Worldline congruence and general covariance

[TrickyDicky]

I've read most of the posts here and I don't think anyone has mentioned that 'hyper-surface orthogonal' is equivalent (mathematically) to saying that it is possible to set up non-rotating local frames (NSIF's). Obviously spinning frames can't be inertial. This is why it can't be done for the Kerr spacetime.

Sorry, I can't understand what you mean here. Is this a reply to Peter Donis or to something I've said?

Edit: I mean I understand what you state and is correct but how does this respond to my question?

 

[PeterDonis]

...it is time traslation invariant. So it is not possible for them to agree on which direction of time is the "future"...

Why not? There is a choice to be made, I agree (and I've said so before), but there is nothing preventing different observers in different states of motion from both making the same choice. And the choice is discrete, not continuous; there is no way to continuously vary the choice from place to place in the spacetime, such that, for example, an observer at r = r1 makes one choice of which half of his local light cone is the "future", and another observer at r = r2 makes the opposite choice; if two such observers make opposite choices, there is no way to set up a continuous coordinate system that respects both choices, there has to be a discontinuity between them. So I don't understand why you think time translation invariance somehow invalidates having a continuous, consistent definition of which direction of time is the "future".

It does mean that all physical processes will "look the same" regardless of which choice of time direction you make; if that's what you were getting at, see my further comments later in this post.

...because first of all time doesn't flow in such manifolds...

Huh? If by "time doesn't flow" you just mean they're time translation invariant, then I don't see how saying "time doesn't flow" adds anything. If by "time doesn't flow" you mean that observers somehow don't experience time, sure they do; there is a well-defined notion of proper time along each timelike worldline. If, again, you mean that physical processes "look the same" in both directions of time, again, see further comments later in this post.

...and second we can't assume they make the same choice of which half of of the light cones is the "future" half because it is not timelike hypersurface orthogonal.

I don't understand how this follows. The presence of "cross" terms in the metric does not prevent a consistent choice, for observers at different events and/or in different states of motion, about which half of the light cones is the "future" half. Also, Schwarzschild spacetime is time translation invariant, but it *does* admit a coordinate chart which is hypersurface orthogonal (because it's static, not just stationary). And yet your other comments about the implications of time translation invariance would seem to apply to Schwarzschild spacetime as much as to Kerr spacetime.

In any case, Kerr spacetime as we know has nothing to do with our manifold and I'm asking about what happens in our non-stationary universe.

I agree the universe, unlike Kerr spacetime, is not stationary. But I think the above points do bear on the non-stationary example as well. The need to consistently choose a "future" half for each light cone in the spacetime exists for a non-stationary spacetime as well. For example, you say that the universe is "expanding". Why? A solution with the direction of time reversed, in which everything is exactly the same except that the universe is contracting instead of expanding, is equally consistent with the Einstein Field Equation. The only reason we say the universe is "expanding" is that we define the "future" half of the light cones according to our own experience of time flow; we remember times when the universe was smaller, and we look forward to times when the universe will be larger.

Similar remarks apply to the second law: we say that entropy is "increasing" because the physical process of memory, for example, requires entropy to increase as a memory is "formed", so again we experience time flow in the same direction as entropy increases.

Perhaps you mean that in a time translation invariant spacetime, there would be no such thing as entropy increasing? That if we really lived in a "pure" Kerr spacetime, the second law would not hold? If so, I would disagree, or more precisely I would insist on rephrasing the claim. If it were really physically possible for a completely time translation invariant spacetime to exist, I don't believe conscious beings could exist in it; so a "pure" Kerr spacetime, for example, would not have entropy increase only because it would not have any real change, or any conscious observers, at all. Strictly speaking, if one really takes time translation invariance seriously, it means that nothing can really change at all, and the existence of any kind of actual "observers" that can experience anything requires change.

But I can certainly imagine an "impure" Kerr spacetime, for example, in which the overall spacetime was (at least in a time averaged sense) time translation invariant, but in which entropy still increased in a given direction. That just means that General Relativity can't model physical processes at the level of detail required to treat things like entropy increase, or to handle processes like those in the brains of conscious beings. The overall GR solution would provide a background within which more detailed models, such as statistical mechanics, would be used to handle things like entropy and the second law. It's true that such a spacetime would not, in the strict sense, be fully time translation invariant, which is why I call it "impure"; but I could certainly see it being, on average, time translation invariant for a period of billions of years, long enough for intelligent life to develop.

I'm rambling, but I think the bottom line is that I still don't see how our physical observations of the universe expanding, the second law, etc., pick out a "preferred frame" in the GR sense. They do pick out a preferred "direction of time", in the sense that they show that the actual spacetime we live in is not stationary; but I don't see how they pick out anything more specific than that. Just saying that the spacetime is not stationary certainly does not pick out a "preferred frame" in the GR sense.

 

[TrickyDicky]

I'm rambling, but I think the bottom line is that I still don't see how our physical observations of the universe expanding, the second law, etc., pick out a "preferred frame" in the GR sense. They do pick out a preferred "direction of time", in the sense that they show that the actual spacetime we live in is not stationary; but I don't see how they pick out anything more specific than that. Just saying that the spacetime is not stationary certainly does not pick out a "preferred frame" in the GR sense.

Let's see if we make an effort to simplify this, just by choosing any coordinate system we are picking a preferred frame of reference (I don't mean the sense of frame as a state of motion here). There's no problem with that. What I'm getting at is that according to general covariance nothing physical, more specifically, no physical law should be modified by picking a certain frame, or a certain manifold slicing. I think we all agree on this.
Now my question is if certain laws like the second law of thermodynamics are tied to a certain slicing of the manifold (the one time hypersurface orthogonal), I would say this one is tied to it but I may be wrong. I haven't found any convincing post that shows otherwise yet.
I think it is perfectly possible to pick a set of coordinates that "slice up" spacetime in a way that makes worldlines intersect so that an observers shared second law of thermodynamics is made impossible.

 

[PeterDonis]

Now my question is if certain laws like the second law of thermodynamics are tied to a certain slicing of the manifold (the one time hypersurface orthogonal),

And I am saying that they are not. The laws you cite are tied to a choice of a direction of time, but that's all. Once that choice is made (e.g., we choose the "future" direction of time to be the one in which entropy increases, or the universe expands), you can still choose any coordinate system you like, so long as you define the direction of your time coordinate appropriately (which just amounts to being consistent and continuous in your labeling of which half of the light cones is the "future" half), and all the laws will still hold in it.

I think it is perfectly possible to pick a set of coordinates that "slice up" spacetime in a way that makes worldlines intersect so that an observers shared second law of thermodynamics is made impossible.

Whether a given pair of worldlines intersect or not is independent of the coordinate system; it's an invariant feature of the spacetime. If a given pair of worldlines intersect in one coordinate system, they intersect in any coordinate system. So I don't understand what you're proposing here.

 

[Mentz114]

Sorry, I can't understand what you mean here. Is this a reply to Peter Donis or to something I've said?

Edit: I mean I understand what you state and is correct but how does this respond to my question?

I latched on to the Kerr mention there - it's not relevant to your question. But now I'm here I'll give my two cents worth -

So can someone explain to me why we share all those physical observations if the congruence itself is not physical but a choice to make calculations in GR easier?

The congruence of comoving observers is not a 'preferred' frame except that it corresponds most closely to us. It would seem that it is a natural frame to calculate in because we have the best chance of matching our observations and calculations.

If u^\mu=(1,0,0,0) is a congruence then so is any Lorentz boosted frame, \Lambda^\mu_\nu u^\nu but they won't all be geodesic, and we believe we are on a geodesic or perhaps our galaxy is on a geodesic and we are part of that bound system.

I don't know if that addresses the question but I admit I can't see what your problem is.

 

[PeterDonis]

The congruence of comoving observers is not a 'preferred' frame except that it corresponds most closely to us. It would seem that it is a natural frame to calculate in because we have the best chance of matching our observations and calculations.

Actually, as I've noted in previous posts, it isn't, strictly speaking. Here on Earth we see a large dipole anisotropy in the CMBR, which indicates that we are *not* anywhere near at rest in the "comoving" frame. Even removing Earth's velocity in orbit about the Sun still leaves a large velocity relative to the "comoving" frame (about 600 km/s IIRC) for the center of mass of the Solar System. I'm not sure even subtracting the Solar System's velocity around the CoM of the Milky Way galaxy would put one at rest, within measurement error, relative to the "comoving" frame.

That said, the observation of the dipole anisotropy in the CMBR allows us to know, pretty accurately, what Lorentz transformation we need to apply to convert our actual raw data into "corrected" data in the comoving frame. We want to do that because calculating in the comoving frame is so much simpler that the effort saved more than makes up for the effort required to convert our data into that frame. So in practical terms you are right, we use the comoving frame because it is the most natural one in which to match data with models.

 

[Mentz114]

Actually, as I've noted in previous posts, it isn't, strictly speaking. Here on Earth we see a large dipole anisotropy in the CMBR, which indicates that we are *not* anywhere near at rest in the "comoving" frame. Even removing Earth's velocity in orbit about the Sun still leaves a large velocity relative to the "comoving" frame (about 600 km/s IIRC) for the center of mass of the Solar System. I'm not sure even subtracting the Solar System's velocity around the CoM of the Milky Way galaxy would put one at rest, within measurement error, relative to the "comoving" frame.

That said, the observation of the dipole anisotropy in the CMBR allows us to know, pretty accurately, what Lorentz transformation we need to apply to convert our actual raw data into "corrected" data in the comoving frame. We want to do that because calculating in the comoving frame is so much simpler that the effort saved more than makes up for the effort required to convert our data into that frame. So in practical terms you are right, we use the comoving frame because it is the most natural one in which to match data with models.

Yep. The FLRW dust cosmologies don't take into account any gravitational interaction between 'dust' particles. I wonder at what scale we can ignore this. Dust particles are clusters of clusters maybe ? It's a difficult problem trying to add a gravitational potential to a cosmological model at any scale.

 

[TrickyDicky]

And I am saying that they are not. The laws you cite are tied to a choice of a direction of time, but that's all. Once that choice is made (e.g., we choose the "future" direction of time to be the one in which entropy increases, or the universe expands), you can still choose any coordinate system you like, so long as you define the direction of your time coordinate appropriately (which just amounts to being consistent and continuous in your labeling of which half of the light cones is the "future" half), and all the laws will still hold in it.

Well I guess that I'll have to find a way to show you that that choice of a direction is what the second law is all about.

Whether a given pair of worldlines intersect or not is independent of the coordinate system; it's an invariant feature of the spacetime. If a given pair of worldlines intersect in one coordinate system, they intersect in any coordinate system. So I don't understand what you're proposing here.

Here I must tell you you have some important misconception about GR and manifolds. You can get to coordinates that make worldlines intersect by a coordinate transformation and any coordinate transformation is allowed in GR. Just think of any transformation that produces timelike worldlines that are not orthogonal to the spacelike hypersurfaces.
If this was an invariant feature of spacetime, why should we use the Weyl postulate to begin with?

 

[tom.stoer]

Here I must tell you you have some important misconception about GR and manifolds. You can get to coordinates that make worldlines intersect by a coordinate transformation and any coordinate transformation is allowed in GR.

For two intersecting worldlines we can calculate their distance s and for the intersection point we will find find s²=0; but this s² is an invariant. Otherwise one observer would see two objects meeting each other (in his reference frame), whereas a second observer would not see something different. This is only possible if some coordinate singularity is introduced.

 

[PAllen]

Here I must tell you you have some important misconception about GR and manifolds. You can get to coordinates that make worldlines intersect by a coordinate transformation and any coordinate transformation is allowed in GR. Just think of any transformation that produces timelike geodesics that are not orthogonal to the spacelike hypersurfaces.
If this was an invariant feature of spacetime, why should we use the Weyl postulate to begin with?

Intersection of two world lines is a collision between test bodies following them. You believe that a coordinate transform can change whether or not two bodies collide ??!

 

[WannabeNewton]

Here I must tell you you have some important misconception about GR and manifolds. You can get to coordinates that make worldlines intersect by a coordinate transformation and any coordinate transformation is allowed in GR. Just think of any transformation that produces timelike geodesics that are not orthogonal to the spacelike hypersurfaces.
If this was an invariant feature of spacetime, why should we use the Weyl postulate to begin with?

Actually, if you initially have a congruence of geodesics then Raychaudri's equation \frac{\mathrm{d} \theta }{\mathrm{d} \tau } = -\frac{1}{3}\theta ^{2} - \sigma _{\mu \nu }\sigma ^{\mu \nu } + \omega _{\mu \nu }\omega ^{\mu \nu } - R_{\mu \nu }U^{\mu }U^{\nu } (where \theta ,\sigma ,\omega is the expansion, shear, and rotation of the congruence respectively and \mathbf{U} is the tangent vector field to the congruence) can be solved to find when there is an intersection and clearly the intersection is invariant; you are solving for a scalar.

 

[PeterDonis]

Well I guess that I'll have to find a way to show you that that choice of a direction is what the second law is all about.

If you mean a choice of direction in time, it should be evident from what I've said already that I agree that choosing a direction in time is what the second law is all about. But there are *lots* of coordinate systems for any given spacetime that all share the same direction of time.

Here I must tell you you have some important misconception about GR and manifolds. You can get to coordinates that make worldlines intersect by a coordinate transformation and any coordinate transformation is allowed in GR. Just think of any transformation that produces timelike geodesics that are not orthogonal to the spacelike hypersurfaces.

As you can see, I'm not the only one that finds this assertion highly questionable. However, you do go on to ask another question:

If this was an invariant feature of spacetime, why should we use the Weyl postulate to begin with?

Let me walk through the steps of reasoning involved as I see them:

(1) We observe that our universe is not stationary; there is a definite "future" direction of time which is the direction in which we experience time (we remember the past, not the future), in which entropy increases (the second law holds), and in which the universe is expanding.

(2) Because of #1, any physical model that applies to our universe as a whole has to use a spacetime which is not stationary. But there are *lots* of possible solutions to the Einstein Field Equation which are not stationary, and in which all of the observations cited in #1 would hold. That set of constraints, by itself, simply doesn't "pin down" the model enough to be workable.

(3) We then observe that, if we correct for the dipole anisotropy in the CMBR, the universe as a whole appears homogeneous and isotropic (provided we average over a large enough distance scale). We also observe that individual galactic clusters in the universe do not appear to interact with each other; each individual cluster's motion is basically independent of all the others.

(4) Combining #3 with #1 focuses our attention on spacetimes in which the universe is (a) not stationary (expanding), and (b) composed of a homogeneous, isotropic perfect fluid in which individual galactic clusters are the fluid "particles", each of which is moving on a geodesic worldline. These are the FRW spacetimes.

(I should note that I'm leaving out two technicalities: first, that as we go back in time to when the universe was much smaller, hotter, and denser, the equation of state of the "fluid" changes. What I just described is the "dust" model, in which there is zero pressure, which applies to the period when the universe is matter-dominated. In the far past, when the universe was radiation-dominated, the equation of state was different, and did not have zero pressure. Second, the best-fit model of the universe today is actually not matter-dominated but dark energy-dominated, and dark energy has an equation of state with *negative* pressure, which is why the expansion of the universe is accelerating. I'm leaving all this out because it doesn't affect the main point for this thread, but I wanted to be clear that the technicalities are there.)

(5) Once we know we're dealing with a FRW spacetime, then the question arises, what is the best coordinate chart to use? Obviously there are still many possibilities, such as the "Solar System centric" coordinates I described in a previous post; but as I and others have noted, the most natural coordinates to use are the standard FRW coordinates. The Weyl postulate basically summarizes *why* these are the most natural coordinates: the postulate amounts to saying that, if the spacetime you are working with admits a chart with the properties given in the postulate, you should use it, because it will be simpler than any other chart you could find.

Note that the Weyl postulate, as I've just described it, does *not* make any actual assertion about the physics. You have to already *know* the physics--that you're working with an FRW spacetime--before you even ask the question what coordinate chart to use, and the Weyl postulate is all about coordinate charts, and nothing else. In so far as the Weyl postulate says that the coordinate chart it recommends is "preferred", that is purely an assertion about convenience, not about physics. It certainly does *not* say that, simply by adopting the Weyl postulate, you can *make* the spacetime into one that satisfies it. You have to already know the spacetime admits a chart satisfying the conditions of the postulate, before you can adopt it.

 

[PeterDonis]

Just to throw one more thing into the foodmixer: one could take an alternative viewpoint in which the Weyl postulate basically comes in at stage #3 of the steps in the reasoning, instead of #5. I say this because Weyl apparently first made the postulate in 1923, when we didn't know about the CMBR at all, let alone that correcting for the dipole anisotropy in it resulted in a highly homogeneous and isotropic set of data. (Not to mention all the other evidence for homogeneity and isotropy.) Under those conditions, the reasoning could go like this:

(3) We don't really know what the large-scale structure of the universe looks like, so let's work from the other end: what is the simplest possible type of model we could construct? The answer is the homogeneous and isotropic FRW-type model, described in the standard FRW coordinate chart, which meets the conditions of the Weyl postulate. In other words, Weyl was basically saying, why not try the simplest possible model and see how well it works?

(4) So we develop this type of model (the FRW models, which were developed following Weyl's statement of the postulate), and start making predictions and comparing them with data. Lo and behold, it turns out the models work well. Now that we have the CMBR data, we can see that they hold to a pretty high degree of accuracy (deviations from isotropy in the CMBR, once the dipole is subtracted, only show up at about 1 part in 100,000, for example).

(5) So we conclude that, as a matter of experimental fact, the actual spacetime in which we live does in fact admit a coordinate chart which satisfies the conditions of the Weyl postulate to a pretty good approximation.

This may be a better description of the actual historical path of reasoning that was followed. But note that, on this view, the Weyl postulate still does not make any claim about a particular coordinate chart being "preferred"; it just observes that one particular type of chart is simpler, so it would be nice if the actual universe met the conditions for a spacetime to admit such a chart, at least to a good approximation.

 

[PeterDonis]

According to Weyl's postulate timelike geodesics should be hypersurface orthogonal

And just for one more observation, the above quote (from the OP in the thread) can be read as mis-stating the postulate. The postulate does not say that *all* timelike geodesics must be hypersurface orthogonal. It also does not state that the particular timelike geodesics that any actual observers (such as us on Earth) or galaxies that we observe, are following must be hypersurface orthogonal. It only postulates that there should be *some* family of timelike geodesics in the spacetime (the "comoving" ones) that are hypersurface orthogonal.

 

[tom.stoer]

The other way round: every timelike geodesic locally defines a hypersurface to which it is orthogonal.

 

[TrickyDicky]

Just to throw one more thing into the foodmixer: one could take an alternative viewpoint in which the Weyl postulate basically comes in at stage #3 of the steps in the reasoning, instead of #5. I say this because Weyl apparently first made the postulate in 1923, when we didn't know about the CMBR at all, let alone that correcting for the dipole anisotropy in it resulted in a highly homogeneous and isotropic set of data. (Not to mention all the other evidence for homogeneity and isotropy.

Just a precision. in 1923, when Weyl came up with his postulate,all models were static, we not only didn't know about CMBR there was no FRW model and not a single clue that ours was a non-stationary universe, the notion of expansion was totally unknown so no physics could be attributed to it.
It follows that your reasoning is not historically accurate, and therefore leaves my question unanswered.

 

[TrickyDicky]

For two intersecting worldlines we can calculate their distance s and for the intersection point we will find find s²=0; but this s² is an invariant. Otherwise one observer would see two objects meeting each other (in his reference frame), whereas a second observer would not see something different. This is only possible if some coordinate singularity is introduced.

Clearly You are not interpreting correctly what I'm saying.
The invariant s^2=0 is also an invariant in a coordinate system without the Weyl condition.

Intersection of two world lines is a collision between test bodies following them. You believe that a coordinate transform can change whether or not two bodies collide ??!

I'm saying nothing that implies what you claim. You must be mixing a 4-dim manifold spacetime with a 3-dim space. Certainly a coordinate transformation doesn't change the physics, but you make it sound as if there weren't collisions in a FRW metric. And if we model a physical collision and make a coordinate transformation to a non-orthogonal hypersurface coordinate system that collision should be also there.
Are you denying that we may construct the hypersurfaces t = constant in any number of ways and that in a general relativity 4-spacetime there is no preferred slicing and hence no preferred "time" coordinate t?

You seem to be asserting that it is impossible to use coordinates that don't use Weyl's condition, if you say that you are denying general covarinace and therefore GR.

 

[TrickyDicky]

I must admit some possible sources of confusion from my part, first the way I phrased the second paragraph in post #38 can lead to misunderstanding, I certainly didn't mean that with a change of coordinates we can produce intersections where there were none. What I meant is that with the Weyl postulate the timelike worldlines of the fundamental observers are assumed to form a bundle or congruence in spacetime that diverges from a point in the (finite or infinitely distant) past or converges to such a point in the future.These worldlines are non-intersecting, except possibly at a singular point in the past or future or both. Thus, there is a unique worldline passing through each (non-singular) spacetime point. These are the worldlines that may intersect if we decide not to enforce the Weyl postulate.
A second source of confusion can derive from the fact that not all worldlines are timelike geodesics but every time like geodesic is a worldline. So when I defined the congruence in #20 I probably shouldn't have used the specific definition of worldline congruence used specifically when using Weyl's postulate without making it clear but since we were discussing the wiki definition of the Weyl's postulate I missed to make the distinction. I made that error in post 38 as well, I'll edit it.

 

[TrickyDicky]

Actually, if you initially have a congruence of geodesics then Raychaudri's equation \frac{\mathrm{d} \theta }{\mathrm{d} \tau } = -\frac{1}{3}\theta ^{2} - \sigma _{\mu \nu }\sigma ^{\mu \nu } + \omega _{\mu \nu }\omega ^{\mu \nu } - R_{\mu \nu }U^{\mu }U^{\nu } (where \theta ,\sigma ,\omega is the expansion, shear, and rotation of the congruence respectively and \mathbf{U} is the tangent vector field to the congruence) can be solved to find when there is an intersection and clearly the intersection is invariant; you are solving for a scalar.

This is correct. As I was trying to clarify, of course intersections should be invariants, I was not arguing anything that contradicts this and if that was what Peter Donis was saying I misunderstood him. General timelike worldlines need not be timelike geodesics, it is only by the Weyl's postulate that by restricting geometrically the congruence we obtain fundamental observers that are following timelike geodesics and whose worldlines can only intersect in the way specified in my previous post, since they are not subjected to any force. That doesn't stop other worldlines subjected to forces to collide of course.

 

[TrickyDicky]

Maybe I should stress that the worldlines we are discussing belong to "ideal" fundamental observers so no physical collisions should be considered.
Simply they observe different physical outcomes depending on whether they use a spacetime slicing or a different one, and this seems an inconsistency.

 

[TrickyDicky]

Maybe using someone else's words from a public webpage about cosmology helps get across my point:

"An immediate repercussion of Weyl's postulate is that the worldlines of galaxies do not intersect, except at asingular point in the finite/infinite past. Moreover, only one geodesic is passing through each point in spacetime, except at the origin. This allows one to define the concept of Fundamental Observer, one for each worldline. Each of these is carrying a standard clock, for which they can synchronize and fix a Cosmic Time by agreeing on the initial time t = t0 to couple a time t to some density value. This guarantees a homogenous Universe at each instant of cosmic Universal Time, and fixes its denition.
While for homogeneous Universe it is indeed feasible to use Weyl's postulate to define a universal time, this is no longer a trivial exercise for a Universe with inhomogeneities. The worldlines will no longer only diverge, as structures contract and collapse worldlines may cross. Also, if we were to tie a cosmic time to a particular density value we would end up with reference frames that would occur rather contrived to us. Also, we would end up with the problem of how to define a density perturbation. We would have a freedom of choice for the reference frame with respect to which we would define it. As usually stated, the density perturbation is dependent upon the chosen gauge, i.e. the chosen metric. This issue came prominently to the fore when Lifschitz tried to solve the perturbed Einstein field equations.
The solution was a proper gauge choice, which has become known as 'synchronous gauge". In essence, it involves a choice for the time and spatial coordinates based upon a homogeneous background Universe."

My claim of inconsistency comes from the fact that if this "solution" gauge is not chosen (and we are not obliged to choose it according to general covariance) we get observational differences in key physical laws. And this shouldn't ocurr.

 

[PeterDonis]

The other way round: every timelike geodesic locally defines a hypersurface to which it is orthogonal.

Yes, but the local hypersurfaces may not "match up" globally. For example, in Kerr spacetime, given a global family of timelike geodesics, you can define a local hypersurface at each event that is orthogonal to each timelike geodesic, but there is no way to patch together the various local hypersurfaces into a family of global hypersurfaces that (a) foliate the entire spacetime, and (b) are orthogonal to every one of the family of timelike geodesics.

 

[PeterDonis]

Just a precision. in 1923, when Weyl came up with his postulate,all models were static, we not only didn't know about CMBR there was no FRW model and not a single clue that ours was a non-stationary universe, the notion of expansion was totally unknown so no physics could be attributed to it.
It follows that your reasoning is not historically accurate, and therefore leaves my question unanswered.

Hmm...Hubble discovered the redshift-distance relation in 1929, so you are correct that I wasn't historically accurate. That makes me wonder what Weyl was thinking in 1923 when he formulated his postulate. Did he already realize that GR without a cosmological constant implied that the universe was non-stationary? Or was he trying to find a stationary fluid-like cosmological model?

As far as answering your question, see below.

Maybe using someone else's words from a public webpage about cosmology helps get across my point:

"An immediate repercussion of Weyl's postulate is that the worldlines of galaxies do not intersect, except at asingular point in the finite/infinite past. Moreover, only one geodesic is passing through each point in spacetime, except at the origin. This allows one to define the concept of Fundamental Observer, one for each worldline. Each of these is carrying a standard clock, for which they can synchronize and fix a Cosmic Time by agreeing on the initial time t = t0 to couple a time t to some density value. This guarantees a homogenous Universe at each instant of cosmic Universal Time, and fixes its denition.
While for homogeneous Universe it is indeed feasible to use Weyl's postulate to define a universal time, this is no longer a trivial exercise for a Universe with inhomogeneities. The worldlines will no longer only diverge, as structures contract and collapse worldlines may cross. Also, if we were to tie a cosmic time to a particular density value we would end up with reference frames that would occur rather contrived to us. Also, we would end up with the problem of how to define a density perturbation. We would have a freedom of choice for the reference frame with respect to which we would define it. As usually stated, the density perturbation is dependent upon the chosen gauge, i.e. the chosen metric. This issue came prominently to the fore when Lifschitz tried to solve the perturbed Einstein field equations.
The solution was a proper gauge choice, which has become known as 'synchronous gauge". In essence, it involves a choice for the time and spatial coordinates based upon a homogeneous background Universe."

My claim of inconsistency comes from the fact that if this "solution" gauge is not chosen (and we are not obliged to choose it according to general covariance) we get observational differences in key physical laws. And this shouldn't ocurr.

I think you're still confusing the model with the actual universe. However, it appears to me that whoever wrote the page you quoted from was either similarly confused, or at least was sloppy in their wording. Can you post an actual link?

Choosing the gauge is something that happens in the model; you can't change the actual physics of the actual universe by choosing a gauge. If the actual universe is not perfectly homogeneous (and it isn't), then our actual physical observations will deviate, at some level of measurement accuracy, from those predicted by a perfectly homogeneous model (and they do). But our actual physical observations, as far as we can tell, are still perfectly covariant, i.e., they co-vary with the chosen coordinate system in precisely the way GR says they should.

The problems the above quote talks about appear to me to be problems in how to construct a more accurate model that takes into account the deviations from perfect homogeneity, while still being able to calculate anything with the model. That does not mean that choosing a different gauge than the "solution" gauge would cause the model to make different physical predictions; it just means that the predictions would be harder to calculate.

The statement that "As usually stated, the density perturbation is dependent upon the chosen gauge, i.e. the chosen metric" does make me wonder, though; was the author really trying to say that a different physical prediction would be made by choosing a different gauge? Or was he just being sloppy and meant to say only that the perturbation takes a simpler form with the proper gauge choice, but is still properly covariant (i.e., choosing a different metric would make it look more complicated, but the perturbations as expressed in each coordinate system could still be transformed into each other by doing the appropriate coordinate transformation)? I'd be interested to read the full web page.

 

[TrickyDicky]

Consider a coordinate transformation in the FRW metric such that it doesn't happen to be timelike hypersurface orthogonal.
How would observers with a metric thus obtained agree about a second law of thermodynamics or about what type of potentials are observed, or about Huygen's principle secondary waves forward direction?
Keep in mind that in the tranformed metric each of them constructs the spacelike hypersurfaces t = constant in a different way depending on their location (cross terms dxdt,dydt.dzdt) and their worldlines can intersect at any point. Also their coordinate system tells them that depending on their position they have a different surface of simultaneity of the local Lorentz frame since it doesn't coincide locally with their spacelike hypersurface.
Line elements obtained by coordinate transformations should be physical law invariant in GR, or not?

 

[PAllen]

Consider a coordinate transformation in the FRW metric such that it doesn't happen to be timelike hypersurface orthogonal.
How would observers with a metric thus obtained agree about a second law of thermodynamics or about what type of potentials are observed, or about Huygen's principle secondary waves forward direction?
Keep in mind that in the tranformed metric each of them constructs the spacelike hypersurfaces t = constant in a different way depending on their location (cross terms dxdt,dydt.dzdt) and their worldlines can intersect at any point. Also their coordinate system tells them that depending on their position they have a different surface of simultaneity of the local Lorentz frame since it doesn't coincide locally with their spacelike hypersurface.
Line elements obtained by coordinate transformations should be physical law invariant in GR, or not?

There are several key things coordinate transforms do not change. How these relate to each of the issues you raise, I am not able to specify in detail. But just to make sure we're on the same page for future discussion:

- the timelike character of any given world line is invariant
- the spacelike character of any given path or surface is invariant (in general coordinates it is false to state something t=0 defines a spacelike hypersurface; the equation for a spacelike hypersurface may be very complex in general coordinates).
- causal connections are invariant (that is, from any given event, which events are on, inside, or outside its light cone are invariant). Note, I believe it is possible to uniquely specify a semi-remannian manifold in terms of its null cone structure. The only freedom here is to globally change which half of all cone are considered future pointing.
- If a law can be stated in terms of tensors and scalars, then it will, of course, hold in all
coordinate systems.

So the only question you raise that doesn't seem obvious to me is the formulation of thermodynamics in terms of geometric objects (tensors, etc.). This is something it just happens I've never read about.

 

[TrickyDicky]

So the only question you raise that doesn't seem obvious to me is the formulation of thermodynamics in terms of geometric objects (tensors, etc.). This is something it just happens I've never read about.

The last question was meant to be rhetorical.
Sadly, the only real question I raise is the one that you don't wnanna go into. But I'd think I give enough tools in the post to answer it.

 

[PeterDonis]

Consider a coordinate transformation in the FRW metric such that it doesn't happen to be timelike hypersurface orthogonal.
How would observers with a metric thus obtained agree about a second law of thermodynamics or about what type of potentials are observed, or about Huygen's principle secondary waves forward direction?

Because these don't have anything to do with whether or not the coordinate system they are using is hypersurface orthogonal. Look at my example of Kerr spacetime again: the coordinates are not hypersurface orthogonal (it can't be, as Kerr spacetime admits no such coordinate chart), but observers in different states of motion, using different coordinate systems, can still agree on which time direction is the future and everything that follows from it. The same comment applies to your example above. Why do you think it wouldn't?

I understand that Kerr spacetime is stationary and FRW spacetime isn't, but that doesn't change the fact that observers in different states of motion can agree on a common time direction, and once they do at one event, if the choice is continuous, they must agree throughout the spacetime.

Keep in mind that in the tranformed metric each of them constructs the spacelike hypersurfaces t = constant in a different way depending on their location (cross terms dxdt,dydt.dzdt)

Yes, the same is true for the different observers in Kerr spacetime.

and their worldlines can intersect at any point.

This part would not be true for Kerr spacetime, at least not if the worldlines were chosen as integral curves of the "time" coordinate. But you're leaving out a key point. Call the standard FRW coordinates chart A, and the non-orthogonal coordinates chart B. Consider the two global families of timelike worldlines, A and B, which are the families of integral curves of the "time" coordinate for their respective charts. Then it is true that no pair of worldlines in family A will intersect, while pairs of worldlines in family B may intersect. But that's no problem, because family A and family B contain *different* worldlines! In fact, they are disjoint: no worldline that appears in one family will appear in the other. So since you're looking at two different sets of worldlines, of course you are going to see different physics.

To test whether GR's rule of general covariance holds, what you would have to do is, for example, *transform* the description of family A into chart B's coordinates. Then you would find that, even though the description of family A's worldlines looked more complicated in chart B, it would still hold in chart B that no pair of A worldlines intersect. Similar remarks would apply if you transformed the description of family B into chart A's coordinates; the same pairs of B worldlines that intersected in chart B, would still intersect when described using chart A, and at the same events (though those events might have different coordinates in chart A).

Also their coordinate system tells them that depending on their position they have a different surface of simultaneity of the local Lorentz frame

Yes, the same is true in Kerr spacetime.

since it doesn't coincide locally with their spacelike hypersurface.

Yes, the same is true in Kerr spacetime.

Line elements obtained by coordinate transformations should be physical law invariant in GR, or not?

Yes, and in Kerr spacetime, they are. The same would apply to your example. See my more detailed comment on that above.

 

[TrickyDicky]

Because these don't have anything to do with whether or not the coordinate system they are using is hypersurface orthogonal. Look at my example of Kerr spacetime again: the coordinates are not hypersurface orthogonal (it can't be, as Kerr spacetime admits no such coordinate chart), but observers in different states of motion, using different coordinate systems, can still agree on which time direction is the future and everything that follows from it. The same comment applies to your example above. Why do you think it wouldn't?

I understand that Kerr spacetime is stationary and FRW spacetime isn't, but that doesn't change the fact that observers in different states of motion can agree on a common time direction, and once they do at one event, if the choice is continuous, they must agree throughout the spacetime.

Well thanks at least you answer the question. I'm afraid though we totally disagree about this quoted part.
A Kerr spacetime is not the best example due to its being stationary, since it doesn't fulfill my first requirement of being obtained from a coordinate transformation of the FRW metric, but even there you should be able to see a second law of thermodynamics would be impossible in such a universe.
You seem to still not fully understand what my set up is (surely my fault), the observers need not be in different states of motion or having each a different coordinate system, if the Kerr metric was a valid example they could all use this metric in the usual coordinates. The problem is that it is not the example I'm referring to, being a different manifold from ours it certainly will have different physics anyway. And it certainly wouldn't have a second law not even having an intrinsic concept of time passage.

 

[PeterDonis]

A Kerr spacetime is not the best example due to its being stationary, since it doesn't fulfill my first requirement of being obtained from a coordinate transformation of the FRW metric, but even there you should be able to see a second law of thermodynamics would be impossible in such a universe.

In a "pure" Kerr spacetime, as I defined it in a previous post, I think I agree with you. But I don't think a second law would be impossible in an "impure" Kerr spacetime, where the Kerr geometry was an "average" background on top of which more complicated microphysics took place. That's kind of out of scope here, though, since it's speculation on my part.

You seem to still not fully understand what my set up is (surely my fault), the observers need not be in different states of motion or having each a different coordinate system

If the different observers in your setup are not supposed to be in different states of motion or using a different coordinate system, then you are right, I don't understand the scenario you are describing. Nor do I understand how observers in the same state of motion could somehow no longer observe the second law to be valid by choosing a coordinate system that wasn't hypersurface orthogonal. Maybe it would help if you posted a link to the full article you quoted from earlier.

 

[TrickyDicky]

If the different observers in your setup are not supposed to be in different states of motion or using a different coordinate system, then you are right, I don't understand the scenario you are describing. Nor do I understand how observers in the same state of motion could somehow no longer observe the second law to be valid by choosing a coordinate system that wasn't hypersurface orthogonal. Maybe it would help if you posted a link to the full article you quoted from earlier.

That quote was from the lecture notes of a course on cosmology and GR from the university of Groningen, it is a regular cosmology course, in my post only the part between "" was from the course, the last paragraph in the post was not part of the notes (just in case you thought so). I only used it to clarify the Weyl postulate, it has nothing to do with the second law.
The part about density perturbations IMO only stresses the fact that the FRW metric needs the Weyl postulate as a precondition to introduce the homogeneity condition. So that if that is not the case a spatially inhomogenous universe is the result. This beg the question if the spatial homogeneity condition from the cosmological principle overrides the principle of general covariance.