Worldline congruence and general covariance

[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!

 

[PeterDonis]

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

It's true that you can define coordinates with either direction of time. For example, I could define "inverted" FRW coordinates where the sign of t was reversed and nothing else was changed; in those coordinates, the expansion would be negative and the universe would be "contracting". However, there would be no way to do a Lorentz transformation at any event between those "inverted" coordinates and standard FRW coordinates (more precisely, between a local patch of one and a local patch of the other). So local Lorentz invariance is enough to ensure that, if we pick a direction of time in one coordinate system, any others that we relate to it must have the same direction of time. I guess if we wanted to be really careful, we would have to say that local Lorentz invariance is part of "general covariance", so general covariance does require you to pick a time orientation. In principle you could pick either one (since the expanding and contracting FRW models are both valid solutions of the EFE), but since we actually observe the universe to be expanding in the same direction of time as we feel ourselves to be "moving", we pick that direction of time as the "future".

 

[TrickyDicky]

It's true that you can define coordinates with either direction of time. For example, I could define "inverted" FRW coordinates where the sign of t was reversed and nothing else was changed; in those coordinates, the expansion would be negative and the universe would be "contracting". However, there would be no way to do a Lorentz transformation at any event between those "inverted" coordinates and standard FRW coordinates (more precisely, between a local patch of one and a local patch of the other). So local Lorentz invariance is enough to ensure that, if we pick a direction of time in one coordinate system, any others that we relate to it must have the same direction of time. I guess if we wanted to be really careful, we would have to say that local Lorentz invariance is part of "general covariance", so general covariance does require you to pick a time orientation. In principle you could pick either one (since the expanding and contracting FRW models are both valid solutions of the EFE), but since we actually observe the universe to be expanding in the same direction of time as we feel ourselves to be "moving", we pick that direction of time as the "future".

Sure, but you are ultimately agreeing here that in the FRW metric, expansion/contraction is coordinate dependent, in a similar way that entropy effects would be (as you say "we" are the ones that pick the correct time coordinate afterwards), and we all know this dependency is generally considered the hallmark of an unphysical effect. (See Ben's post)

 

[PeterDonis]

Sure, but you are ultimately agreeing here that in the FRW metric, expansion/contraction is coordinate dependent

I think you are using the term "coordinate dependent" here in a different sense than Ben was using it in his post. I would use the term "direction of time dependent", which is just another way of saying "time asymmetric". But the time asymmetry can be described in entirely coordinate independent terms, except for the choice of which side of the asymmetry is to be called the "future" side, i.e., except for picking a direction of time. (For example, in the FRW spacetime--we'll assume we're talking about the case where it doesn't recollapse, to avoid any issues with the closed, recollapsing version being "time symmetric"--the time asymmetry can be described in coordinate-independent terms simply by saying that there is a curvature singularity at one "end" of time, but not the other. The only question then is whether we call that end the "past" or the "future" end. See below.)

But as I've said several times already, the only thing required to pick a direction of time is to pick which half of the light cone is the "future" half, and in any spacetime meeting a very general set of conditions, which I outlined earlier (and FRW spacetime, as well as any other spacetime that's been considered as physically reasonable, as far as I know, meets those conditions), once you've made that choice at any particular event, you can continuously extend it throughout the spacetime. And since the light cones are invariant geometric features of the spacetime (i.e., they are not coordinate dependent), the choice of which half of the light cones is the "future" half will also be coordinate independent, except for the (trivial, in my view--but see below) fixing of the sign of the time coordinate.

So I guess what this boils down to is: the fact that a particular spacetime is time asymmetric is not coordinate dependent. And given a choice of which half of the light cones is the "future" half, the physics in such a spacetime is not coordinate dependent. That choice itself could be considered "coordinate dependent" in the sense that it fixes the sign of the time coordinate; but I don't think this is a big issue, because it's inherent in the very fact that the spacetime is time asymmetric.

I should also stress that I am not saying there are no interesting physical questions left once we've done everything I describe above. It is definitely an interesting physical question *why* we find that the direction of time we experience in our consciousness is the same direction of time in which the universe is expanding. (There is also the question of why it's the same direction of time in which the second law holds, but I answered that in an earlier post: our conscious perception of time depends on the formation of memories, and the formation of memories requires entropy increase.) But that question has nothing to do with general covariance, precisely because we can formulate it in coordinate-independent terms (I pretty much just did; if someone insists on pedantic exactitude, just rephrase what I said above in terms of light cones), so it will arise regardless of how we assign coordinates. Even if we adopt coordinates in which the sign of time is reversed (so we say the universe is "contracting" instead of "expanding"), the same question still arises: we just phrase it as "why is the universe contracting in the same direction of time in which we remember things?" instead of "why is the universe expanding in the same direction of time in which we anticipate things?" There's no physical difference between these versions of the question; it's just a difference in wording.

 

[TrickyDicky]

I think you are using the term "coordinate dependent" here in a different sense than Ben was using it in his post. I would use the term "direction of time dependent", which is just another way of saying "time asymmetric". But the time asymmetry can be described in entirely coordinate independent terms, except for the choice of which side of the asymmetry is to be called the "future" side, i.e., except for picking a direction of time. (For example, in the FRW spacetime--we'll assume we're talking about the case where it doesn't recollapse, to avoid any issues with the closed, recollapsing version being "time symmetric"--the time asymmetry can be described in coordinate-independent terms simply by saying that there is a curvature singularity at one "end" of time, but not the other. The only question then is whether we call that end the "past" or the "future" end. See below.)

But as I've said several times already, the only thing required to pick a direction of time is to pick which half of the light cone is the "future" half, and in any spacetime meeting a very general set of conditions, which I outlined earlier (and FRW spacetime, as well as any other spacetime that's been considered as physically reasonable, as far as I know, meets those conditions), once you've made that choice at any particular event, you can continuously extend it throughout the spacetime. And since the light cones are invariant geometric features of the spacetime (i.e., they are not coordinate dependent), the choice of which half of the light cones is the "future" half will also be coordinate independent, except for the (trivial, in my view--but see below) fixing of the sign of the time coordinate.

So I guess what this boils down to is: the fact that a particular spacetime is time asymmetric is not coordinate dependent. And given a choice of which half of the light cones is the "future" half, the physics in such a spacetime is not coordinate dependent. That choice itself could be considered "coordinate dependent" in the sense that it fixes the sign of the time coordinate; but I don't think this is a big issue, because it's inherent in the very fact that the spacetime is time asymmetric.

I should also stress that I am not saying there are no interesting physical questions left once we've done everything I describe above. It is definitely an interesting physical question *why* we find that the direction of time we experience in our consciousness is the same direction of time in which the universe is expanding. (There is also the question of why it's the same direction of time in which the second law holds, but I answered that in an earlier post: our conscious perception of time depends on the formation of memories, and the formation of memories requires entropy increase.) But that question has nothing to do with general covariance, precisely because we can formulate it in coordinate-independent terms (I pretty much just did; if someone insists on pedantic exactitude, just rephrase what I said above in terms of light cones), so it will arise regardless of how we assign coordinates. Even if we adopt coordinates in which the sign of time is reversed (so we say the universe is "contracting" instead of "expanding"), the same question still arises: we just phrase it as "why is the universe contracting in the same direction of time in which we remember things?" instead of "why is the universe expanding in the same direction of time in which we anticipate things?" There's no physical difference between these versions of the question; it's just a difference in wording.

I find your position reasonable.
I just find the "time direction dependency" an awkward point. This direction property seems defining for the time coordinate.

 

[PeterDonis]

This direction property seems defining for the time coordinate.

I'm curious why it seems this way. There's no corresponding requirement for the space coordinates; nobody objects if I flip the direction of the x axis, let's say. Why should the time axis, as a coordinate, be any different? Is it just because our conscious experience picks out a direction of time and it seems "wrong" to pick a sign for the time coordinate that doesn't match that direction?

 

[TrickyDicky]

I'm curious why it seems this way. There's no corresponding requirement for the space coordinates; nobody objects if I flip the direction of the x axis, let's say. Why should the time axis, as a coordinate, be any different? Is it just because our conscious experience picks out a direction of time and it seems "wrong" to pick a sign for the time coordinate that doesn't match that direction?

Yeah, this is one of the old good questions that doesn't seem to have an answer within physics yet.