Worldline congruence and general covariance

[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.

 

[PeterDonis]

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...I haven't dipped into Hawking & Ellis in quite a while, but I suspect there is something in there about this.

Well, thanks to Google and Wikipedia, I don't even have to crack open Hawking and Ellis.

Check out the Wiki page on causality conditions:

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

There's a fair bit of technical jargon here, but the upshot appears to me to be that my quote above is basically correct. The key causality condition is "stably causal", which is described on the Wiki page; this condition basically entails that there are no closed causal (timelike or null) curves in both the spacetime itself, and in any "nearby" spacetimes that can be produced from it by a small perturbations (this is where the "stably" part comes from). If a spacetime meets this condition, then there is a global time function on the spacetime, which prevents the sort of thing TrickyDicky was describing from happening. Note that there are *no* symmetry conditions imposed in any of the relevant theorems; the spacetime does not have to be homogeneous, isotropic, spherically symmetric, stationary, etc., etc. It just has to be stably causal.

 

[PAllen]

Well, thanks to Google and Wikipedia, I don't even have to crack open Hawking and Ellis.

Check out the Wiki page on causality conditions:

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

There's a fair bit of technical jargon here, but the upshot appears to me to be that my quote above is basically correct. The key causality condition is "stably causal", which is described on the Wiki page; this condition basically entails that there are no closed causal (timelike or null) curves in both the spacetime itself, and in any "nearby" spacetimes that can be produced from it by a small perturbations (this is where the "stably" part comes from). If a spacetime meets this condition, then there is a global time function on the spacetime, which prevents the sort of thing TrickyDicky was describing from happening. Note that there are *no* symmetry conditions imposed in any of the relevant theorems; the spacetime does not have to be homogeneous, isotropic, spherically symmetric, stationary, etc., etc. It just has to be stably causal.

Since the Kerr black hole has CTCs, and we presume the universe has rotating black holes, unless the hypothesis that the Kerr solution is not real world accurate in its interior, the real universe is not causally stable.

Personally I do believe the Kerr active region is not realistic and that the universe has no ctc's, so is almost certainly causally stable.

Thanks for the research, Peter!

 

[PeterDonis]

Since the Kerr black hole has CTCs, and we presume the universe has rotating black holes, unless the hypothesis that the Kerr solution is not real world accurate in its interior, the real universe is not causally stable.

Personally I do believe the Kerr active region is not realistic and that the universe has no ctc's, so is almost certainly causally stable.

I found an interesting paper by Matt Visser on arxiv that discusses this:

http://arxiv.org/abs/0706.0622

From p. 13:

Thus \nabla t is certainly a timelike vector in the region r > 0, implying that this portion of the manifold is “stably causal”, and that if one restricts attention to the region r > 0 there is no possibility of forming timelike curves. However, if one chooses to work with the maximal analytic extension of the Kerr spacetime, then the region r < 0 does make sense (at least mathematically), and certainly does contain closed timelike curves. (See for instance the discussion in Hawking and Ellis.) Many (most?) relativists would argue that this r < 0 portion of the maximally extended Kerr spacetime is purely of mathematical interest and not physically relevant to astrophysical black holes.

Note that the "r" he is talking about is not the "standard" radial coordinate, which is why he can say that having r < 0 makes sense. But the r < 0 region does not correspond to the entire Kerr interior; as far as I can tell, r < 0 would be a region "inside" the ring singularity. However, the pathological effects of the CTC region are not confined to this "r < 0" portion; later on (pp. 35-36), there's this:

[Y]ou should not physically trust in the inner horizon or the inner ergosurface. Although they are certainly there as mathematical solutions of the exact vacuum Einstein equations, there are good physics reasons to suspect that the region at and inside the inner horizon, which can be shown to be a Cauchy horizon, is grossly unstable — even classically — and unlikely to form in any real astrophysical collapse.

Aside from issues of stability, note that although the causal pathologies [closed timelike curves] in the Kerr spacetime have their genesis in the maximally extended r < 0 region, the effects of these causal pathologies can reach out into part of the r > 0 region, in fact out to the inner horizon at r = r− — so the inner horizon is also a chronology horizon for the maximally extended Kerr spacetime. Just what does go on deep inside a classical or semiclassical black hole formed in real astrophysical collapse is still being debated — see for instance the literature regarding “mass inflation” for some ideas. For astrophysical purposes it is certainly safe to discard the r < 0 region, and almost all relativists would agree that it is safe to discard the entire region inside the inner horizon r < r− .

The bit about the inner horizon being a Cauchy horizon basically means you can solve for the entire spacetime outside that horizon without having to know what goes on inside it, and indeed without even assuming that the region inside it it exists. So there seems to be a fairly general opinion that, indeed, the CTC region of Kerr spacetime, and in fact the entire region inside the inner horizon where causal pathologies can reach, is not physically realistic.

 

[TrickyDicky]

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.

But you surely realize that the Raychaudhuri equation assumes the Weyl condition. Take a look at the Wiki entry of the Raych. eq. and se how the Ray scalar is constructed from a timelike vector field that can be interpreted as a congruence of nonintersecting world lines( therefore spacelike hypersurfce orthogonal) so the starting point of that equation is a certain preferred manifold slicing.
About de Sitter space, if you look at the wiki entry under the subtitle static cordinates and observe the metric you'll notice it doesn't follow the Weyl postulate in those coordinates.

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.

Precisely having CTC's is one consequence of not having causal stability and that is what not using the Weyl postulate will lead to. But if you think about it the second law of thermodynamics demands a well defied causality.
So there's no way out of the fact that one needs a specific spacetime coordinate slicing up to have causal stability or even the notion of causality and without that certain physical laws lose their usual meaning like those where a causally stable consensus on when entropy is increasing is needed.
Once again let's not forget that in general relativity any slicing up of the spacetime manifold should be physically indistinguishible from any other.

 

[PeterDonis]

But you surely realize that the Raychaudhuri equation assumes the Weyl condition. Take a look at the Wiki entry of the Raych. eq. and se how the Ray scalar is constructed from a timelike vector field that can be interpreted as a congruence of nonintersecting world lines( therefore spacelike hypersurfce orthogonal) so the starting point of that equation is a certain preferred manifold slicing.

A congruence of nonintersecting timelike worldlines does not have to be hypersurface orthogonal. Remember I commented earlier that there are different possible meanings of the word "congruence"; the one used on the Raychaudhuri equation page is the "standard" one, as given on this Wiki page:

http://en.wikipedia.org/wiki/Congruence_(general_relativity)

(Btw, the standard definition requires the family of worldlines to be non-intersecting.) A congruence of timelike worldlines is only hypersurface orthogonal if the vorticity vanishes, but the Raychaudhuri equation is completely general and applies to any congruence.

About de Sitter space, if you look at the wiki entry under the subtitle static cordinates and observe the metric you'll notice it doesn't follow the Weyl postulate in those coordinates.

Well, de Sitter spacetime is a vacuum solution, so I'm not sure how one would apply the Weyl postulate to it, since the Weyl postulate talks about the mass-energy in the universe being a perfect fluid, not a vacuum.

However, if we allow the postulate to apply to a zero-density perfect fluid, so to speak, then de Sitter spacetime *is* perfectly homogeneous and isotropic; as the Wiki page notes, it is "maximally symmetric", so it does satisfy the Weyl postulate. You agreed earlier in this thread that homogeneity and isotropy are coordinate-independent, so the fact that de Sitter spacetime doesn't "look" homogeneous and isotropic in static coordinates does not mean it isn't; it just means those coordinates don't match up with the symmetry.

Btw, regarding the expansion scalar of de Sitter spacetime, since dS is a vacuum solution with a positive cosmological constant, its expansion scalar will be positive (i.e., dS is expanding in the coordinate-invariant sense). The fact that it "looks" static in a particular set of coordinates is an illusion.

Precisely having CTC's is one consequence of not having causal stability and that is what not using the Weyl postulate will lead to.

No, that is what having a spacetime that is not stably causal will lead to. But can you show that a spacetime must satisfy the conditions of the Weyl postulate in order to be stably causal? The Weyl postulate is an extremely restrictive symmetry condition, and being stably causal is an extremely general property that does not require the spacetime to have any particular symmetry.

But if you think about it the second law of thermodynamics demands a well defied causality.

No argument here.

So there's no way out of the fact that one needs a specific spacetime coordinate slicing up to have causal stability or even the notion of causality

Again, can you show this explicitly? As I pointed out in my previous post, the global causality theorems in GR make *no* assumptions about any symmetries of the spacetime, and they certainly don't depend on using any particular slicing up of the spacetime.

 

[PeterDonis]

a zero-density perfect fluid, so to speak

Actually, I shouldn't have said "zero-density" here, since the positive cosmological constant in dS spacetime can be considered to be a non-zero energy density. The key point is that there is no "normal" matter or radiation in dS spacetime. But the cosmological constant "energy density" can be treated as a perfect fluid, so the Weyl postulate analysis can be applied; I shouldn't have implied that that was questionable.

 

[TrickyDicky]

A congruence of nonintersecting timelike worldlines does not have to be hypersurface orthogonal. Remember I commented earlier that there are different possible meanings of the word "congruence"; the one used on the Raychaudhuri equation page is the "standard" one, as given on this Wiki page:

http://en.wikipedia.org/wiki/Congruence_(general_relativity)

(Btw, the standard definition requires the family of worldlines to be non-intersecting.) A congruence of timelike worldlines is only hypersurface orthogonal if the vorticity vanishes, but the Raychaudhuri equation is completely general and applies to any congruence.

I think you have some confusion about this.
Let's see, a general timelike congruence certainly doesn't have to be hypersurface orthogonal. I think we should agree about this.
But the timelike congruence used in the Ray eq. is not the general timelike congruence as it is explicit in the wiki page, it is a non-intersecting worldlines congruence.
Please explain to me how do you get timelike worldlines in a 4-manifold that are not 3-hypersurface orthogonal to not intersect.

However, if we allow the postulate to apply to a zero-density perfect fluid, so to speak, then de Sitter spacetime *is* perfectly homogeneous and isotropic; as the Wiki page notes, it is "maximally symmetric", so it does satisfy the Weyl postulate.

It doesn't, you are conflating the Weyl principle and the cosmological principle again. When I said the Weyl postulate is a precondition of homogeneity I was only referring to an FRW metric, not to a general spacetime. As you say probably the very fact that it is an empty universe doesn't allow to use the Weyl's postulate in the usual formulation for particle fluids.
But usually even in emty models test particles are used that have timelike worldlines, when using those test particles in the static coordinates of de Sitter spacetime you get intersecting worldlines.

Btw, regarding the expansion scalar of de Sitter spacetime, since dS is a vacuum solution with a positive cosmological constant, its expansion scalar will be positive (i.e., dS is expanding in the coordinate-invariant sense). The fact that it "looks" static in a particular set of coordinates is an illusion.

Then again, what is expanding in an empty universe? See what I wrote above about the Ray eq.

No, that is what having a spacetime that is not stably causal will lead to. But can you show that a spacetime must satisfy the conditions of the Weyl postulate in order to be stably causal? The Weyl postulate is an extremely restrictive symmetry condition, and being stably causal is an extremely general property that does not require the spacetime to have any particular symmetry.

The Weyl postulate does not require spacetime to have any particular symmetry, remember it's just a way of slicing up the manifold.

Again, can you show this explicitly? As I pointed out in my previous post, the global causality theorems in GR make *no* assumptions about any symmetries of the spacetime, and they certainly don't depend on using any particular slicing up of the spacetime.

I'd say those theorems make a lot of assumptions, see their wiki entry.

 

[TrickyDicky]

can you show that a spacetime must satisfy the conditions of the Weyl postulate in order to be stably causal?

How do you define causality if there is no way to reach consensus about a particular time? I mean if every observer has his own timelike congruence not related to the one of other observers by a common spacelike hypersurface, how do you get them to agree on causality?
Maybe some observers are able to agree but you can't guarantee in general, (you can guarantee it in a flat spacetime like Minkowski's though) that there won't be some observers whose light cones will have the future cone pointing in opposite directions depending on the geometry of the manifold at hand.

 

[PeterDonis]

you can't guarantee in general, (you can guarantee it in a flat spacetime like Minkowski's though) that there won't be some observers whose light cones will have the future cone pointing in opposite directions depending on the geometry of the manifold at hand.

If the spacetime is stably causal, yes, you can, because there must be a *global time function* on the spacetime. As the Wiki page on causality conditions that I linked to before says, this is a scalar function on the spacetime whose gradient is everywhere timelike and future-directed. You are correct that this, by itself, does not ensure that observers can globally agree on "what time it is", so to speak. However, it *does* ensure that there can't be any "flips" in which half of the light cone is the "future" half, because of the continuity of the gradient. And that, by itself, is enough to ensure a stable notion of causality. Causality does not require global agreement on a time coordinate; it only requires a stable, continuous light cone structure with no "flips" in direction, and the "stably causal" requirement ensures that. And note that, if all we know is that the spacetime is stably causal, we can't say much else about it: for example, we can't say that a stably causal spacetime must have any particular symmetry, or even that a family of non-intersecting timelike worldlines that covers the spacetime must exist.

There is a stronger requirement, called globally hyperbolic. A spacetime is globally hyperbolic if and only if there is a Cauchy surface for the spacetime. A Cauchy surface is a spacelike hypersurface that is intersected by every causal (inextensible, timelike or null) curve exactly once. So a Cauchy surface is like a global "instant of time". It can be shown that, if we have one Cauchy surface, the entire spacetime can be foliated by Cauchy surfaces, each representing a different "instant of time". And if we combine this with the gradient of the global time function (which we have because any globally hyperbolic spacetime is stably causal), we find that we have a family of timelike worldlines such that every event in the spacetime lies on exactly one worldline in the family. So now we have something that looks like our intuitive notion of "space" and "time". But as we've seen, we don't even need that to ensure causality.

What we still do *not* have, even with a globally hyperbolic spacetime, is a family of worldlines with any other special property, such as hypersurface orthogonality. In other words, we have a family of timelike worldlines and a slicing of the spacetime into spacelike hypersurfaces (Cauchy surfaces); but it may be that the worldlines are not orthogonal (or not everywhere orthogonal) to the hypersurfaces. We can't ensure orthogonality without imposing additional requirements on the spacetime, such as adopting the Weyl postulate. But already, as you can see, we have ensured a lot about causality, without ever having to touch the Weyl postulate.

How do you define causality if there is no way to reach consensus about a particular time? I mean if every observer has his own timelike congruence not related to the one of other observers by a common spacelike hypersurface, how do you get them to agree on causality?

Since you mentioned Minkowski spacetime (in what I quoted earlier in this post), I should note that the statements just quoted seem odd, since even in Minkowski spacetime you can have observers in relative motion that do not agree on "a particular time" (because of relativity of simultaneity) but do agree on causality, because, as you note, we can always guarantee in flat spacetime that there is a stable notion of the "future" half of the light cones. In the terminology I used above, flat Minkowski spacetime is guaranteed to be globally hyperbolic (because any surface of constant time t in any inertial coordinate system is obviously a Cauchy surface).

Also, I think we're having some terminology confusion again, in particular around the word "congruence". I used the term "family of worldlines" above to avoid getting into terminology issues, but we can go into more detail about them if needed.

 

[TrickyDicky]

If the spacetime is stably causal, yes, you can, because there must be a *global time function* on the spacetime...


What we still do *not* have, even with a globally hyperbolic spacetime, is a family of worldlines with any other special property, such as hypersurface orthogonality. In other words, we have a family of timelike worldlines and a slicing of the spacetime into spacelike hypersurfaces (Cauchy surfaces); but it may be that the worldlines are not orthogonal (or not everywhere orthogonal) to the hypersurfaces. We can't ensure orthogonality without imposing additional requirements on the spacetime, such as adopting the Weyl postulate. But already, as you can see, we have ensured a lot about causality, without ever having to touch the Weyl postulate.

We may have ensured a lot about causality , but the whole point is that the Weyl postulate is not a requirement on spacetime, it's just a way to slice it in order to obtain some coordinates, even a supposedly "globally hyperbolic" spacetime can get acausal observers when the Weyl postulate is not used.

 

[PeterDonis]

I'm responding separately here to what appear to me to be more terminology issues.

Let's see, a general timelike congruence certainly doesn't have to be hypersurface orthogonal. I think we should agree about this.

Yep.

But the timelike congruence used in the Ray eq. is not the general timelike congruence as it is explicit in the wiki page, it is a non-intersecting worldlines congruence.

And, as I noted, when you check the standard definition of "congruence", it requires the worldlines to be non-intersecting. There may be a more general term for sets of worldlines some of which may intersect, but it isn't "congruence".

Please explain to me how do you get timelike worldlines in a 4-manifold that are not 3-hypersurface orthogonal to not intersect.

Well, I've mentioned Kerr spacetime several times now. The family of timelike geodesics which are "co-rotating" (I realize this is a hand-waving definition, hopefully you understand what I mean--if not I'll go into more detail) with the black hole, outside the horizon (more precisely, the "outer horizon", Kerr spacetime also has an "inner horizon" but that's not relevant here), forms a congruence of non-intersecting timelike worldlines that are not hypersurface orthogonal.

Perhaps the term "hypersurface orthogonal" is causing confusion. Consider the timelike congruence in Kerr spacetime that I just described. Obviously, if I pick any individual event on one of the worldlines in the congruence, I can find a local patch of spacelike hypersurface that is orthogonal to it. But that local patch of hypersurface will *not* be a piece of the global hypersurface of constant coordinate time t, because the worldline is not orthogonal to that global hypersurface. They "twist" around the hole. But that doesn't require them to intersect, since at any given radius r all the worldlines in the congruence are "twisting" around the hole at the same angle, so to speak.

It doesn't, you are conflating the Weyl principle and the cosmological principle again. When I said the Weyl postulate is a precondition of homogeneity I was only referring to an FRW metric, not to a general spacetime.

A historical question: wasn't the Weyl postulate specifically invented for cosmology? In what other contexts have you seen it used?

As you say probably the very fact that it is an empty universe doesn't allow to use the Weyl's postulate in the usual formulation for particle fluids.

Actually, I amended that somewhat in a follow-up post. In spacetimes with a non-zero cosmological constant (CC), the CC can be thought of as having a stress-energy tensor which is "fluid-like" (though it has a rather strange equation of state). So a fluid model would be applicable to cases like de Sitter spacetime.

The Weyl postulate does not require spacetime to have any particular symmetry, remember it's just a way of slicing up the manifold.

It's a way of slicing up the manifold, yes, but it's a way that assumes that the manifold has a particular symmetry that matches the slicing. The symmetry may not need to be *exact*; our universe appears to be only approximately homogeneous and isotropic, but one can still use the Weyl postulate to set up "comoving" coordinates that match the *average* behavior of the cosmological fluid pretty well, i.e., the "average" galaxy moves on a worldline that is hypersurface orthogonal. But if our universe weren't even approximately homogeneous and isotropic, you would not be able to choose a slicing that made even an "average" galaxy's worldline hypersurface orthogonal; it might not be possible at all, not even approximately.

I'd say those theorems make a lot of assumptions, see their wiki entry.

I'm not sure what you mean by "a lot of assumptions". Can you give any assumptions in particular that seem problematic to you?

Also, my main point was simply that, whatever assumptions there are, they do not include any assumptions about symmetry or the presence of families of worldlines or slicings of spacetime with any particular properties.

 

[PeterDonis]

For some reason I didn't see the following until after my previous post appeared:

We may have ensured a lot about causality , but the whole point is that the Weyl postulate is not a requirement on spacetime, it's just a way to slice it in order to obtain some coordinates

Yes, but it's also an assumption about the symmetry properties of the spacetime (which may be approximate, as I said in my previous post). If the spacetime does not have the requisite properties it may not be possible to find a slicing/set of coordinates that meets the requirements of the Weyl postulate.

even a supposedly "globally hyperbolic" spacetime can get acausal observers when the Weyl postulate is not used.

I'm not sure what you mean by "acausal observers", but I'll assume you mean observers whose local light cones can be "flipped" as you described earlier. If that's what you meant, then the claim just quoted is not correct. In fact, I see two things wrong with it.

First, the causal structure of a spacetime is invariant; it does not depend on any particular choice of coordinates, slicing, etc. So if "acausal observers" are present (or not), they are present (or not) whether or not you choose coordinates based on the kind of slicing described by the Weyl postulate. So whether or not you "use" the Weyl postulate can't have any effect on whether or not there are "acausal observers" present.

Second, as I said in my previous post, no "flip" in the light cone structure is possible even if the spacetime is just stably causal (not even globally hyperbolic). And, as I said in my previous post, a stably causal spacetime (or even a globally hyperbolic spacetime) may not admit a slicing that meets the Weyl postulate requirements, even approximately. So there is plenty of room for spacetimes that do not allow any "acausal observers" but are not even globally hyperbolic; and there is plenty of room for spacetimes that don't allow "acausal observers", *are* globally hyperbolic, but do not allow the Weyl postulate to be used (because they don't admit a slicing that satisfies it).

 

[DrGreg]

Please explain to me how do you get timelike worldlines in a 4-manifold that are not 3-hypersurface orthogonal to not intersect.

Well, I've mentioned Kerr spacetime several times now. The family of timelike geodesics which are "co-rotating" (I realize this is a hand-waving definition, hopefully you understand what I mean--if not I'll go into more detail) with the black hole, outside the horizon (more precisely, the "outer horizon", Kerr spacetime also has an "inner horizon" but that's not relevant here), forms a congruence of non-intersecting timelike worldlines that are not hypersurface orthogonal.

There is an even simpler example in flat spacetime: the worldlines of observers who are all at rest relative to a uniformly rotating disk. (Roughly speaking, the reason you can't form orthogonal hypersurfaces is the Sagnac Effect.) There is something about hypersurface orthogonality in the Wikipedia articles Born coordinates, Stationary spacetime, Ehrenfest paradox.

 

[TrickyDicky]

There is an even simpler example in flat spacetime: the worldlines of observers who are all at rest relative to a uniformly rotating disk. (Roughly speaking, the reason you can't form orthogonal hypersurfaces is the Sagnac Effect.) There is something about hypersurface orthogonality in the Wikipedia articles Born coordinates, Stationary spacetime, Ehrenfest paradox.

I forgot to say "in a curved manifold so that leaves flat spacetimes out, but I must remind you guys that the Kerr metric was ruled out as a valid example from the start of the thread (see post #30 by Mentz). So I'm leaving out rotating frames too, the reason is that by definition they can't be hypersurface orthogonal, and what the weyl's postulate demands is a manifold that has at least the possibility of such slicing.

 

[TrickyDicky]

Yes, but it's also an assumption about the symmetry properties of the spacetime (which may be approximate, as I said in my previous post). If the spacetime does not have the requisite properties it may not be possible to find a slicing/set of coordinates that meets the requirements of the Weyl postulate.

That's right, this is why Kerr spacetime is not a valid example. (See above post)

I'm not sure what you mean by "acausal observers", but I'll assume you mean observers whose local light cones can be "flipped" as you described earlier. If that's what you meant, then the claim just quoted is not correct. In fact, I see two things wrong with it.

First, the causal structure of a spacetime is invariant; it does not depend on any particular choice of coordinates, slicing, etc. So if "acausal observers" are present (or not), they are present (or not) whether or not you choose coordinates based on the kind of slicing described by the Weyl postulate. So whether or not you "use" the Weyl postulate can't have any effect on whether or not there are "acausal observers" present.

Second, as I said in my previous post, no "flip" in the light cone structure is possible even if the spacetime is just stably causal (not even globally hyperbolic). And, as I said in my previous post, a stably causal spacetime (or even a globally hyperbolic spacetime) may not admit a slicing that meets the Weyl postulate requirements, even approximately. So there is plenty of room for spacetimes that do not allow any "acausal observers" but are not even globally hyperbolic; and there is plenty of room for spacetimes that don't allow "acausal observers", *are* globally hyperbolic, but do not allow the Weyl postulate to be used (because they don't admit a slicing that satisfies it).

Precisely the point of this thread is to solve an incongruence in our own model of spacetime.
I've presented an example that might lead to consider all those causal structures not as invariants of the spacetimes but determined by a certain coordinate choice.

The problem is that so far the incongruence presented in the OP has not been refuted, hypersurface orthogonality (a coordinate condition) seems to be a requisite to keep the causality of the phsical laws (a few but very important ones) that are not time traslation invariant.

 

[PeterDonis]

I must remind you guys that the Kerr metric was ruled out as a valid example from the start of the thread (see post #30 by Mentz). So I'm leaving out rotating frames too, the reason is that by definition they can't be hypersurface orthogonal, and what the weyl's postulate demands is a manifold that has at least the possibility of such slicing.

I guess I'm not clear about why you are making such a restriction, since the whole point of this thread is to examine whether or not the restriction is actually necessary in order to ensure causality. It would seem necessary to look at spacetimes that do *not* admit a Weyl postulate slicing in order to address that question; nobody is disputing that a spacetime that *does* admit a Weyl postulate slicing has stable causality. The debate is entirely about the causal status of spacetimes that *don't* meet the Weyl postulate restriction.

I suspect what you really meant to say here is that Kerr spacetime is stationary (and for rotating frames in flat spacetime, the spacetime itself is static), and you want to talk about *non*-stationary spacetimes that may or may not meet the Weyl postulate restriction. That's fine, but everything I said about what is required for stable causality still applies to non-stationary spacetimes. See below.

I've presented an example that might lead to consider all those causal structures not as invariants of the spacetimes but determined by a certain coordinate choice.

Is the "example" you have in mind your speculation about the second law and universe expansion possibly being coordinate dependent? Or that somehow the presence of "cross terms" in the metric might make the second law not hold? If so, see next comment.

The problem is that so far the incongruence presented in the OP has not been refuted, hypersurface orthogonality (a coordinate condition) seems to be a requisite to keep the causality of the phsical laws (a few but very important ones) that are not time traslation invariant.

You are wrong, this "incongruence" has been refuted, by the theorems I referred to. Those theorems are certainly not restricted to stationary spacetimes; they apply perfectly well to non-stationary spacetimes. The theorems state that *any* spacetime that is stably causal (i.e., no closed causal curves and stable against small perturbations) has a global time function. That by itself is enough to ensure that physical observations that depend on "the direction of time", like the second law and universe expansion, are general covariant in the way that GR asserts.

The stably causal condition by itself is *not* enough to ensure that there is a Cauchy surface, which would guarantee a global "slicing" of the spacetime; that requires global hyperbolicity. And even global hyperbolicity is not enough to ensure that there is a slicing that is hypersurface orthogonal (i.e., no "cross terms" in the metric), which is what the Weyl postulate requires. So, as I said in a previous post, there is plenty of room for spacetimes in which the second law, universe expansion, etc. are general covariant, but which do *not* admit a slicing that satisfies the Weyl postulate. The "incongruence" you speak of is refuted.

 

[PeterDonis]

The theorems state that *any* spacetime that is stably causal (i.e., no closed causal curves and stable against small perturbations) has a global time function. That by itself is enough to ensure that physical observations that depend on "the direction of time", like the second law and universe expansion, are general covariant in the way that GR asserts.

I should have added, and the existence of a global time function is also enough to guarantee that the direction of time is "stable", i.e., it doesn't "flip over" in the way TrickyDicky described.

 

[TrickyDicky]

I guess I'm not clear about why you are making such a restriction, since the whole point of this thread is to examine whether or not the restriction is actually necessary in order to ensure causality. It would seem necessary to look at spacetimes that do *not* admit a Weyl postulate slicing in order to address that question; nobody is disputing that a spacetime that *does* admit a Weyl postulate slicing has stable causality. The debate is entirely about the causal status of spacetimes that *don't* meet the Weyl postulate restriction.
Is the "example" you have in mind your speculation about the second law and universe expansion possibly being coordinate dependent? Or that somehow the presence of "cross terms" in the metric might make the second law not hold? If so, see next comment.You are wrong, this "incongruence" has been refuted, by the theorems I referred to. Those theorems are certainly not restricted to stationary spacetimes; they apply perfectly well to non-stationary spacetimes. The theorems state that *any* spacetime that is stably causal (i.e., no closed causal curves and stable against small perturbations) has a global time function. That by itself is enough to ensure that physical observations that depend on "the direction of time", like the second law and universe expansion, are general covariant in the way that GR asserts.

I should have added, and the existence of a global time function is also enough to guarantee that the direction of time is "stable", i.e., it doesn't "flip over" in the way TrickyDicky described.

I'm afraid you haven't explicitly shown any theorems and the postulates or assumptions they build upon in a formal way, certainly mentioning some definitions in the wikipage about causality conditions doesn't account as refuting anything IMHO.

The global time function (or the cosmic time, to use the term Hawking used in his 1968 paper about stably causal spacetimes) assumes an agreement about a future-directed timelike which is precisely what you don't necessarily have in a curved non-stationary manifold like ours unless you slice it according to the condition that the ’particles’ in the universe lie on a congruence of time-like geodesics, that is the perfect fluid condition is a necessary assumption for the "global time function" .
When to the previous condition you add that the time-like geodesics diverge from from a point in the finite (or infinite) past, you get the globally hyperbolic manifold.
Now the confusing point in the Raych. eq. is that the congruence used there is indeed somewhat more general than the one just mentioned because as it says in the wikipage the timelike worldlines are not necessarily geodesics, but certainly in the GR solution for our universe they are geodesics, don't you think?
Basically the fundamental reason they have to be geodesics in GR and thus satisfy Weyl's postulate is that given the vanishing torsion of GR the worldlines can't twist around each other, the vorticity-free property is imposed on them because of the symmetric connection.

 

[PeterDonis]

I'm afraid you haven't explicitly shown any theorems and the postulates or assumptions they build upon in a formal way, certainly mentioning some definitions in the wikipage about causality conditions doesn't account as refuting anything IMHO.

Um, yes, I understand that Wikipedia in and of itself is not an authoritative source, but if you check the references on that page you will see that the theorem about any stably causal spacetime having a global time function was first proved by Hawking in a published paper (which I see you refer to in your statement below). The definitions of the various causality conditions are taken from relativity textbooks such as Hawking & Ellis. So this is not just stuff that someone on Wikipedia made up; the Wiki page just provides a nice short summary. If you really want me to bombard you with references, I'll start collecting links.

The global time function (or the cosmic time, to use the term Hawking used in his 1968 paper about stably causal spacetimes) assumes an agreement about a future-directed timelike which is precisely what you don't necessarily have in a curved non-stationary manifold like ours unless you slice it according to the condition that the ’particles’ in the universe lie on a congruence of time-like geodesics, that is the perfect fluid condition is a necessary assumption for the "global time function" .

It's true that the word "future" presumes a choice about which half of each light cone is the "future" half. But the point of the global time function is that, once you've established that the "future" half of the light cone points in the direction of the time function's gradient (i.e., time increases towards the future) at a single event in the spacetime, you've established it everywhere. (And if the time function's gradient points into the "past", then you just invert the sign of the time function to get another time function whose gradient points into the future.) Your proposed scenario of the direction of time "flipping around" from one observer to another is therefore ruled out if there is a global time function.

Also, none of this depends on a particular slicing of the manifold, or a perfect fluid condition, or anything else. It applies to any stably causal spacetime, which includes plenty of spacetimes that don't even admit slicings like the ones you describe.

When to the previous condition you add that the time-like geodesics diverge from from a point in the finite (or infinite) past, you get the globally hyperbolic manifold.

Global hyperbolicity has nothing to do with whether timelike geodesics diverge from a point in the past. All it means is that there's a Cauchy surface. Schwarzschild spacetime, for example, is globally hyperbolic, and geodesics don't diverge from a point there. The FRW spacetime is globally hyperbolic, yes, but that has nothing to do with the divergence of worldlines from the initial singularity.

Now the confusing point in the Raych. eq. is that the congruence used there is indeed somewhat more general than the one just mentioned because as it says in the wikipage the timelike worldlines are not necessarily geodesics, but certainly in the GR solution for our universe they are geodesics, don't you think?

There are congruences in the GR solution for our universe that are not geodesic congruences. It is true that the particular "comoving" congruence in FRW spacetimes is a geodesic congruence. But the Raychaudhuri equation covers any congruence, geodesic or not.

Basically the fundamental reason they have to be geodesics in GR and thus satisfy Weyl's postulate is that given the vanishing torsion of GR the worldlines can't twist around each other, the vorticity-free property is imposed on them because of the symmetric connection.

No, the vorticity in the Raychaudhuri equation has nothing to do with the fact that GR uses a torsion-free connection. The vorticity in the R equation is a "twist" in congruences of worldlines; the torsion-free connection is part of the derivation of the curvature tensor from the metric. They're two different things.

 

[TrickyDicky]

It's true that the word "future" presumes a choice about which half of each light cone is the "future" half. But the point of the global time function is that, once you've established that the "future" half of the light cone points in the direction of the time function's gradient (i.e., time increases towards the future) at a single event in the spacetime, you've established it everywhere. (And if the time function's gradient points into the "past", then you just invert the sign of the time function to get another time function whose gradient points into the future.) Your proposed scenario of the direction of time "flipping around" from one observer to another is therefore ruled out if there is a global time function.

When you say "once you've stablished..." , I guess you don't even realize that the way you stablish that in FRW manifolds is thru the Weyl's principle, now if you argue this, you need to go back to read some cosmological relativity texts.

Also, none of this depends on a particular slicing of the manifold, or a perfect fluid condition, or anything else. It applies to any stably causal spacetime, which includes plenty of spacetimes that don't even admit slicings like the ones you describe.

Please, we know there are very physically weird spacetime solutions of GR so let's keep the discssion strictly within the scope of spacetimes compatible with what we observe in our universe, the OP was about our spacetime and the models of our own spacetime.

Global hyperbolicity has nothing to do with whether timelike geodesics diverge from a point in the past. All it means is that there's a Cauchy surface. Schwarzschild spacetime, for example, is globally hyperbolic, and geodesics don't diverge from a point there. The FRW spacetime is globally hyperbolic, yes, but that has nothing to do with the divergence of worldlines from the initial singularity.

If youvread more about cosmology you'd see you're wrong here, a Cauchy surface is basically a spacelike hypersurface that acts as cosmic time and intersected by worldlines just once, sound familiar? Now put that in an expanding spacetime and guess what you get: timelike geodesics diverging. Cool, ain't it?
The Schwarzschild spacetime is static. And as I keep telling I'm restricting the analysis to FRW cosmologies. My claims about Weyl's pstulate are not general but referred to a very specific type of spacetime and GR.

There are congruences in the GR solution for our universe that are not geodesic congruences. It is true that the particular "comoving" congruence in FRW spacetimes is a geodesic congruence. But the Raychaudhuri equation covers any congruence, geodesic or not.

Very true, but since we want to apply the equation to FRW universes, guess what you find:a a timelike geodesic congruence, a.k.a the Weyl's postulate

No, the vorticity in the Raychaudhuri equation has nothing to do with the fact that GR uses a torsion-free connection. The vorticity in the R equation is a "twist" in congruences of worldlines; the torsion-free connection is part of the derivation of the curvature tensor from the metric. They're two different things.

Read carefully, I said the absence of vorticity, not the vorticity.
Wrong again. There is a very interesting explanation by John Baez in the web, I'll try to find the link, but basically the symmetric connection forces geodesic in GR to not twist.

http://math.ucr.edu/home/baez/gr/torsion.html

 

[PeterDonis]

When you say "once you've stablished..." , I guess you don't even realize that the way you stablish that in FRW manifolds is thru the Weyl's principle, now if you argue this, you need to go back to read some cosmological relativity texts.

Would you mind pointing me to a reference that describes how the Weyl postulate is used to establish which half of the light cone is the "future" half? The Weyl postulate deals with the assumption of homogeneity and isotropy, and the "comoving" worldlines of fluid elements being hypersurface orthogonal. It says nothing about which direction of time is "future" vs."past". If you are saying that the Weyl postulate somehow decrees that the "expanding" direction of time is the future, that may be the convention in cosmology because we observe the actual universe as a whole to be expanding; however, there are perfectly valid collapsing FRW models that obey the Weyl postulate, in the sense of having a congruence of "comoving" timelike worldlines that are hypersurface orthogonal. They are just converging instead of diverging. So physically, I don't see how the condition of hypersurface orthogonality picks out a preferred direction of time, even in a non-stationary spacetime; both the "expanding" and "contracting" versions of the FRW spacetimes are valid, physically speaking.

Please, we know there are very physically weird spacetime solutions of GR so let's keep the discssion strictly within the scope of spacetimes compatible with what we observe in our universe, the OP was about our spacetime and the models of our own spacetime.

As I said before, if you're going to argue that the Weyl postulate is *required* to establish causality, you need to show that it is *necessary*, which means you need to consider models where it doesn't hold and see if causality is still there. Nobody is disputing that the Weyl postulate is *sufficient* to establish causality. If we're going to restrict discussion to spacetimes compatible with what we actually observe, then there's nothing to be discussed, because we actually observe that the Weyl postulate holds to a certain approximation.

If youvread more about cosmology you'd see you're wrong here, a Cauchy surface is basically a spacelike hypersurface that acts as cosmic time and intersected by worldlines just once, sound familiar?

Yes. But remember that the presence of a Cauchy surface is a stronger condition than just the presence of a global time function.

Now put that in an expanding spacetime and guess what you get: timelike geodesics diverging. Cool, ain't it?

Here's what you said in the previous post of yours that I was responding to:

The global time function (or the cosmic time, to use the term Hawking used in his 1968 paper about stably causal spacetimes) assumes an agreement about a future-directed timelike which is precisely what you don't necessarily have in a curved non-stationary manifold like ours unless you slice it according to the condition that the ’particles’ in the universe lie on a congruence of time-like geodesics, that is the perfect fluid condition is a necessary assumption for the "global time function" .
When to the previous condition you add that the time-like geodesics diverge from from a point in the finite (or infinite) past, you get the globally hyperbolic manifold.

I understood you to be arguing that (a) you need the Weyl postulate with a perfect fluid to have a "global time function", and (b) you need that plus geodesics diverging to get global hyperbolicity. Both of those claims are false. (Even if we restrict attention *only* to non-stationary "expanding" spacetimes, they're false. If we restrict attention to only spacetimes that meet the Weyl postulate requirements, then as I said above, I don't see the point of this whole discussion.) If I misunderstood you and those claims aren't what you were saying, then what exactly were you saying? If you were only saying that the Weyl postulate with an expanding universe is *consistent* with a global time function and global hyperbolicity, of course I agree; but you appeared to be making a much stronger claim than that.

Very true, but since we want to apply the equation to FRW universes, guess what you find:a a timelike geodesic congruence, a.k.a the Weyl's postulate

For the Weyl postulate to hold the congruence has to be hypersurface orthogonal, i.e., vorticity-free. The Raychaudhuri equation is not limited to that case, even in FRW spacetimes; there are plenty of timelike worldline congruences in such spacetimes that are non-geodesic and/or not hypersurface orthogonal. See next comment.

Read carefully, I said the absence of vorticity, not the vorticity.
Wrong again. There is a very interesting explanation by John Baez in the web, I'll try to find the link, but basically the symmetric connection forces geodesic in GR to not twist.

http://math.ucr.edu/home/baez/gr/torsion.html

I'm quite familiar with that web page (and I agree it's a very good one). In particular, I read the part where it says:

Relatively few people understand why in GR we assume the connection --- the gadget we use to do parallel translation --- is torsion-free.

Do you understand what the bolded phrase means? It means that in GR, there is no twisting of a vector when you parallel transport it along a worldline. It says nothing about twisting of a congruence of worldlines relative to one another, which is what the vorticity in the Raychaudhuri equation refers to. Again, from the Baez page:

If, no matter how we choose P and Q and v, the time derivative of the distance between C(t) and D(t) at t = 0 is ZERO, up to terms proportional to epsilon^2, then the torsion is zero!

Again, the bolded phrase is crucial. Parallel transport deals with the first time derivative; but the vorticity in the Raychaudhuri equation, which is a particular piece of the curvature tensor, deals with the *second* time derivative, the part that would be proportional to epsilon squared, and which is *not* constrained by the torsion-free connection. So it is perfectly possible, as I said above, to have a congruence of timelike worldlines in an expanding FRW spacetime that has non-zero vorticity; the torsion-free nature of the connection used in GR does not prohibit that.

 

[PeterDonis]

...the vorticity in the Raychaudhuri equation, which is a particular piece of the curvature tensor, deals with the *second* time derivative, the part that would be proportional to epsilon squared, and which is *not* constrained by the torsion-free connection.

On re-reading, I should clarify the above. The vorticity is a property of a congruence of worldlines, not of the spacetime itself, so I shouldn't have said it was a "piece of the curvature tensor". What I should have said is that the vorticity is related to the curvature tensor; or perhaps a better way of stating it would be that the vorticity of a congruence of worldlines can be used to deduce properties of the curvature tensor. The key point, that the vorticity is not constrained by the torsion-free connection in GR, still stands.

 

[TrickyDicky]

Would you mind pointing me to a reference that describes how the Weyl postulate is used to establish which half of the light cone is the "future" half?

It turns out it does establish it. I gave you a reference with the explicit original wording of the postulate: "The particles of the substratum (representing the nebulae) lie in spacetime on a bundle of geodesics diverging from a point in the (finite or infinite) past". Remember this was 1923 without notion of expanding universe (Friedman had published his paper a few months earlier but at the time Weyl wrote his postulate he had not read it).
So the "future" half is established by the diverging direction.

The Weyl postulate deals with the assumption of homogeneity and isotropy, and the "comoving" worldlines of fluid elements being hypersurface orthogonal. It says nothing about which direction of time is "future" vs."past". If you are saying that the Weyl postulate somehow decrees that the "expanding" direction of time is the future, that may be the convention in cosmology because we observe the actual universe as a whole to be expanding; however, there are perfectly valid collapsing FRW models that obey the Weyl postulate, in the sense of having a congruence of "comoving" timelike worldlines that are hypersurface orthogonal. They are just converging instead of diverging. So physically, I don't see how the condition of hypersurface orthogonality picks out a preferred direction of time, even in a non-stationary spacetime; both the "expanding" and "contracting" versions of the FRW spacetimes are valid, physically speaking.

I've explained to you earlier that it doesn't deal with that assumption, is totally independent of it, it's just that in the FRW cosmology acts as a necessary precondition to the cosmology principle assumption.
In fact hypersurface orthogonality was an addition to the original Weyl's postulate made by Robertson when introducing the FRW metric, it is just a logic outcome of using the original postulate in an expanding FRW metric context.

As I said before, if you're going to argue that the Weyl postulate is *required* to establish causality, you need to show that it is *necessary*, which means you need to consider models where it doesn't hold and see if causality is still there. Nobody is disputing that the Weyl postulate is *sufficient* to establish causality. If we're going to restrict discussion to spacetimes compatible with what we actually observe, then there's nothing to be discussed, because we actually observe that the Weyl postulate holds to a certain approximation.

You have a confusion about what I argue and what I don't (and I admit it can be due to my sloppy way of argumenting). I'll try to clarify:I say that Weyl's postulate establish causality only in the case of the FRW cosmology.
You seemed to be arguing that Weyl postulate was not "sufficient" to establish causality above.

For the Weyl postulate to hold the congruence has to be hypersurface orthogonal, i.e., vorticity-free.

See above comment.

The Raychaudhuri equation is not limited to that case, even in FRW spacetimes; there are plenty of timelike worldline congruences in such spacetimes that are non-geodesic and/or not hypersurface orthogonal.

I don't agree with you here, as I explained the very "constructor" of the FRW metric, Robertson, used the Weyl p. as precondition and added the hypersurface orthogonality bit to the postulate.
I already agreed that the Raychaudhuri equation refers to a more general congruence than the used in the Weyl's postulate. But I explained that within torsion-free GR it amounts to the same one.

Do you understand what the bolded phrase means? It means that in GR, there is no twisting of a vector when you parallel transport it along a worldline. It says nothing about twisting of a congruence of worldlines relative to one another, which is what the vorticity in the Raychaudhuri equation refers to.

Let's see if we can reach some mutual understanding. Do you agree that due to torsion-free timelike geodesics are not allowed to twist in GR (rotate around their axis)?
Now let's quote the wikipedia page on the Raychaudhuri equation:"let \vec{X} be a timelike geodesic unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal. For example, this situation can arise in studying the world lines of the dust particles in cosmological models which are exact dust solutions of the Einstein field equation (provided that these world lines are not twisting about one another, in which case the congruence would have nonzero vorticity)."
I understand this last phrase to mean that worldlines twisting around each other would have nonzero vorticity, even if the wording is a bit confusing.
I infer from this that you are not correct when you say that vorticity is totally unrelated to torsion-free GR.
Also according to the quoted wiki paragraph I'd say it is not possible to have a congruence of timelike worldlines in an expanding FRW spacetime that has non-zero vorticity as you claim, that is precisely what the Weyl's postulate and hypersurface orthogonality in expanding FRW metric prohibit.

 

[PeterDonis]

I gave you a reference with the explicit original wording of the postulate: "The particles of the substratum (representing the nebulae) lie in spacetime on a bundle of geodesics diverging from a point in the (finite or infinite) past". Remember this was 1923 without notion of expanding universe (Friedman had published his paper a few months earlier but at the time Weyl wrote his postulate he had not read it).
So the "future" half is established by the diverging direction.

I agree that this reference establishes that Weyl, when he proposed the postulate, *claimed* that the "future" direction of time was established by the diverging direction. I'm not sure I agree that that claim is still physically valid, in the light of what we know today. Weyl was not only unaware of the expanding universe and the FRW models of same; he was also unaware of the "time reversed" versions of those models, the collapsing FRW models, for example the one used in the classic Oppenheimer-Snyder paper in 1939.

You have a confusion about what I argue and what I don't (and I admit it can be due to my sloppy way of argumenting). I'll try to clarify:I say that Weyl's postulate establish causality only in the case of the FRW cosmology.
You seemed to be arguing that Weyl postulate was not "sufficient" to establish causality above.

No, I am arguing that the Weyl postulate is not *necessary* to establish causality in the case of "expanding universe" cosmologies. I say "expanding universe" since it's more general than "FRW cosmology", which could be taken to restrict attention only to spacetimes that satisfy the Weyl postulate; and as I've said several times now, the whole question is whether such a restriction is *necessary* to establish causality, which means to answer the question you have to consider models that don't meet the restriction, and see whether causality still holds; if, as I claim, it does, then the Weyl postulate is not necessary for causality. I explicitly said in previous posts that the fact that the Weyl postulate is *sufficient* to establish causality is not in question.

I already agreed that the Raychaudhuri equation refers to a more general congruence than the used in the Weyl's postulate. But I explained that within torsion-free GR it amounts to the same one.

No, it doesn't. See below.

Let's see if we can reach some mutual understanding. Do you agree that due to torsion-free timelike geodesics are not allowed to twist in GR (rotate around their axis)?

No. See below.

Now let's quote the wikipedia page on the Raychaudhuri equation:"let \vec{X} be a timelike geodesic unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal. For example, this situation can arise in studying the world lines of the dust particles in cosmological models which are exact dust solutions of the Einstein field equation (provided that these world lines are not twisting about one another, in which case the congruence would have nonzero vorticity)."

Just to clarify, this part of the Wiki page is discussing a particular application of the Raychaudhuri equation, not the equation in general.

I understand this last phrase to mean that worldlines twisting around each other would have nonzero vorticity, even if the wording is a bit confusing.

I understand it the same way, provided that "worldlines twisting around each other" is interpreted correctly; see below. I agree the wording is not optimal (which is often the case with Wikipedia).

I infer from this that you are not correct when you say that vorticity is totally unrelated to torsion-free GR.

This is because you are confusing vorticity with the torsion of the connection; as I said in my last post, they are two different things. To see why, look again at that John Baez web page on torsion in GR that you linked to. It describes a thought experiment (unfortunately I don't know how to make Baez' ASCII art look the same here as it does on his page, so I'll leave out the drawings):

Take a tangent vector v at P. Parallel translate it along a very short curve from P to Q, a curve of length epsilon. We get a new tangent vector w at Q. Now let two particles free-fall with velocities v and w starting at the points P and Q. They trace out two geodesics...

Okay. Now, let's call our two geodesics C(t) and D(t), respectively. Here we use as the parameter t the proper time: the time ticked out by stopwatches falling along the geodesics. (We set the stopwatches to zero at the points P and Q, respectively.)

Now we ask: what's the time derivative of the distance between C(t) and D(t)? Note this "distance" makes sense because C(t) and D(t) are really close, so we can define the distance between them to be the arclength along the shortest geodesic between them.

If, no matter how we choose P and Q and v, the time derivative of the distance between C(t) and D(t) at t = 0 is ZERO, up to terms proportional to epsilon2, then the torsion is zero! And conversely! (One can derive this from the definition of torsion, assuming our recipe for parallel transport is metric preserving.)

If v got "rotated" a bit when we dragged it over to Q...then the time derivative of the distance would not be zero (it'd be proportional to epsilon). In this case the torsion would not be zero.

This thought experiment gives us a recipe for generating a congruence of timelike worldlines: start with some chosen worldline V, and pick a spacelike curve S that intersects V at point P, and call V's tangent vector at P, v. We also specify that V is a geodesic, so that its tangent vector at P is sufficient to specify it throughout the spacetime.

Now parallel transport v along curve S. Take any point Q of S, and call the parallel transported version of v at Q, w. Now find the timelike geodesic intersecting S at Q whose tangent vector at Q is w. The set of all such timelike geodesics, intersecting S, will form a congruence (with one caveat: I haven't worked out exactly what conditions the spacetime as a whole has to satisfy for this to be true, in the sense that the worldlines don't intersect unless the spacetime as a whole has a singularity, such as the initial singularity in FRW spacetime; see further comments below). And the torsion-free nature of the connection in GR does guarantee that this particular congruence will have vanishing vorticity.

However, the congruence I've just described is not necessarily the *only* congruence that might have a worldline intersecting spacelike surface S at point V with tangent vector v. There might be other such congruences, either because worldline V itself belongs to more than one congruence, or because there are other congruences that are non-geodesic but contain worldlines intersecting S at P with tangent vector v (for non-geodesic worldlines, the tangent vector at a point is not sufficient to specify a single worldline). The torsion-free connection does *not* prevent this. What the torsion-free connection does allow us to say is this: consider point Q on spacelike surface S, where the parallel transported tangent vector of worldline V is w. There may be another worldline passing through Q, call it Z, whose tangent vector z at Q is *different* from w; and Z may be part of some *other* congruence of timelike worldlines that includes either V, or some other worldline with the same tangent vector v at P. If that is the case, then this second congruence will have *non-zero* vorticity.

Now for the caveat: as I said above, I have not worked out specifically what conditions the spacetime as a whole has to satisfy for the recipe given above to produce a congruence of non-intersecting timelike geodesics. I believe that global hyperbolicity is sufficient; I suspect that even stable causality might be sufficient. If either of those is correct, then what I've said above will hold in a far more general set of spacetimes, even "expanding" non-stationary ones, than those which satisfy the Weyl postulate. (In fact, even in spacetimes which do satisfy the Weyl postulate, such as expanding FRW spacetimes, the torsion-free connection does not force all congruences of timelike geodesics to be vorticity-free; see next comment below.)

Also according to the quoted wiki paragraph I'd say it is not possible to have a congruence of timelike worldlines in an expanding FRW spacetime that has non-zero vorticity as you claim, that is precisely what the Weyl's postulate and hypersurface orthogonality in expanding FRW metric prohibit.

No, they don't. The postulate does not claim that *all* congruences of timelike worldlines in expanding FRW spacetime must be hypersurface orthogonal; it only claims that there *exists* such a congruence (the congruence of worldlines of "comoving" observers), and that that congruence describes the worldlines of the "particles" of the cosmological fluid. In other words, it claims that the cosmological fluid has vanishing vorticity; but there are plenty of other congruences of worldlines, which could describe families of observers who are *not* comoving with the fluid, and which could have non-zero vorticity.

 

[PeterDonis]

There may be another worldline passing through Q, call it Z, whose tangent vector z at Q is *different* from w; and Z may be part of some *other* congruence of timelike worldlines that includes either V, or some other worldline with the same tangent vector v at P. If that is the case, then this second congruence will have *non-zero* vorticity.

I should expand on this a little more. In fact, it could even be the case that there is another worldline passing through Q, call it Y, whose tangent vector at Q *is* w (the same as the geodesic W passing through Q which is part of the first congruence), but which is not a geodesic and therefore is not the same as W. Even in *this* case, the congruence containing V and Y can have non-zero vorticity. This possibility is what I was thinking of when I said that vorticity is related to curvature: if points P and Q are separated by distance epsilon, as in Baez' scenario, then even though worldlines V and Y have "the same" tangent vectors along surface S (i.e., one is the parallel transported version of the other), so the first time derivative of the "distance" between V and Y is zero at surface S, the *second* time derivative of that distance (the term proportional to epsilon squared instead of epsilon) can be non-zero, because worldlines V and Y curve differently, and so they might twist around each other taken as a whole, even though they are "parallel" for an instant as they cross surface S. Again, the torsion-free connection in GR does not prevent this. (And the different curvature of V and Y might tell us something about the curvature of the spacetime as well.)

 

[TrickyDicky]

This is because you are confusing vorticity with the torsion of the connection

I don't claim that vorticity and the torsion of the connection are the same thing

This thought experiment gives us a recipe for generating a congruence of timelike worldlines: start with some chosen worldline V, and pick a spacelike curve S that intersects V at point P, and call V's tangent vector at P, v. We also specify that V is a geodesic, so that its tangent vector at P is sufficient to specify it throughout the spacetime.

Now parallel transport v along curve S. Take any point Q of S, and call the parallel transported version of v at Q, w. Now find the timelike geodesic intersecting S at Q whose tangent vector at Q is w. The set of all such timelike geodesics, intersecting S, will form a congruence (with one caveat: I haven't worked out exactly what conditions the spacetime as a whole has to satisfy for this to be true, in the sense that the worldlines don't intersect unless the spacetime as a whole has a singularity, such as the initial singularity in FRW spacetime; see further comments below). And the torsion-free nature of the connection in GR does guarantee that this particular congruence will have vanishing vorticity.

This is what I'm saying, no more.

However, the congruence I've just described is not necessarily the *only* congruence that might have a worldline intersecting spacelike surface S at point V with tangent vector v. There might be other such congruences, either because worldline V itself belongs to more than one congruence, or because there are other congruences that are non-geodesic but contain worldlines intersecting S at P with tangent vector v (for non-geodesic worldlines, the tangent vector at a point is not sufficient to specify a single worldline). The torsion-free connection does *not* prevent this. What the torsion-free connection does allow us to say is this: consider point Q on spacelike surface S, where the parallel transported tangent vector of worldline V is w. There may be another worldline passing through Q, call it Z, whose tangent vector z at Q is *different* from w; and Z may be part of some *other* congruence of timelike worldlines that includes either V, or some other worldline with the same tangent vector v at P. If that is the case, then this second congruence will have *non-zero* vorticity

Once again this is trivial and I have said anything contrary to this, let's not distract from the OP.
It is quite obvious that a slicing of the manifold that models our spacetime that has cross terms of the type dxdt, dydt, dzdt, doesn?t guarantee the presence of a synchronous cosmic time and therefore doesn't guarantee an agreement on the second law for observers using such asynchronous time coordinate.
It doesn't matter at all whether you consider the universe is contracting or expanding as long as everybody agrees on which one is the case, since choosing the Wey¡s slicing what guarantees is the agreement on that not the particular direction one chooses -this said, I found very few people that like you seems ready to argue that our universe is contracting ;)

Not choosing this particular slicing of spacetime allows the disagreement among differently located observers.
I found this on the web that actually suits well part of what I'm trying to clarify in the OP
"Weyl's cosmic time thus becomes a global, standard clock time that applies to every observer in the universe, making possible simultaneity of events. Unfortunately, this kind of cosmic time flies in the face of relativity, where time is always relative, depending on things like particle velocity and gravitational effects. Consequently, Weyl's postulate appears to prevent a completely covariant treatment of the simple cosmological models that utilize his postulate "

No, they don't. The postulate does not claim that *all* congruences of timelike worldlines in expanding FRW spacetime must be hypersurface orthogonal; it only claims that there *exists* such a congruence (the congruence of worldlines of "comoving" observers), and that that congruence describes the worldlines of the "particles" of the cosmological fluid. In other words, it claims that the cosmological fluid has vanishing vorticity; but there are plenty of other congruences of worldlines, which could describe families of observers who are *not* comoving with the fluid, and which could have non-zero vorticity.

Sure, those happen not to be geodesics,wich are the type of worldlines that I'm referring to from the start.

 

[PeterDonis]

I don't claim that vorticity and the torsion of the connection are the same thing

Ok, good. I wasn't sure based on your previous posts, but now I understand better where you were coming from.

It is quite obvious that a slicing of the manifold that models our spacetime that has cross terms of the type dxdt, dydt, dzdt, doesn?t guarantee the presence of a synchronous cosmic time...

True, in the sense that such a slicing will not correspond to a global "comoving" frame.

...and therefore doesn't guarantee an agreement on the second law for observers using such asynchronous time coordinate.

False. One does not need to be at rest in a global "comoving" frame in order to agree on the second law; the spacetime does not even have to *admit* a global "comoving" frame. All that needs to be true is that all observers agree on the direction of time, in the sense of agreeing on which half of each local light cone is the "future" half, and on the definition of that direction being continuous throughout the spacetime. That is guaranteed by a much weaker set of conditions than the presence of a global "comoving" frame, as I showed in previous posts.

It doesn't matter at all whether you consider the universe is contracting or expanding as long as everybody agrees on which one is the case, since choosing the Wey¡s slicing what guarantees is the agreement on that not the particular direction one chooses -this said, I found very few people that like you seems ready to argue that our universe is contracting ;)

I wasn't arguing that our actual universe is contracting, just that there are valid FRW-type models in which the future direction of time is the contracting direction. I agree that the important point is global agreement on the direction of time, as I said above.

I found this on the web that actually suits well part of what I'm trying to clarify in the OP
"Weyl's cosmic time thus becomes a global, standard clock time that applies to every observer in the universe, making possible simultaneity of events. Unfortunately, this kind of cosmic time flies in the face of relativity, where time is always relative, depending on things like particle velocity and gravitational effects. Consequently, Weyl's postulate appears to prevent a completely covariant treatment of the simple cosmological models that utilize his postulate "

This looks to me like an equivocation on the word "simultaneity". It is true that the time coordinate of a global "comoving" frame can be used to set up a global sense of simultaneity. However, it is *not* true that this sense of simultaneity will coincide with the *local* sense of simultaneity (meaning the simultaneity of the local Lorentz frame) of *every* observer in the universe, whatever their state of motion. And the claim that having the global "simultaneity" somehow contradicts relativistic covariance requires the latter to be true, not the former. So the claim is false.

Another way to put this is to imagine an observer who is not at rest in the "comoving" frame of the universe, and suppose that he wants to set his clock by the global "cosmic time". He will find that he has to build in a correction to the clock's rate; the "natural" rate of ticking of his clock, which is determined by his proper time, will *not* be the same as the rate of ticking of cosmic time (which he could check by exchanging light signals with another observer who *is* at rest in the "comoving" frame, and whose proper time is the same as cosmic time). In other words, "cosmic" time is *not* the same as proper time for any observer who is not at rest in the "comoving" frame. And that means that the presence of the "comoving" frame, and the decision to adopt its time as the global "cosmic" time, does *not* contradict relativistic covariance; that would only be contradicted if observers not at rest in the comoving frame somehow found that their proper time *was* the same as cosmic time, and they won't.

For example: the worldline of the Earth is *not* a "comoving" worldline; we see a large dipole anisotropy in the CMBR, for example. Therefore, the global sense of simultaneity that is provided by the global "comoving" frame for our actual universe is *not* the same as the local sense of simultaneity here on Earth. That is, a pair of events which are simultaneous according to the global "cosmic time" of the "comoving" frame are *not* simultaneous to us here on Earth. The difference is small, and it is normally not an issue in cosmology because we don't need a level of accuracy where the difference would be significant, but it's there. Our proper time here on Earth is *not* the same as cosmic time. We could, if we chose, decide to adopt a "cosmic time standard", so that we recorded the times of events, for the record, as their "cosmic" times instead of according to our local Earth proper time; but we would then have to build corrections into all our clocks, precisely because relativistic covariance works. Only if we found that our clocks somehow kept "cosmic time" *without* needing correction would we have any reason to doubt relativistic covariance.

(And it's also worth noting, as I've said before, that we here on Earth observe the second law to hold and the universe to be expanding, even though we are not at rest in the "comoving" frame.)

Can you provide a link to the full article you quoted?

 

[TrickyDicky]

True, in the sense that such a slicing will not correspond to a global "comoving" frame.


False. One does not need to be at rest in a global "comoving" frame in order to agree on the second law; the spacetime does not even have to *admit* a global "comoving" frame. All that needs to be true is that all observers agree on the direction of time, in the sense of agreeing on which half of each local light cone is the "future" half, and on the definition of that direction being continuous throughout the spacetime. That is guaranteed by a much weaker set of conditions than the presence of a global "comoving" frame, as I showed in previous posts.

I see I can't manage to make you understand what I mean. I never said anything about needing to be at rest in the comoving "frame". If a coordinate system with cross terms is used there is not even a defined comoving frame to have the possibility wrt which be at rest.
It's about using different coordinates, not about frames in the sense of state of motion.
You say: "All that needs to be true is that all observers agree on the direction of time..." and yet you don't realize that using a different coordinate system with crossed terms is what precisely would prevent you from having that agreement.

Another way to put this is to imagine an observer who is not at rest in the "comoving" frame of the universe, and suppose that he wants to set his clock by the global "cosmic time". He will find that he has to build in a correction to the clock's rate; the "natural" rate of ticking of his clock, which is determined by his proper time, will *not* be the same as the rate of ticking of cosmic time (which he could check by exchanging light signals with another observer who *is* at rest in the "comoving" frame, and whose proper time is the same as cosmic time). In other words, "cosmic" time is *not* the same as proper time for any observer who is not at rest in the "comoving" frame. And that means that the presence of the "comoving" frame, and the decision to adopt its time as the global "cosmic" time, does *not* contradict relativistic covariance; that would only be contradicted if observers not at rest in the comoving frame somehow found that their proper time *was* the same as cosmic time, and they won't.

For example: the worldline of the Earth is *not* a "comoving" worldline; we see a large dipole anisotropy in the CMBR, for example. Therefore, the global sense of simultaneity that is provided by the global "comoving" frame for our actual universe is *not* the same as the local sense of simultaneity here on Earth. That is, a pair of events which are simultaneous according to the global "cosmic time" of the "comoving" frame are *not* simultaneous to us here on Earth. The difference is small, and it is normally not an issue in cosmology because we don't need a level of accuracy where the difference would be significant, but it's there. Our proper time here on Earth is *not* the same as cosmic time. We could, if we chose, decide to adopt a "cosmic time standard", so that we recorded the times of events, for the record, as their "cosmic" times instead of according to our local Earth proper time; but we would then have to build corrections into all our clocks, precisely because relativistic covariance works. Only if we found that our clocks somehow kept "cosmic time" *without* needing correction would we have any reason to doubt relativistic covariance.

You keep using this example as if it were relevant to the discussion. It is not, it doesn't matter at all that we may use a different local time as long as it is still calculated in terms of the global cosmic time, if it is referenced to cosmic time it means we are using the comoving observers slicing. The coreect example would be using a coordinate system that doesn't allow to be referenced to comoving observers, that is one with the cross terms, in this coordinate system's metric there can't be no agreement between certain separate observers as to what direction time goes.

Can you provide a link to the full article you quoted?

It's not an article, it's just some guy with a blog on the web, I only included it to see if using someone else's words helped, Obviously, it didn't:

 

[PeterDonis]

You say: "All that needs to be true is that all observers agree on the direction of time..." and yet you don't realize that using a different coordinate system with crossed terms is what precisely would prevent you from having that agreement

You're right, I don't "realize" why that would have to be true. It amounts to saying that no two observers in relative motion can agree on the direction of time. That's obviously absurd. See following comments.

You keep using this example as if it were relevant to the discussion. It is not, it doesn't matter at all that we may use a different local time as long as it is still calculated in terms of the global cosmic time, if it is referenced to cosmic time it means we are using the comoving observers slicing.

No, it doesn't. Consider the Earth example again. Our local proper time on Earth, and the simultaneity associated with it, automatically implies a slicing of spacetime that is different from the "comoving" one. That has to be the case because we are not at rest in the "comoving" frame. Relative motion, and the consequent change in the local surfaces of simultaneity, is all that is required to change the slicing that "local" time is based on. But relative motion, by itself (and even if it includes non-zero vorticity and consequent "cross terms" in the metric--see next comment), is *not* enough to change the perceived direction of time.

The coreect example would be using a coordinate system that doesn't allow to be referenced to comoving observers, that is one with the cross terms, in this coordinate system's metric there can't be no agreement between certain separate observers as to what direction time goes.

Yes, there can. You keep confusing agreement on the *direction* of time, which only requires agreement on which half of the light cones is the "future" half, with agreement on the *surfaces of simultaneity*, which is a much stronger restriction, and is *not* required for agreement on causality, the second law, etc.

 

[TrickyDicky]

Once againg you are confusing frames, motion and coordinates.
Quote:"You keep confusing agreement on the *direction* of time, which only requires agreement on which half of the light cones is the "future" half, with agreement on the *surfaces of simultaneity*, which is a much stronger restriction, and is *not* required for agreement on causality, the second law, etc."
In your opinion how exactly is agreement on which half of the light cones is the "future" half achieved in the FRW metric?