any coordinate system, DOES have some distinguishing features:

1. Every point has a preferred frame, given by the timelike Killing vector. An observer in this preferred frame is called "static",
2. There is a preferred point in space, the "center of the universe", where the singularity is. This point is picked out because it is the one point left invariant by SO(3) rotations generated by the spacelike Killing vectors,
3. There is a trapped null surface, the event horizon. This is where the timelike Kiling vector momentarily becomes null (as it transitions to being spacelike on the interior). (Note that the interior portion of Schwarzschild is not static, because it doesn't have a timelike Killing vector.)

None of these features has anything to do with general covariance. These are geometrical properties of the spacetime itself, and they will show up in any coordinate description. General covariance is precisely the idea that any real, physical feature of the spacetime must have exactly this property: that it can be defined and exists independently of any coordinate system.

So a preferred frame may exist and be generally-covariant, if it is an actual geometrical feature and can be defined independently of coordinates.

Here there are some apparently contradictory statements about coordinate independence, physical features and geometry.
Do you claim that general covariance (diffeomorphism invariance of the manifold) is unrelated to the geometrical properties of the spacetime?

 

[TrickyDicky]

That is basically achieved by definition. Given a completely arbitrary metric in one coordinates, its expression in all other coordinates are specified by the transformation rule for covariance. The definition of the this transform quite trivially guarantees that any computation of an invariant based on the metric tensor comes out the same. All observable physics is supposed to be defined in GR as some flavor of invariant or coordinate independent geometric quantity (generally involving the world line of the measuring instrument, so there is observer dependence but not coordinate dependence). Conversely, anything that can only be expressed in a coordinate dependent way cannot be a legitimate observable in GR.

Thanks, this is my understanding too.

 

[PeterDonis]

Thanks, this is my understanding too.

And mine. However, I would also answer "yes" to this question:

Do you claim that general covariance (diffeomorphism invariance of the manifold) is unrelated to the geometrical properties of the spacetime?

The only clarification I would make is that, as Ben Niehoff said, geometrical properties of the spacetime must be expressible in generally covariant form. But there is no requirement that a generally covariant spacetime have any particular set of geometric properties, so in that sense those properties are unrelated to general covariance.

I don't find anything to object to in Ben Niehoff's post, and I don't think anything in it contradicts anything in PAllen's post. At least, I don't if we are talking about manifolds that are possible solutions of the Einstein Field Equation, since that seems to me to be a requirement for something to be called a "spacetime". Can you give an example of a manifold that is a solution to the EFE but is *not* diffeomorphism invariant?

 

[TrickyDicky]

It does NOT mean that "the geometry of spacetime must have no distinguishing features".

You stress this as if it had anything to do with something I have said. I have never implied anything like this.

1. Every point has a preferred frame

None of these features has anything to do with general covariance.

So a preferred frame may exist and be generally-covariant, if it is an actual geometrical feature and can be defined independently of coordinates.

I don't get this, first you say this feature has nothing to do with general covariance and then you say is generally covariant.

 

[TrickyDicky]

However, I would also answer "yes" to this question:
Originally Posted by TrickyDicky
Do you claim that general covariance (diffeomorphism invariance of the manifold) is unrelated to the geometrical properties of the spacetime?

The only clarification I would make is that, as Ben Niehoff said, geometrical properties of the spacetime must be expressible in generally covariant form. But there is no requirement that a generally covariant spacetime have any particular set of geometric properties, so in that sense those properties are unrelated to general covariance.

Admittedly this is a subtle and tricky point, and the one that i would like to clarify so that I (and maybe others) can make some progress.
As you say there is no requirement that spacetime have any particular set of geometric properties, but if I understood there is a requirement that the spacetime be generally covariant. That general covariance might require certain geometrical properties.

Can you give an example of a manifold that is a solution to the EFE but is *not* diffeomorphism invariant?

I'm trying to understand how time asymmetric physical laws can be expressed in generally covariant form with the FRW metric.

 

[TrickyDicky]

This quote from MTW shows others have been confused by this:


"Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion."
I'm not sure the blame should be on 1917 mathematics though.

 

[PeterDonis]

I'm trying to understand how time asymmetric physical laws can be expressed in generally covariant form with the FRW metric.

That part seems simple to me, since there is already a time asymmetric quantity in the FRW metric: the scale factor a(t). (One could perhaps argue that a(t) is time symmetric for the case of a closed universe, since there is a time of "maximum expansion" and a(t) is symmetric about that time; but that only applies to a closed universe, so a(t) is certainly time asymmetric for an open universe.) This seems to me to be a good "geometric object", because it can be defined independently of coordinates; it is the distance between "comoving" worldlines, which are themselves defined independently of coordinates, as it varies along the family of spacelike hypersurfaces orthogonal to those worldlines, which are also defined independently of coordinates. So just transform the FRW metric to any coordinate system other than the "comoving" one, and see how a(t) transforms; that will give at least one example of how a time asymmetric quantity can be expressed in generally covariant form. I don't have time to work this example explicitly right now, but it certainly seems doable.

Granted, a(t) by itself isn't exactly a "physical law"; but since expressing a time asymmetric physical law basically boils down to expressing time asymmetric physical quantities, it seems like the above approach would work. For example, the second law is written in terms of entropy, and I can think of at least one candidate for a "geometric object" that could represent entropy: a function of the stress-energy tensor (since entropy is related to energy). Then just figure out how this geometric object transforms under a change of coordinates, and you can translate the second law into any coordinates you like.

 

[PAllen]

And mine. However, I would also answer "yes" to this question:



The only clarification I would make is that, as Ben Niehoff said, geometrical properties of the spacetime must be expressible in generally covariant form. But there is no requirement that a generally covariant spacetime have any particular set of geometric properties, so in that sense those properties are unrelated to general covariance.

I don't find anything to object to in Ben Niehoff's post, and I don't think anything in it contradicts anything in PAllen's post. At least, I don't if we are talking about manifolds that are possible solutions of the Einstein Field Equation, since that seems to me to be a requirement for something to be called a "spacetime". Can you give an example of a manifold that is a solution to the EFE but is *not* diffeomorphism invariant?

I also see not the slightest disagreement between my understanding and what Ben Niehoff wrote. I would not have bothered saying anything if I saw Ben's post (which covered a lot more ground than mine) before mine (perils of simul-posting).

In particular, I don't see general covariance as posing any limitations whatsoever on manifold geometry.

The "no prior geometry" or "no absolute geometric objects" or "symmetry group of the theory is the MMG" that various authors have used to provide content missing from general covariance is indeed a thorny subject that I think is not yet fully resolved. I've been following this off and on for 15 years and keep seeing new papers overturning the results of prior paper. It is a small research niche. My current opinion is that there is not yet a bulletproof formalization of a symmetry that leads uniquely to Einstein's gravity without either implicitly assuming it or allowing other theories as well.

Note also that there is a degenerate sense that any metric is a solution of Einstein's equations: just derive the Einstein tensor for it and call it the stress energy tensor (this has somehow gotten to be called Synge's method, though he argued against it not in favor of it). The various flavors of "energy conditions" then try to rule out ludicrous solutions (some include the feature that inertial bodies of non-zero size follow spacelike trajectories !). Unfortunately, this avenue is not fully satisfactory yet either. Conditions tight enough to rule out nonsense also rule out physically plausible solutions.

 

[TrickyDicky]

That part seems simple to me, since there is already a time asymmetric quantity in the FRW metric: the scale factor a(t). (One could perhaps argue that a(t) is time symmetric for the case of a closed universe, since there is a time of "maximum expansion" and a(t) is symmetric about that time; but that only applies to a closed universe, so a(t) is certainly time asymmetric for an open universe.) This seems to me to be a good "geometric object", because it can be defined independently of coordinates; it is the distance between "comoving" worldlines, which are themselves defined independently of coordinates, as it varies along the family of spacelike hypersurfaces orthogonal to those worldlines, which are also defined independently of coordinates. So just transform the FRW metric to any coordinate system other than the "comoving" one, and see how a(t) transforms; that will give at least one example of how a time asymmetric quantity can be expressed in generally covariant form. I don't have time to work this example explicitly right now, but it certainly seems doable.

Granted, a(t) by itself isn't exactly a "physical law"; but since expressing a time asymmetric physical law basically boils down to expressing time asymmetric physical quantities, it seems like the above approach would work. For example, the second law is written in terms of entropy, and I can think of at least one candidate for a "geometric object" that could represent entropy: a function of the stress-energy tensor (since entropy is related to energy). Then just figure out how this geometric object transforms under a change of coordinates, and you can translate the second law into any coordinates you like.

As you admit a(t) is not a physical law, is just a scale factor, and the stress-energy tensor is not entropy, so I guess this is not as simple as you think.

 

[TrickyDicky]

In particular, I don't see general covariance as posing any limitations whatsoever on manifold geometry.

Well, I thought that we have agreed that whatever geometrical features a manifold has they are generally covariant (independent of the coordinates). So general covariance of the metric looks very much like a type of geometrical constraint.It would be a way to make sure that the metric is really invariant for any general coordinate transformation
Let's not forget that when looking for plausible solutions of the EFE we always start imposing some geometrical limitations on the metric that seem justified by observation like for instance spherical symmetry (spatial isotropy).

Note also that there is a degenerate sense that any metric is a solution of Einstein's equations: just derive the Einstein tensor for it and call it the stress energy tensor (this has somehow gotten to be called Synge's method, though he argued against it not in favor of it). The various flavors of "energy conditions" then try to rule out ludicrous solutions (some include the feature that inertial bodies of non-zero size follow spacelike trajectories !). Unfortunately, this avenue is not fully satisfactory yet either. Conditions tight enough to rule out nonsense also rule out physically plausible solutions.

This is correct, this is why some geometrical limitations have to be imposed on the metric, I'm just saying that general covariance can be understood as one of this limitations that have to be taken into account in the construction of form of the metric.

 

[TrickyDicky]

After rereading a few times Ben Niehoff's post I finally came to understand it and also agree with what it explains.
Only thing I still can't figure out is where Ben gathered that I think "the geometry of spacetime must have no distinguishing features". But this is pretty irrelevant.

What I can't see answered in that post is how do we make sure the form of the metric that we are trying to find as a solution of the EFE is generally covariant? Or even if we need to make sure of that according with the GR theory.

 

[Ben Niehoff]

After rereading a few times Ben Niehoff's post I finally came to understand it and also agree with what it explains.
Only thing I still can't figure out is where Ben gathered that I think "the geometry of spacetime must have no distinguishing features". But this is pretty irrelevant.

Earlier you seemed to be claiming that a preferred time direction in the FLRW universe was incompatible with general covariance. Hopefully now you see they have nothing to do with each other.

What I can't see answered in that post is how do we make sure the form of the metric that we are trying to find as a solution of the EFE is generally covariant? Or even if we need to make sure of that according with the GR theory.

I'm not sure if I follow. Do you mean, how can we assume the metric has some specific form, given that we know the spacetime has certain symmetries?

If so, then the point is this: If the spacetime has certain symmetries, then we know that there must exist some coordinate system, in some open patch, in which the metric has a form making those symmetries manifest. Since the EFE are generally covariant (by design, being constructed of geometric quantities), we know that it is sufficient to obtain a solution in such a coordinate system. Extending the solution outside of the open patch can be done by geodesic continuation. This is what happens when we extend Schwarzschild using Kruskal-Szekeres coordinates.

One caveat here: we can't always make ALL the symmetries manifest at the same time. This is true even in flat Minkowski space: In Cartesian coordinates, the translational symmetries are manifest, but the spherical symmetry is not; conversely in spherical coordinates, the spherical symmetry is manifest, but the translational symmetries are not.

 

[TrickyDicky]

Earlier you seemed to be claiming that a preferred time direction in the FLRW universe was incompatible with general covariance. Hopefully now you see they have nothing to do with each other.

Ok, now I see where that came from. I am indeed still not totally sure about that but I tend to agree.

I'm not sure if I follow. Do you mean, how can we assume the metric has some specific form, given that we know the spacetime has certain symmetries?

If so, then the point is this: If the spacetime has certain symmetries, then we know that there must exist some coordinate system, in some open patch, in which the metric has a form making those symmetries manifest. Since the EFE are generally covariant (by design, being constructed of geometric quantities), we know that it is sufficient to obtain a solution in such a coordinate system.

Actually, this is not exactly what I meant with my question.
If you look at my answer to PAllen, there I suggest that the fact that there are solutions of the EFE that are totally unphysical should warn us that not every metric that is a cosmological solution of the EFE is generally covariant, thus my question: must the metric be generally covariant according to GRT? And if so how do we ascertain that? You seem to imply that the mere fact that the EFE are generally covariant (are tensor equations and tensors are generally covariant objects) assures that the particular metrics that are cosmological solutions of these EFE are generally covariant, is this the case?

 

[TrickyDicky]

So a preferred frame may exist and be generally-covariant, if it is an actual geometrical feature and can be defined independently of coordinates.

Ok, but clearly in the case of the FRW metric the preferred frame is not coordinate independent. So according to your own statement is not genearally covariant, and its consequences should be unphysical. I don't get this.

 

[PAllen]

If you look at my answer to PAllen, there I suggest that the fact that there are solutions of the EFE that are totally unphysical should warn us that not every metric that is a cosmological solution of the EFE is generally covariant, thus my question: must the metric be generally covariant according to GRT? And if so how do we ascertain that? You seem to imply that the mere fact that the EFE are generally covariant (are tensor equations and tensors are generally covariant objects) assures that the particular metrics that are cosmological solutions of these EFE are generally covariant, is this the case?

How to decide plausibility of EFE solutions is certainly an issue. General covariance has nothing to do with the solution to this issue. Specify any metric at all on a topological manifold in some set of coordinate patches. Then achieve general covariance purely by definition: the metric expressed in any other coordinates is that given by the tensor transformation law. No metric, is excluded by this definition, even the most physically implausible ones. Similarly, any arbitrary metric can be treated as an EFE solution (as I explained in my earlier post).

Even formulations like "no prior geometry" don't rule out any metrics. They just aim to rule out other metric theories besides the EFE (i.e. different relationships between geometry and physics).

To rule out unphysical metrics you need to add additional constraints on what are plausible stress energy tensors. These are the various "energy conditions". These are limitations motivated by physical interpretation of the stress energy tensor; they have no relation at all to general covariance or theory constraints like "no prior geometry".

 

[PeterDonis]

As you admit a(t) is not a physical law, is just a scale factor, and the stress-energy tensor is not entropy

Yes, I apologize for dashing that last post off quickly and not being very careful in consequence. a(t) itself, as it appears in the standard form of the FRW metric, is a scale factor, but it is related to a geometric invariant. I believe the proper geometric invariant is the expansion of the congruence of "comoving" worldlines, and that in FRW coordinates, the expansion is expressed, in terms of the scale factor, as:

$$\theta = \frac{1}{a} \frac{da}{dt}$$

(I say "I believe" because I haven't been able to find a reference that explicitly gives the formula for the expansion for FRW spacetime in FRW coordinates, and I haven't had time to do the calculation myself. The above formula is what looks right to me based on my understanding of what the expansion means physically.)

The fact that "the universe is expanding" is then expressed as the fact that $\theta$ is always positive, and since $\theta$ is a geometric invariant, if it is positive in one coordinate system (such as the standard FRW coordinates), it is positive in any coordinate system (more precisely, in any coordinate system with the same direction of time as standard FRW coordinates). So we have a way to express a time asymmetric fact, that the universe is expanding, in a generally covariant form. If we can do that, we should similarly be able to express a time asymmetric physical law in a generally covariant form.

With regard to entropy and the stress-energy tensor, I said entropy may be a *function* of the stress-energy tensor, not that it was the same as the stress-energy tensor. But that was dashed off quickly too; I need to consider this more before making any further suggestion about how to represent entropy, specifically, in a generally covariant form. One thing that has occurred to me is that the FRW model may not be capable of capturing entropy as we normally understand it, since the FRW model assumes a cosmological fluid of uniform density, and a lot of the entropy in our actual universe is due to gravitational clumping of matter into galaxies, stars, and especially black holes. So even if the second law can be captured in generally covariant form, it may not be possible to use the FRW spacetime to do it; a more complicated model may be needed.

 

[TrickyDicky]

How to decide plausibility of EFE solutions is certainly an issue. General covariance has nothing to do with the solution to this issue. Specify any metric at all on a topological manifold in some set of coordinate patches. Then achieve general covariance purely by definition: the metric expressed in any other coordinates is that given by the tensor transformation law. No metric, is excluded by this definition, even the most physically implausible ones. Similarly, any arbitrary metric can be treated as an EFE solution (as I explained in my earlier post).

Even formulations like "no prior geometry" don't rule out any metrics. They just aim to rule out other metric theories besides the EFE (i.e. different relationships between geometry and physics).

To rule out unphysical metrics you need to add additional constraints on what are plausible stress energy tensors. These are the various "energy conditions". These are limitations motivated by physical interpretation of the stress energy tensor; they have no relation at all to general covariance or theory constraints like "no prior geometry".

I see what you mean, I suddenly realized that by definition of manifold the general covariance is automatically obtained for the EFE solutions. No need to impose it or ascertain it.
Well it seems I was the one with the blind spot.

 

[TrickyDicky]

if it is positive in one coordinate system (such as the standard FRW coordinates), it is positive in any coordinate system (more precisely, in any coordinate system with the same direction of time as standard FRW coordinates).

I still have problems with the bolded phrase. This seems like a coordinate condition.

With regard to entropy and the stress-energy tensor, I said entropy may be a *function* of the stress-energy tensor, not that it was the same as the stress-energy tensor. But that was dashed off quickly too; I need to consider this more before making any further suggestion about how to represent entropy, specifically, in a generally covariant form. One thing that has occurred to me is that the FRW model may not be capable of capturing entropy as we normally understand it, since the FRW model assumes a cosmological fluid of uniform density, and a lot of the entropy in our actual universe is due to gravitational clumping of matter into galaxies, stars, and especially black holes. So even if the second law can be captured in generally covariant form, it may not be possible to use the FRW spacetime to do it; a more complicated model may be needed.

This is very close to my thinking abou this issue.
See my last question to Ben Niehoff.

 

[Haelfix]

I sense some confusion about the following..

Forget about diffeomorphism invariance for a second.

Imagine a solution to a theory of classical physics, eg one that has gallilean invariance. Clearly we know that rotational invariance is a subgroup of this invariance group, and indeed the laws of physics must be and are invariant under rotations.

However a particular solution of the equations of motion need not be! For instance, if you are talking about the Earth orbiting the sun, one can see that the solution breaks the rotational invariance of the physics (eg you can't flip Earth and sun). In that case, the initial conditions of the Earth picks out a preferred coordinate system. However you can still change coordinate systems (eg cartesian to polar for instance)!

So when we talk about solutions to the Einstein field equations, a similar thing occurs. Any state that is specified by a given Cauchy data or a four metric will in general break the full diffeormorphism invariance of the theory down to a finite number of isometries that remain unbroken. So for instance the Minkowski metric breaks the diffeomorphism group down to 10 generators (4 translations, 3 rotations, and 3 boosts). That doesn't mean you can't change coordinates in Minkowski space! All it means is that you have to take the pullback!