Worldline congruence and general covariance

[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?

 

[PeterDonis]

Once againg you are confusing frames, motion and coordinates.

It seems to me that you are often using language which invites confusion as to what you are trying to say. For example, consider the quote of yours that I was responding to that prompted your latest post:

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.

You speak of "a coordinate system that doesn't allow to be referenced to comoving observers", but the worldlines of such comoving observers are coordinate-independent, geometric objects in the spacetime; they don't somehow disappear or become non-describable when I adopt coordinates in which the comoving observers are not at rest. The comoving worldlines are still there, and they can still be described even in a different coordinate systems; their description just won't look as simple. But all their invariant features (such as, for example, the fact that they are orthogonal to a particular set of spacelike hypersurfaces) can still be calculated and verified in any coordinate system.

You also speak of "that coordinate system's metric", but the metric, as a geometric object, is coordinate-independent; what changes from one coordinate system to another is only the *expression* of the metric in terms of the coordinate differentials. Changing coordinates certainly doesn't change the invariants that depend on the metric, and those invariants include the causal structure of the spacetime, i.e., the light cones and their sense of orientation. So if I have agreement, with reference to one coordinate system, as to which half of the light cones is the "future" half, and that sense of the light cones is continuous throughout the spacetime (which it must be if the spacetime is stably causal--see below), then since that is part of the causal structure, it is coordinate-independent; changing coordinate systems doesn't change it, so if I have such agreement in one coordinate system, I have it in any coordinate system, including one with "cross terms" in its expression for the metric.

Reading your quote above as it stands, it appears to deny what I just wrote. If you did not intend to do that, then what did you mean when you wrote what I quoted above?

In your opinion how exactly is agreement on which half of the light cones is the "future" half achieved in the FRW metric?

To answer the question as you stated it, obviously in our actual universe we experience time to flow in the direction that the universe is expanding. So the half of the light cones in which the universe is larger is obviously the "future" half. It's worth noting once more, though, that as I said above, obtaining such agreement does *not* require adopting the "comoving" coordinate system; agreement on the sense of direction of the light cones can be obtained in any coordinate system, and since it is invariant once obtained, it will hold in any coordinate system. For example, we can choose, here on Earth, which half of our light cones is the "future" half based on which direction of time has the universe expanding, and if we compared our choice with that of a "comoving" observer, made using the same criterion, we would find agreement.

However, as I've pointed out repeatedly now, the question you asked in the quote above is the wrong question to ask. The question you should be asking is:

Is there a way to get agreement on which half of the light cones is the future half, in a non-stationary spacetime that does *not* meet the conditions of the Weyl postulate?

Answer: yes, as long as it is stably causal, so a global time function can be defined. The gradient of the global time function is everywhere timelike, and we can choose its increasing direction as the future direction of time; or, if we decide that the decreasing direction makes more sense, we can simply invert the sign of the global time function, to get a new global time function whose gradient is likewise timelike, but points in the opposite direction. In other words, any global time function can be used to obtain a global agreement on which half of the light cones is the "future" half. So it can be done in any stably causal spacetime, which includes spacetimes that do not meet the conditions of the Weyl postulate.

You might also ask another question: Does the global time function guarantee that, once we've obtained agreement on which half of the light cones is the "future" half in a stably causal spacetime, that future direction won't "flip over" from one observer to another along a spacelike surface? We know that can't happen in a spacetime that satisfies the Weyl postulate; so the question is, could it happen in a spacetime that is stably causal and has a global time function, but does not satisfy the Weyl postulate?

Answer: no, it can't happen. Here's why: pick a spacelike hypersurface, and suppose that at some point on it, point A, the gradient of the global time function picks out the "future" half of the light cone as pointing one way. Now ask: what would have to happen for the gradient of the global time function to point the other way at some other point, B, on the same spacelike hypersurface? That could only happen in one of two ways: at some point, C, between A and B, the gradient would either have to go to zero, or else it would have to be tangent to the surface. But since the gradient is everywhere timelike, neither of those things can happen: a zero vector is not timelike (because it has a zero norm, and a timelike vector can't have a zero norm); and the surface is spacelike, so a timelike vector can't be tangent to it. So the global time function, or more precisely the fact that its gradient is everywhere timelike, guarantees that the direction of time can't "flip".

One other thing to remember: as I've said before, the existence of a global time function, as defined above (i.e, a scalar with a gradient that is everywhere timelike and future-directed), does *not* guarantee that the spacetime must satisfy the Weyl postulate. It doesn't even guarantee the existence of a Cauchy surface, and even a spacetime with a Cauchy surface may not satisfy the Weyl postulate; a Cauchy surface implies the existence of a global slicing of the spacetime, but it does not, by itself, guarantee that there is a congruence of "comoving" worldlines which are everywhere orthogonal to the slicing. The phrase "time function" by itself is ambiguous, and the quotes you've given have shown that some authors use it to imply a much tighter constraint than the standard definition does; in this post (and indeed in all my posts in this thread), I am using the term only to refer to its standard definition.

 

[TrickyDicky]

To answer the question as you stated it, obviously in our actual universe we experience time to flow in the direction that the universe is expanding. So the half of the light cones in which the universe is larger is obviously the "future" half.

This is the right question because apparently is the one you can't answer.
How is that so obvious to you, that we "experience", built into the FRW metric, mathematically?

 

[PeterDonis]

This is the right question because apparently is the one you can't answer.
How is that so obvious to you, that we "experience", built into the FRW metric, mathematically?

It isn't. The FRW metric is equally valid, mathematically, for either direction of time (expanding or contracting), as I've said several times. We have to make a choice, based on actual observation, that the expanding model better fits the data for cosmology. If we observed the universe to be contracting in the "future" direction of time, we would choose a contracting FRW spacetime as our model, and the math would work just as well. The math alone can't make the choice.

Having said that, I noted in my last post that none of the questions about the direction of time depend on the use, or even the existence, of a set of "comoving" worldlines that are hypersurface orthogonal. So what does the question you just asked, and I just answered, have to do with the Weyl postulate? Wouldn't the same question apply just as well if the actual data showed an expanding universe that wasn't homogeneous and isotropic (so the Weyl postulate was not satisfied and we had to adopt a somewhat different expanding spacetime model, one that didn't have a congruence of "comoving" worldlines that was hypersurface orthogonal)?

 

[TrickyDicky]

It isn't. The FRW metric is equally valid, mathematically, for either direction of time (expanding or contracting), as I've said several times. We have to make a choice, based on actual observation, that the expanding model better fits the data for cosmology. If we observed the universe to be contracting in the "future" direction of time, we would choose a contracting FRW spacetime as our model, and the math would work just as well. The math alone can't make the choice.

Having said that, I noted in my last post that none of the questions about the direction of time depend on the use, or even the existence, of a set of "comoving" worldlines that are hypersurface orthogonal. So what does the question you just asked, and I just answered, have to do with the Weyl postulate? Wouldn't the same question apply just as well if the actual data showed an expanding universe that wasn't homogeneous and isotropic (so the Weyl postulate was not satisfied and we had to adopt a somewhat different expanding spacetime model, one that didn't have a congruence of "comoving" worldlines that was hypersurface orthogonal)?

Once again the postulate was written before there was observations leading to think of expansion. And it already provided a cosmic time and a way to agree about time direction, not about how to label that agreement. The key word here is agreement, not whether we call it future or past.

If you don't know what my question has to do with the Weyl's postulate, I'm afraid we need to leave it here until you do.

 

[PeterDonis]

Once again the postulate was written before there was observations leading to think of expansion. And it already provided a cosmic time and a way to agree about time direction, not about how to label that agreement

I agree that it provided a cosmic time. I don't know that I agree that it provided a way to agree about time direction. I saw that you quoted Weyl as saying that in his model, the geodesics diverged from a single point a finite time in the past. The problem is that mathematically, the time reverse of that model, which has the geodesics converging on a point a finite time in the future, is just as valid. Hypersurface orthogonality alone doesn't pick out a direction of time; both models have comoving worldlines that are hypersurface orthogonal. And hypersurface orthogonality is the only condition I see in the Weyl postulate. Did Weyl give any reason for preferring the expanding model over the contracting one, given that hypersurface orthogonality does not pick out either one over the other? Or, whether Weyl gave an argument or not, do *you* have an argument that somehow gets from hypersurface orthogonality to expansion being preferred over contraction? Or is there some other reason for picking the expanding model if one doesn't already know, by observation, that the universe is expanding?

One other question: how does any of this relate to general covariance? I see the question about the direction of time, but I don't see how it has anything to do with general covariance. Even if general covariance holds (which it does), the question about the direction of time is still there.