Worldline congruence and general covariance

[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.

 

[PeterDonis]

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.

I understand, but I would still be interested to see the paragraph you quoted in context. As I commented before, the part you quoted appears, at the very least, to be using language rather loosely. Maybe in context there are clarifications elsewhere in the notes that make it clearer what they are trying to say.

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.

Once again, I think you're confusing the model with the actual universe. The actual universe is not exactly homogeneous; we know that. If you are trying to say that adopting the Weyl postulate somehow requires one to believe that the actual universe *is* exactly homogeneous, I think that's obviously wrong. Homogeneity is a useful approximation we adopt to make the model tractable, and that's all. Also, adopting homogeneity as an assumption in the model doesn't require us to write the model down in the standard FRW coordinates; we could do so in any coordinate system we want, and we would still be able to verify that, when we calculate physical invariants, they come out the same as when we write the model down in standard FRW coordinates. Since homogeneity and isotropy can be defined entirely in terms of physical invariants, this means the standard FRW model written down in any coordinate chart will still be homogeneous and isotropic, and will predict the same physics. So in that sense I don't see how the homogeneity condition could possibly override the principle of general covariance.

If you are trying to say that somehow an inhomogeneous model would make different physical predictions, well, yes, of course it would. The FRW model makes predictions on the assumption that the mass-energy in the universe can be modeled as a perfectly homogeneous and isotropic perfect fluid. Since it isn't, the FRW predictions will deviate from actual observations at some level of accuracy. Obviously, if we construct a more complicated model in which the mass-energy in the model universe follows some pattern that is not completely homogeneous and isotropic, that model will make different predictions than the standard FRW model; and if we've chosen our model of the inhomogeneities well, the more complicated model's predictions might match the data better than a simple FRW model does. But I still don't see how any of that overrides or contradicts the principle of general covariance. The predictions of the two models are different because they contain different stress-energy tensors, so the RHS of the Einstein Field Equation changes; hence the LHS (and therefore the geometry of the spacetime in the model) has to change too. But that will be true even if we insist on writing down both models in exactly the same coordinate chart. It has nothing to do with general covariance.

One final note: even if an inhomogeneous model makes different physical predictions, the differences will be in the specific worldlines of specific pieces of matter. I don't see how the inhomogeneity would change the expansion of the universe, or the second law being true, or anything like that. (I guess that, to be precise, I should say that I don't see how any inhomogeneous model that matched the data at least as well as a homogeneous FRW model would change the expansion of the universe, etc.) The reason I say this is that I don't see how the expansion of the universe or the second law would depend on *perfect* homogeneity; the amount of homogeneity and isotropy we actually observe would seem to be plenty good enough.

 

[TrickyDicky]

I understand, but I would still be interested to see the paragraph you quoted in context. As I commented before, the part you quoted appears, at the very least, to be using language rather loosely. Maybe in context there are clarifications elsewhere in the notes that make it clearer what they are trying to say.

http://www.astro.rug.nl/~weygaert/tim1publication/cosmo2009/cosmo2009.robertsonwalker.pdf

Once again, I think you're confusing the model with the actual universe. The actual universe is not exactly homogeneous; we know that. If you are trying to say that adopting the Weyl postulate somehow requires one to believe that the actual universe *is* exactly homogeneous, I think that's obviously wrong. Homogeneity is a useful approximation we adopt to make the model tractable, and that's all.

I'm not trying to say that, this is trivial.

Also, adopting homogeneity as an assumption in the model doesn't require us to write the model down in the standard FRW coordinates; we could do so in any coordinate system we want, and we would still be able to verify that, when we calculate physical invariants, they come out the same as when we write the model down in standard FRW coordinates. Since homogeneity and isotropy can be defined entirely in terms of physical invariants, this means the standard FRW model written down in any coordinate chart will still be homogeneous and isotropic.

This is wrong but I think maybe it should be clarified in the cosmology sub-forum. Basically matter distribution in the universe is not considered an invariant in the sense of a physical law, is more like a symmetry condition imposed on the metric, and related to the initial conditions.
A coordinate change that involves losing hypersurface orthogonality certainly will alter the homogeneity condition, only fundamental observers with the Weyl condition see spatial homogeneity hypersurfaces.

One final note: even if an inhomogeneous model makes different physical predictions, the differences will be in the specific worldlines of specific pieces of matter. I don't see how the inhomogeneity would change the expansion of the universe, or the second law being true, or anything like that. (I guess that, to be precise, I should say that I don't see how any inhomogeneous model that matched the data at least as well as a homogeneous FRW model would change the expansion of the universe, etc.) The reason I say this is that I don't see how the expansion of the universe or the second law would depend on *perfect* homogeneity; the amount of homogeneity and isotropy we actually observe would seem to be plenty good enough.

This deviates from my OP that was more specifically about Weyl's principle, I have never mentioned anything about "perfect homogeneity".

 

[PeterDonis]

http://www.astro.rug.nl/~weygaert/tim1publication/cosmo2009/cosmo2009.robertsonwalker.pdf

Thanks for the link.

This is wrong but I think maybe it should be clarified in the cosmology sub-forum. Basically matter distribution in the universe is not considered an invariant in the sense of a physical law, is more like a symmetry condition imposed on the metric, and related to the initial conditions.

I would say "a symmetry condition imposed on the stress-energy tensor", but since that implies a similar symmetry condition on the Einstein tensor (which involves derivatives of the metric), it pretty much comes to the same thing.

However, that brings up a question: when we say the stress-energy tensor, or the metric, obeys a symmetry condition, is that an invariant? Or do we only say it holds in the particular coordinate system where the symmetry is manifest? For example, can we correctly say the FRW metric is homogeneous and isotropic, period, or can we only say it's homogeneous and isotropic in the standard FRW coordinates, but not in some other coordinates?

The reason I bring this up is that, when I said that homogeneity and isotropy can be defined in terms of physical invariants, I was assuming that the former was the correct usage (homogeneity and isotropy are features of the invariant geometry, independent of what coordinate chart we use to describe it). When you say my statement you quoted is "wrong", you appear to be assuming that the latter (that we can only say that, for example, the FRW metric is homogeneous and isotropic if we express it in the standard FRW "comoving" coordinates) is the correct usage. That usage seems wrong to me, though, because it doesn't seem right to me to say a geometry only has a certain symmetry (and homogeneity and isotropy are symmetries) in a certain set of coordinates; as I understand symmetry, it is supposed to be an invariant feature of the geometry itself.

A coordinate change that involves losing hypersurface orthogonality certainly will alter the homogeneity condition, only fundamental observers with the Weyl condition see spatial homogeneity hypersurfaces.

Here, again, you seem to be taking the position that a geometry can only be said to have a symmetry if it is described using the particular coordinate chart that matches the symmetry. That doesn't seem right to me. I agree that only "comoving" observers in an FRW spacetime will *see* their hypersurfaces of simultaneity as homogeneous and isotropic; other, non-comoving observers will not. But the FRW geometry itself still has the symmetries of homogeneity and isotropy, even if those symmetries are not explicitly manifest to observers who are not "comoving".

This deviates from my OP that was more specifically about Weyl's principle, I have never mentioned anything about "perfect homogeneity".

Not in so many words, but you did say this:

I say that the congruence seems very physical because if we didn't have it I find it hard to understand things like the second law of thermodynamics and the consensus we all have on the direction of time, or the fact that we only observe retarded potentials,or the very fact that the universe is expanding for everyone-if we didn't have the congruence of timelike geodesics, the universe could be expanding for some contracting for others and neither for others depending on the coordinate system they used- all these seem to be "physical" consequences of having worldline congruence.

This seems to me like you are saying that only "comoving" observers would observe the second law to be true and would see the universe as expanding, or something very much like it. In other words, it seems like you are saying that perfect homogeneity is required. Since that seems extreme, and since it seems unlikely to me that you would take such an extreme position, I'm trying to understand what you were actually saying, and where the Weyl postulate comes into it, since the Weyl postulate basically amounts to the assumption of perfect homogeneity once again, so if perfect homogeneity is not required that amounts to saying that the Weyl postulate is just a calculational convenience after all.

 

[TrickyDicky]

Here, again, you seem to be taking the position that a geometry can only be said to have a symmetry if it is described using the particular coordinate chart that matches the symmetry. That doesn't seem right to me. I agree that only "comoving" observers in an FRW spacetime will *see* their hypersurfaces of simultaneity as homogeneous and isotropic; other, non-comoving observers will not. But the FRW geometry itself still has the symmetries of homogeneity and isotropy, even if those symmetries are not explicitly manifest to observers who are not "comoving".

I'm really not taking that position.
I also agree that the geometry itself should have the same symmetries regardless if they are not manifest to observers not comoving. So ignore my final sentence in post #60, it just slipped my mind.
Remembert the OP was not about homogeneity which as mentioned before is not a physical law but about the second law of thermodynamics.

 

[PAllen]

I've said a few times I'm not familiar with how to treat thermodynamics in GR in general, but more specifically, in coordinate independent terms. I would like to ask a few question hopefully related to the concerns of this thread. One prelude is that a clear advance in GR theory was the ability to state truly coordinate independent definition of asymptotic flatness; and to describe features like stationary and static character of spacetimes in terms of e.g. killing vectors rather than conditions that needed to be checked in preferred coordinates.

1) Is there, and what is the nature of a description of 'expanding space' in fully coordinate independent terminology (i.e. without saying: if true, there must exist such and such preferred coordinates demonstrating some property)? This seems non-trivial to me in that my understanding is that Minkowski flat spacetime can be made to appear expanding with appropriate coordinate choices. I have no strong feel for this question.

2) We know that, in fact, there is one stupid, trivial class of coordinate transforms that lead to violation of the second law of thermodynamics - global time reversal transforms. What strong arguments, preferably coordinate independent ones, can be offered that no other type of continuous transform can lead to local violations of the second law? My feeling is that any transform that leaves the sense of all lightcones unchanged, leaves thermodynamics unchanged. Does anyone know of a more formal statement and argument of this nature?

 

[PeterDonis]

I'm really not taking that position.
I also agree that the geometry itself should have the same symmetries regardless if they are not manifest to observers not comoving. So ignore my final sentence in post #60, it just slipped my mind.

Ok.

Remember the OP was not about homogeneity which as mentioned before is not a physical law but about the second law of thermodynamics.

Well, it was also about the Weyl postulate, and I'm still having trouble seeing how that fits in. All the physical questions about the second law, expanding universe, etc. are the same in an inhomogeneous universe as in a homogeneous one, and the same for non-comoving observers as for comoving ones. See my comments to PAllen below.

1) Is there, and what is the nature of a description of 'expanding space' in fully coordinate independent terminology (i.e. without saying: if true, there must exist such and such preferred coordinates demonstrating some property)?

I don't think there can be a description of "expanding", specifically, because GR is time symmetric; if we have a solution to the EFE that we call "expanding" (say, the expanding FRW spacetime), then the time reverse of it is also a solution and will be "contracting" (say, the contracting FRW models that are used to model the interior of stars collapsing into black holes, as in the classic Oppenheimer-Snyder paper). The only difference between the two solutions is which direction we, the people making the models, perceive as the "future" direction of time. That depends on our memories, so it depends on the second law, as I've said before. The only way to link this to the expansion of the universe would be to find some argument for why the second law should only hold if the future direction of time is the one in which the universe is getting larger.

2) We know that, in fact, there is one stupid, trivial class of coordinate transforms that lead to violation of the second law of thermodynamics - global time reversal transforms. What strong arguments, preferably coordinate independent ones, can be offered that no other type of continuous transform can lead to local violations of the second law? My feeling is that any transform that leaves the sense of all lightcones unchanged, leaves thermodynamics unchanged. Does anyone know of a more formal statement and argument of this nature?

I don't. But I would point out that when you say a time reversal violates the second law, this is true if you keep everything about the solution the same (i.e., time reversal reverses the sign of entropy change). But it's possible, as I implied above, that there might be a different solution that had everything else time reversed (at least, at a macroscopic level; obviously if you exactly time reversed every individual particle you would have to reverse entropy change), but still had entropy increasing in the new "future" direction of time (i.e., in the opposite direction from the original solution). I can't think of an argument that would rule this out a priori.

 

[PeterDonis]

I don't think there can be a description of "expanding", specifically, because GR is time symmetric...

I should qualify this. I believe there is an invariant definition of "expanding", but it depends on assuming that you've already decided which time direction is the "future". (Time reversing the definition then becomes an invariant definition of "contracting".) An invariant definition of "expanding" would look at frame-independent observables like the Hubble redshift-distance relation. I haven't been able to find a nice, compact formulation of such a definition, though; the best I've found is the discussion in Ned Wright's Cosmology FAQ:

http://www.astro.ucla.edu/~wright/cosmology_faq.html

If, OTOH, you don't have any other means of telling which time direction is the "future" (e.g., suppose we didn't have memories, didn't experience the passage of time, entropy was constant, every cyclic process never changed, etc.--this may not actually be possible but consider it just as a hypothetical), then you would still have two invariant descriptions of "size change" that were time reverses of each other, but you wouldn't be able to tell which one was describing "expansion" and which was describing "contraction".

 

[PeterDonis]

Another item that just occurred to me: I believe the Raychaudhuri Equation can also be used to define an invariant notion of "expansion" (or "contraction"), and that Hawking and Penrose used this in the proofs of the singularity theorems.

 

[TrickyDicky]

I've said a few times I'm not familiar with how to treat thermodynamics in GR in general, but more specifically, in coordinate independent terms. I would like to ask a few question hopefully related to the concerns of this thread. One prelude is that a clear advance in GR theory was the ability to state truly coordinate independent definition of asymptotic flatness; and to describe features like stationary and static character of spacetimes in terms of e.g. killing vectors rather than conditions that needed to be checked in preferred coordinates.

1) Is there, and what is the nature of a description of 'expanding space' in fully coordinate independent terminology (i.e. without saying: if true, there must exist such and such preferred coordinates demonstrating some property)? This seems non-trivial to me in that my understanding is that Minkowski flat spacetime can be made to appear expanding with appropriate coordinate choices. I have no strong feel for this question.

Apparently there is no such description, rather as you say there are examples that the expanding "property" is coordinate dependent: for instance the Milne model that is a patch of Minkowki spacetime is static or expanding depending on the cordinates. Something very similar happens to the de Sitter geometry, it is static or expanding depending on the coordinate choice.

2) We know that, in fact, there is one stupid, trivial class of coordinate transforms that lead to violation of the second law of thermodynamics - global time reversal transforms. What strong arguments, preferably coordinate independent ones, can be offered that no other type of continuous transform can lead to local violations of the second law? My feeling is that any transform that leaves the sense of all lightcones unchanged, leaves thermodynamics unchanged. Does anyone know of a more formal statement and argument of this nature?

As Peter Donis points out, I woudn't consider the trivial case you mention a second law violation because observers can still agree on what they call increase of entropy.

The type of metric I was picturing was one in which the time-space cross-terms produce a location dependent time and therefore an absence of synchronous time , in this way observers situated in different locations can't agree on time and there will be some located at certain points such that they will have reversed time arrow respect to each other. In such situation they couldn't agree about increase of entropy.

 

[PeterDonis]

Apparently there is no such description, rather as you say there are examples that the expanding "property" is coordinate dependent: for instance the Milne model that is a patch of Minkowki spacetime is static or expanding depending on the cordinates. Something very similar happens to the de Sitter geometry, it is static or expanding depending on the coordinate choice.

To expand on my previous post, the expansion scalar, which is mentioned in the page on the Raychaudhuri equation I linked to, is an invariant and offers a reasonable definition of "expanding" (or "contracting") that is general covariant. In Minkowski spacetime the expansion scalar is zero, which to me means that the "expansion" in the Milne model under a certain set of coordinates is only apparent. Off the top of my head I don't know what the expansion scalar looks like for De Sitter spacetime, I'll have to look it up.

The type of metric I was picturing was one in which the time-space cross-terms produce a location dependent time and therefore an absence of synchronous time , in this way observers situated in different locations can't agree on time and there will be some located at certain points such that they will have reversed time arrow respect to each other. In such situation they couldn't agree about increase of entropy.

I think this is only possible if there are closed timelike curves in the spacetime, or if there is some kind of discontinuity in the light cone structure. Just having time-space cross terms present is not enough by itself (I've already pointed out Kerr spacetime as a counterexample; another is Painleve coordinates in Schwarzschild spacetime). I don't see how just having cross terms present plus being non-stationary would be enough either; the cross terms would add vorticity and shear (again, using the terms as they appear in the Raychaudhuri equation), but would not allow the kind of "reversed time" you are talking about, at least not without, as I said above, some kind of discontinuity in the light cone structure. I haven't dipped into Hawking & Ellis in quite a while, but I suspect there is something in there about this.