Worldline congruence and general covariance

[TrickyDicky]

According to Weyl's postulate timelike geodesics should be hypersurface orthogonal, this in itself seems to clash with the GR principle that there should be no physically preferred frame or slicing of the spacetime manifold (general covariance).
Usually there is much insistence in textbooks that the preferred frame choice of standard cosmology, the comoving chart, is not a physical one but a practical one in terms of calculations, which is sometimes difficult to discern (the physical versus practical subtletty).
The worldline congruence indeed sometimes seems quite physical, mathematically is the same thing that we apply to stationary models when we want to make them static (by making the timelike killing fields hypesurface orthogonal so there is no time-space mixed components) but in our case there is no killing fields of course.
I say that the congruence seems very physical because if we didn't have it I find it hard to understand things like the second law of thermodynamics and the consensus we all have on the direction of time, or the fact that we only observe retarded potentials,or the very fact that the universe is expanding for everyone-if we didn't have the congruence of timelike geodesics, the universe could be expanding for some contracting for others and neither for others depending on the coordinate system they used- all these seem to be "physical" consequences of having worldline congruence.
So can someone explain to me why we share all those physical observations if the congruence itself is not physical but a choice to make calculations in GR easier?

 

[tom.stoer]

According to Weyl's postulate timelike geodesics should be hypersurface orthogonal, this in itself seems to clash with the GR principle that there should be no physically preferred frame or slicing of the spacetime manifold (general covariance).

There is no clash.

Consider a point plus the local light cone. Through this point there are infinitly many timelike geodesics (inside the lightcone), each defining a spacelike hypersurface. Of course all those timelike geodesics and all hypersurfaces (as the spacelike parts of reference frames) are equivalent. The coordinate transformations between these (locally defined) reference frames are (locally) Lorentz transformations.

So there is no preferred referece frame.

 

[TrickyDicky]

There is no clash.

Consider a point plus the local light cone. Through this point there are infinitly many timelike geodesics (inside the lightcone), each defining a spacelike hypersurface. Of course all those timelike geodesics and all hypersurfaces (as the spacelike parts of reference frames) are equivalent. The coordinate transformations between these (locally defined) reference frames are (locally) Lorentz transformations.

So there is no preferred referece frame.

You seem to be missing my point. What you just wrote is in accordance with what I wrote should happen in GR. I'm not claiming there is a preferred frame.
I was asking about our empirical observation of the congruence in the above mentioned ways, have you considered answering the question at the end of the OP?

 

[PeterDonis]

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

I'm not sure I understand. A local Lorentz transformation can't change the direction of time; only a Lorentz transformation combined with a time reversal can. So, for example, if we look at all the possible local Lorentz frames passing through the event of me writing this post at this instant, even though all those frames will look very different (they will have very different hypersurfaces of simultaneity, they will see the CMBR very differently, etc.), all those frames will agree on the direction of time, the second law of thermodynamics, the fact that the universe is expanding, etc.

 

[tom.stoer]

So, for example, if we look at all the possible local Lorentz frames passing through the event of me writing this post at this instant, even though all those frames will look very different (they will have very different hypersurfaces of simultaneity, they will see the CMBR very differently, etc.), all those frames will agree on the direction of time, the second law of thermodynamics, the fact that the universe is expanding, etc.

minor correction: all those frames will agree on the interior of the light cone whereas the directions of time which are perpendicular to the spatial hypersurfaces will not be identical

 

[TrickyDicky]

I'm not sure I understand. A local Lorentz transformation can't change the direction of time; only a Lorentz transformation combined with a time reversal can. So, for example, if we look at all the possible local Lorentz frames passing through the event of me writing this post at this instant, even though all those frames will look very different (they will have very different hypersurfaces of simultaneity, they will see the CMBR very differently, etc.), all those frames will agree on the direction of time, the second law of thermodynamics, the fact that the universe is expanding, etc.

I'm talking about GR, not SR, global frames not local frames, Lorentz transformations and light cones have nothing to do with my post.

 

[PeterDonis]

minor correction: all those frames will agree on the interior of the light cone whereas the directions of time which are perpendicular to the spatial hypersurfaces will not be identical

I should have been more precise, yes. By "direction of time" I meant only which half of the local light cone is the "future" half, and which is the "past" half. A Lorentz transformation by itself leaves that invariant, so all of the local inertial frames at a given event will agree on it, even though they will not agree on which particular timelike vector within the light cone is "the direction of time" orthogonal to their particular hypersurface of simultaneity.

 

[PeterDonis]

I'm talking about GR, not SR, global frames not local frames, Lorentz transformations and light cones have nothing to do with my post.

But all the local frames have to fit consistently into a global frame, and neighboring local frames have to match up with each other in a smooth, continuous manner, regardless of which global congruence of worldlines I choose. So if I have two different global congruences of worldlines, call them A and B, and what I said holds locally at a given event E, then it applies locally to the Lorentz transformation between the worldline from A that passes through E, and the worldline from B that passes through E. And if this is true at E, then it must also be true for events in the local neighborhood of E, by continuity. So eventually, by continuity, we get that what I said must hold, locally, at *every* event E. And if what I said holds for every event, locally, then it must hold, globally, for the entire congruences A and B. In other words, A and B must also agree globally on things like the second law of thermodynamics, whether the universe is expanding, etc.

 

[TrickyDicky]

But all the local frames have to fit consistently into a global frame, and neighboring local frames have to match up with each other in a smooth, continuous manner, regardless of which global congruence of worldlines I choose. So if I have two different global congruences of worldlines, call them A and B, and what I said holds locally at a given event E, then it applies locally to the Lorentz transformation between the worldline from A that passes through E, and the worldline from B that passes through E. And if this is true at E, then it must also be true for events in the local neighborhood of E, by continuity. So eventually, by continuity, we get that what I said must hold, locally, at *every* event E. And if what I said holds for every event, locally, then it must hold, globally, for the entire congruences A and B. In other words, A and B must also agree globally on things like the second law of thermodynamics, whether the universe is expanding, etc.

What you write is true, but I'm talking about the global congruence, locally (in SR) there is clearly a preferred coordinate frame, the uniform motion frame, this is not the case in GR.
By the way if you argument that if something holds for every local event then it must hold globally is correct why don't you apply it to the conservation of energy issue of the other thread?

 

[TrickyDicky]

Peter, You seem to be thinking in terms of local events but th OP was about timelike geodesics.

 

[PeterDonis]

What you write is true, but I'm talking about the global congruence, locally (in SR) there is clearly a preferred coordinate frame, the uniform motion frame, this is not the case in GR.

I'm not sure I see the distinction you're making. In SR, it's true that the local "preference" for inertial frames clearly implies a global "preference" for congruences of inertial worldlines. But that does not mean that anyone congruence of inertial worldlines is "preferred" over any other. So there is a sort of "preferred" *type* of frame in SR, but there is not a single "preferred" frame.

In any case, what I was saying doesn't even require restricting consideration to "preferred" *types* of frames. In Minkowski spacetime, *all* timelike observers will agree on which half of the light cone is "future" and which is "past", not just inertial observers. The same goes for GR. So in our universe, for example, *all* timelike observers will agree on whether the universe is expanding, regardless of their state of motion, inertial or not. Same for the second law of thermodynamics. So I don't see that our observation of these physical phenomena "picks out" any particular global congruence of timelike worldlines over any other.

By the way if you argument that if something holds for every local event then it must hold globally is correct why don't you apply it to the conservation of energy issue of the other thread?

I may have just done so; check out the post I made a few minutes ago.

 

[TrickyDicky]

I'm not sure I see the distinction you're making. In SR, it's true that the local "preference" for inertial frames clearly implies a global "preference" for congruences of inertial worldlines. But that does not mean that anyone congruence of inertial worldlines is "preferred" over any other. So there is a sort of "preferred" *type* of frame in SR, but there is not a single "preferred" frame.

In any case, what I was saying doesn't even require restricting consideration to "preferred" *types* of frames. In Minkowski spacetime, *all* timelike observers will agree on which half of the light cone is "future" and which is "past", not just inertial observers. The same goes for GR. So in our universe, for example, *all* timelike observers will agree on whether the universe is expanding, regardless of their state of motion, inertial or not. Same for the second law of thermodynamics. So I don't see that our observation of these physical phenomena "picks out" any particular global congruence of timelike worldlines over any other.

You are throwing out the window the very reason GR was introduced, namely curvature. What you are saying is valid for flat spacetimes, not for curved ones.

I may have just done so; check out the post I made a few minutes ago.

I can't find that post.
Do you mean that you consider energy globally conserved in GR? You seem to be making the same mistake as pointed out above here, that works for flat spacetimes, not for curved ones.

 

[PeterDonis]

You are throwing out the window the very reason GR was introduced, namely curvature. What you are saying is valid for flat spacetimes, not for curved ones.

I don't understand why you're saying this. I was using Minkowski spacetime as an example, but what I was saying applies to curved spacetimes as well. For example, what I was saying is valid for an FRW model of our own universe; it amounts to saying that you don't have to be at rest relative to the "preferred" cosmological frame, the one in which the universe looks homogeneous and isotropic, in order to agree which half of each local light cone is the "future" half, or to agree that the universe is expanding, the second law of thermodynamics holds, etc. Certainly we here on Earth are not at rest relative to the preferred cosmological frame--we see a large dipole anisotropy in the CMBR, indicating that we are moving relative to such a frame. And yet we see the universe as expanding, we see the second law of thermodynamics hold, etc.

So I certainly didn't intend what I was saying to only apply to flat spacetimes. I guess I'm not really understanding the problem you pose in the OP, since I don't see any "preferred" frame, or global timelike congruence of worldlines, emerging from the issues you pose.

 

[PeterDonis]

I can't find that post.

I meant this one:

https://www.physicsforums.com/showpost.php?p=3496654&postcount=7

Do you mean that you consider energy globally conserved in GR?

It depends on the spacetime. The ADM energy can be defined and is conserved in an asymptotically flat spacetime. The Komar mass (which is also a conserved total energy) can be applied in a stationary spacetime. Both of these definitions basically boil down to finding a time translation symmetry for the spacetime and then using Noether's theorem.

But for a spacetime like the FRW models used in cosmology, nobody has figured out how to define a conserved total energy (none of the known ones, such as those I mentioned above, work), and the fact that the spacetime has no time translation symmetry makes it highly probable that there is no way to define a conserved energy (since Noether's theorem doesn't apply).

 

[TrickyDicky]

I guess I'm not really understanding the problem you pose in the OP, since I don't see any "preferred" frame, or global timelike congruence of worldlines, emerging from the issues you pose.

Clearly this is what happens, because the examples you use are not related to what is asked in the OP.

Let's see if I can make myself clearer, do you agree that a worldline in Minkowski spacetime is different from a worldline in curved GR spacetime? I hope so.
In that case you'll understand that in SR you can have globally inertial coordinates while in GR due to curvature that is not possible, they're only local.
Therefore the fact that in SR you can have worldlines with worldline congruence is trivial from the fact that as you say every event has Lorentz invariance.
Do you see how this is not trivial in GR, and therefore the Weyl postulate has no justification in the theory, it is just assumed according to most sources as a practical assumption that makes calculations easier, and yet we (that live in a GR universe, not a Minkowski one) have all those physical observations of the congruence.

 

[PAllen]

According to Weyl's postulate timelike geodesics should be hypersurface orthogonal, this in itself seems to clash with the GR principle that there should be no physically preferred frame or slicing of the spacetime manifold (general covariance).
Usually there is much insistence in textbooks that the preferred frame choice of standard cosmology, the comoving chart, is not a physical one but a practical one in terms of calculations, which is sometimes difficult to discern (the physical versus practical subtletty).
The worldline congruence indeed sometimes seems quite physical, mathematically is the same thing that we apply to stationary models when we want to make them static (by making the timelike killing fields hypesurface orthogonal so there is no time-space mixed components) but in our case there is no killing fields of course.
I say that the congruence seems very physical because if we didn't have it I find it hard to understand things like the second law of thermodynamics and the consensus we all have on the direction of time, or the fact that we only observe retarded potentials,or the very fact that the universe is expanding for everyone-if we didn't have the congruence of timelike geodesics, the universe could be expanding for some contracting for others and neither for others depending on the coordinate system they used- all these seem to be "physical" consequences of having worldline congruence.
So can someone explain to me why we share all those physical observations if the congruence itself is not physical but a choice to make calculations in GR easier?

My understanding of the Weyl postulate is a bit different from what you describe (or, at least, my understanding of what you describe). I see it used in a couple of different ways:

1) As a restriction on interesting solutions. Completely analogous to hypothesizing spherical symmetry, then seeing if you can find a class of solutions meeting this. So you hypothesize the Weyl postulate and see if you can find solutions satisfying it (FRW is one example). A physical (rather than mathematical question) is whether such solutions match our universe. They seem to, pretty well.

2) Given a solution derived under some other assumptions, determining whether it satisfies the postulate by finding out whether there is a family of hypersurfaces meeting the condition. This use is different from (1) in that if assumed as a constraint, and you don't make a mistake, you know that the solutions satisfy it.

I have never seen it used purely as computational device without addressing its validity in one of the prior senses. Can you post an example of someone doing this?

[EDIT: addressing another question, the existence of a foliation with hypersurfaces meeting the postulate does not require you to use this chart for any particular physical calculation. The postulate requires the existence of a chart of a certain type, general covariance guarantees you can use other charts of the same manifold for computations if you prefer. The computational convenience issue is simply the observation that doing so will generally make your job harder. ]

 

[PeterDonis]

Let's see if I can make myself clearer, do you agree that a worldline in Minkowski spacetime is different from a worldline in curved GR spacetime? I hope so.

No problem here.

In that case you'll understand that in SR you can have globally inertial coordinates while in GR due to curvature that is not possible, they're only local.

Agreed.

Therefore the fact that in SR you can have worldlines with worldline congruence is trivial from the fact that as you say every event has Lorentz invariance.

Ok, I see a couple of points where I didn't see where you were coming from.

First of all, saying that "every event has Lorentz invariance" is ambiguous. When I said it, I was referring to *local* Lorentz invariance; every event has that in a curved spacetime as well as a flat one. However, in a curved spacetime, if I take the worldline of a particular freely falling particle (i.e., a particular timelike geodesic), the local Lorentz frames at different events on that worldline will not "line up" with each other, whereas in a flat spacetime, they do. So if we mean by "every event has Lorentz invariance" that the local Lorentz frames at different events must all "line up" globally, as I think you are using the term, then obviously that can only happen in a flat spacetime. (I realize "line up" is a fuzzy term, hopefully you understand what I mean. If not, I can try to deploy the appropriate more precise mathematical definitions, but that will take some time.)

That brings up the second point: it looks like you are using "congruence" differently from the way I thought you were. It appears that by "worldline congruence" you mean that the local Lorentz frames at different events along a geodesic worldline "line up" as just described. But the term "congruence" has a (different) standard meaning, as defined here:

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

Here's the key definition:

In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation.

Obviously you can have congruences of timelike curves in a curved spacetime by this definition, and these types of congruences are what I was talking about. Furthermore, in a curved spacetime, the local Lorentz frames along worldlines belonging to a timelike geodesic congruence (e.g., the worldlines of particles freely falling from "infinity" towards a black hole) do not have to "line up" as they do in flat spacetime, so although they belong to a "congruence" by the above definition, they are not "congruent" in the sense I think you were using the term.

However, I admit I'm not entirely sure I understand how you are using the term "congruence", or rather I'm not sure you're using it consistently. See next comment.

Do you see how this is not trivial in GR, and therefore the Weyl postulate has no justification in the theory, it is just assumed according to most sources as a practical assumption that makes calculations easier, and yet we (that live in a GR universe, not a Minkowski one) have all those physical observations of the congruence.

Here it looks to me like you are using "congruence" in a different way, to denote the fact that (after correcting for our own motion relative to the "comoving" frame of the universe, which we do by correcting for the dipole anisotropy we observe in the CMBR) we observe the universe to be homogeneous and isotropic, which is consistent with the Weyl postulate. Here's the Wikipedia definition of the postulate, from here:

http://en.wikipedia.org/wiki/Weyl's_postulate

In relativistic cosmology, Weyl's postulate stipulates that in a fluid cosmological model, the world lines of the fluid particles, which act as the source of the gravitational field and which are often taken to model galaxies, should be hypersurface orthogonal. That is, the world lines should be everywhere orthogonal to a family of spatial hyperslices.

Sometimes, the additional hypothesis is added that the world lines form timelike geodesics.

I think there is a key phrase left out here, namely that the worldlines should be everywhere orthogonal to a family of spatial hyperslices in each of which the spatial metric is homogeneous and isotropic. You can always find a family of spatial hyperslices that is orthogonal to *any* set of timelike worldlines, but the spatial metric induced on each hyperslice might look really weird if the set of worldlines is not well chosen. For a fluid model to work, or at least to be simple enough to be useful, you basically have to have homogeneity and isotropy, so the stress-energy tensor is manageable. (Btw, I don't see what the "additional hypothesis" adds at all, since *any* family of worldlines that is orthogonal to a family of spatial hyperslices has to be timelike.)

For example, in the standard FRW models of the universe, the family of timelike worldlines that is used to define the model is orthogonal to a family of spatial hyperslices in each of which the spatial metric is homogeneous and isotropic. The timelike worldlines that have this property are called "comoving", and objects moving on "comoving" worldlines are "at rest" in the global FRW coordinate system in which the FRW metrics are standardly written.

But the Solar System, for example, does *not* move along such a "comoving" worldline, and so the universe does *not* look homogeneous and isotropic from a position at rest in the Solar System. For example, we see a large dipole anisotropy in the CMBR, as I noted above. We have to correct for that anisotropy to confirm that, to a "comoving" observer, the universe *does* look homogenous and isotropic.

Suppose, then, that we set up a cosmological coordinate system that was "Solar System centric", i.e., the worldline of the Solar System is taken to be one of a family of timelike geodesics that are "at rest" in the coordinate system. (We'll idealize by assuming that we can define the "Solar System" worldline all the way back to the Big Bang.) The rest of the family of worldlines is defined by assuming that they are all "at rest" with respect to the Solar System, along the hypersurface of simultaneity of the Solar System at the present instant. We could set up such a coordinate system, and find a family of spatial hyperslices that were everywhere orthogonal to the entire family of "Solar System centric" worldlines. And of course this would be a *different* set of spatial hyperslices than those used in the standard FRW coordinates.

Obviously, the spatial metric of the hyperslices in a "Solar System centric" cosmological coordinate system would not be homogeneous and isotropic. But that would not change the fact that the universe would be seen to be expanding, or the validity of the second law of thermodynamics, etc., etc. Nor would it change the fact of local Lorentz invariance at each event; we can still set up the Solar System's local Lorentz frame, even though it is "moving" relative to the local Lorentz frame of a "comoving" observer whose worldline is just passing through the Solar System at this instant. (In fact, the transformation we do on our CMBR data to "convert" it to a "comoving" frame is precisely a Lorentz transformation, using a boost to remove our velocity relative to a "comoving" observer. If we were not able to apply such a boost, without any additional transformation, and obtain "comoving" data that was actually homogeneous and isotropic, the FRW models would not be as useful as they are.)

So I'm still not seeing how the Weyl postulate contradicts GR's insistence that there is no "preferred frame". The Weyl postulate is, as noted, a practical convenience; it amounts to saying that calculations are easier when you can pick a coordinate system that matches up with some symmetry of the spacetime. In the FRW case, the symmetry is homogeneity and isotropy of the cosmological fluid; and the fact that we can "correct" our observations for the CMBR dipole anisotropy and find that homogeneity and isotropy hold to a good approximation is good evidence that the actual spacetime of our universe has, to a good approximation, the symmetry that makes the FRW coordinates useful. But we are not *required* to adopt those coordinates; we could calculate the same answers in the "Solar System centric" frame if we wanted, it would just be more complicated.

I also note that in the definition of the Weyl postulate quoted above, the words "in a fluid cosmological model" appear. Of course GR does not *require* you to use a fluid model, and if you don't, the Weyl postulate has nothing to say. So that's another way in which the Weyl postulate does not appear to require a "preferred frame". It's just a way of describing how to construct a manageable fluid model.

 

[TrickyDicky]

My understanding of the Weyl postulate is a bit different from what you describe (or, at least, my understanding of what you describe). I see it used in a couple of different ways:

1) As a restriction on interesting solutions. Completely analogous to hypothesizing spherical symmetry, then seeing if you can find a class of solutions meeting this. So you hypothesize the Weyl postulate and see if you can find solutions satisfying it (FRW is one example). A physical (rather than mathematical question) is whether such solutions match our universe. They seem to, pretty well.

2) Given a solution derived under some other assumptions, determining whether it satisfies the postulate by finding out whether there is a family of hypersurfaces meeting the condition. This use is different from (1) in that if assumed as a constraint, and you don't make a mistake, you know that the solutions satisfy it.

[EDIT: addressing another question, the existence of a foliation with hypersurfaces meeting the postulate does not require you to use this chart for any particular physical calculation. The postulate requires the existence of a chart of a certain type, general covariance guarantees you can use other charts of the same manifold for computations if you prefer. The computational convenience issue is simply the observation that doing so will generally make your job harder. ]

I'm fine with this uses.

I have never seen it used purely as computational device without addressing its validity in one of the prior senses. Can you post an example of someone doing this?

I allude to the computational device use in a different sense. Have you never heard that given general covariance of the theory one can choose whatever coordinates, but that we use the comoving chart just because it makes calculations easier, not because of any physical preference?
That is what I'm contrasting with physical observables that can be associated to a physical worldline congruence.
Maybe the observables I chose have nothing to do with the preferred slicing?

 

[PAllen]

I allude to the computational device use in a different sense. Have you never heard that given general covariance of the theory one can choose whatever coordinates, but that we use the comoving chart just because it makes calculations easier, not because of any physical preference?
That is what I'm contrasting with physical observables that can be associated to a physical worldline congruence.
Maybe the observables I chose have nothing to do with the preferred slicing?

Yes to your first question. Your last question, I think, gets at the heart of the matter you want to discuss. If there were claimed physical observables that were inherently tied to a preferred slicing, that would be a major problem. I assume there aren't any. However, I think I see that you are asking something like the following:

- All book discussions of the the second law in cosmology use a (computationally) preferred slicing. Can someone justify that these things can really be dealt with in an arbitrary coordinate chart?

I can understand wanting to be satisfied on this point. Unfortunately, I can't help you further as thermodynamics in GR is outside of my expertise.

 

[TrickyDicky]

However, I admit I'm not entirely sure I understand how you are using the term "congruence"

I'm always referring to timelike geodesics hypersurface orthogonal.

I think there is a key phrase left out here, namely that the worldlines should be everywhere orthogonal to a family of spatial hyperslices in each of which the spatial metric is homogeneous and isotropic.

It is not left out, that is just a different principle (cosmological principle) that should not be mixed with the Weyl postulate, even though most people conflates these two, and in much of what you post it seems you find it hard to differentiate them too. It is true that the FRW models demand both.

So I'm still not seeing how the Weyl postulate contradicts GR's insistence that there is no "preferred frame". The Weyl postulate is, as noted, a practical convenience; it amounts to saying that calculations are easier when you can pick a coordinate system that matches up with some symmetry of the spacetime. But we are not *required* to adopt those coordinates; we could calculate the same answers in the "Solar System centric" frame if we wanted, it would just be more complicated.

This is basically my point, the theory says just that, my question is why the physical observations mentioned in the OP seem to hinting at a physical requirement rather than to just practical convenience.

 

[TrickyDicky]

Yes to your first question. Your last question, I think, gets at the heart of the matter you want to discuss. If there were claimed physical observables that were inherently tied to a preferred slicing, that would be a major problem. I assume there aren't any. However, I think I see that you are asking something like the following:

- All book discussions of the the second law in cosmology use a (computationally) preferred slicing. Can someone justify that these things can really be dealt with in an arbitrary coordinate chart?

I can understand wanting to be satisfied on this point. Unfortunately, I can't help you further as thermodynamics in GR is outside of my expertise.

Thanks Pallen, you got it.

 

[PAllen]

Here's the Wikipedia definition of the postulate, from here:

http://en.wikipedia.org/wiki/Weyl's_postulate



I think there is a key phrase left out here, namely that the worldlines should be everywhere orthogonal to a family of spatial hyperslices in each of which the spatial metric is homogeneous and isotropic. You can always find a family of spatial hyperslices that is orthogonal to *any* set of timelike worldlines, but the spatial metric induced on each hyperslice might look really weird if the set of worldlines is not well chosen. For a fluid model to work, or at least to be simple enough to be useful, you basically have to have homogeneity and isotropy, so the stress-energy tensor is manageable. (Btw, I don't see what the "additional hypothesis" adds at all, since *any* family of worldlines that is orthogonal to a family of spatial hyperslices has to be timelike.)

The wikipedia formulation does seem imprecise to me, but they get at the issue the spatial hypersurfaces should have a 'nice metric' a little later with the following phrase:

"One consequence of this hypothesis is that if it holds true, we can introduce a comoving chart such that the metric tensor contains no terms of form dt dx, dt dy, or dt dz."

Only I think(?) it isn't a consequence, but must be formulated into the definition.

As to the additional hypothesis, my understanding is the key word is not 'timelike' but 'geodesic'. This distinguishes pressure free dust solutions from more general solutions.

 

[PeterDonis]

The wikipedia formulation does seem imprecise to me, but they get at the issue the spatial hypersurfaces should have a 'nice metric' a little later with the following phrase:

"One consequence of this hypothesis is that if it holds true, we can introduce a comoving chart such that the metric tensor contains no terms of form dt dx, dt dy, or dt dz."

These terms don't appear in the induced metric on a spatial hyperslice; I think the latter metric could still be "messy" (i.e., not even close to the spatial metric of a perfect fluid, for example) even if the metric for the spacetime as a whole lacked these "cross terms". However, it may be that I was wrong to think the Weyl postulate requires the induced spatial metric on each hyperslice to be homogeneous and isotropic; it may be that it's meant to apply to a wider class of solutions (although in that case I'm not sure the term "fluid model" would be valid for all possible solutions that the postulate could apply to).

As to the additional hypothesis, my understanding is the key word is not 'timelike' but 'geodesic'. This distinguishes pressure free dust solutions from more general solutions.

Ah, I see; that makes sense.

 

[PeterDonis]

I'm always referring to timelike geodesics hypersurface orthogonal.

Ok.

It is not left out, that is just a different principle (cosmological principle) that should not be mixed with the Weyl postulate, even though most people conflates these two, and in much of what you post it seems you find it hard to differentiate them too. It is true that the FRW models demand both.

On reflection, I think I agree; see my previous post in response to PAllen.

This is basically my point, the theory says just that, my question is why the physical observations mentioned in the OP seem to hinting at a physical requirement rather than to just practical convenience.

Well, there's at least one piece of evidence that the physical observations you cite don't pick out a preferred frame: as I have pointed out already, we, here in the Solar System, are *not* at rest in the "comoving" FRW coordinates! And yet we see the universe as expanding, we see the second law of thermodynamics being observed, etc. So if those observations do pick out a "preferred" frame, it is *not* the "comoving" one, it's the "Solar System centric" one! But that doesn't seem very plausible; it seems much more likely that the physical observations you cite hold for *any* possible frame (i.e., for any congruence of timelike geodesics that agrees with ours, and with the "comoving" one, on which half of the local light cones is the "future" half).

 

[TrickyDicky]

it seems much more likely that the physical observations you cite hold for *any* possible frame (i.e., for any congruence of timelike geodesics that agrees with ours, and with the "comoving" one, on which half of the local light cones is the "future" half).

Exactly, they hold for any timelike hypersurface orthogonal congruence and of course for any that is referred to the comoving chart like the "solar system centric" one because it is also hypersurface orthogonal.
But what I find puzzling (though I seem to be the only one that does) is that they don't hold for frames that are not hypersurface orthogonal, because according to general covariance they should, GR admits any coordinate transformation, including the ones with crossed-terms (i.e. dxdt,dydt,dzdt). And the second law is not just any law, it seems to be quite fundamental. But it's not just this as I said in the OP it also affects observations in EM (retarded vs advanced potentials), the congruence in the sense of expansion, the congruence in sharing all the same time arrow, even in the weak interaction we wouldn't have an agreement on CP-violation if we chose a coordinate system with timelike geodesics not hypersurface orthogonal, but such a slicing should be physically indistinguishible with the commonly used according to GR.
Maybe someone is aware of how this is dealt with in GR.

 

[PeterDonis]

But what I find puzzling (though I seem to be the only one that does) is that they don't hold for frames that are not hypersurface orthogonal

What makes you think they don't?

 

[TrickyDicky]

What makes you think they don't?

The math. If you have two people using non-timelike orthogonal hypersurface coordinates, (or one using it and other using regular comoving chart) they'll disagree about the interior and exterior of their light cones and about all I mentioned before.
However, actually we all share the same observations about those physical situations (they are physical laws invariants).

 

[PeterDonis]

The math. If you have two people using non-timelike orthogonal hypersurface coordinates, (or one using it and other using regular comoving chart) they'll disagree about the interior and exterior of their light cones and about all I mentioned before.

Maybe we're having a terminology problem again. Can you give an example of "non-timelike orthogonal hypersurface coordinates"? As far as I know, *any* valid set of coordinates has to agree on the interior and exterior of light cones, since those are invariant. Valid coordinate systems could disagree about the direction of time (which half of the light cone was "future" and which was "past"), and therefore about whether the universe was expanding or whether the second law of thermodynamics was valid, but that has nothing to do with whether the coordinates are hypersurface orthogonal; it's just a binary choice that is made when setting up coordinates. Also, being able to find a set of hypersurface orthogonal coordinates is a property of the spacetime; not all spacetimes admit such a set of coordinates.

Maybe I should give an example to make it clearer where I'm coming from. Kerr spacetime does not admit a hypersurface orthogonal coordinate system; no matter how you set up coordinates, there will be at least one $dt d\phi$ "cross term" in the metric that can't be eliminated. But, for example, observers who are at rest in a coordinate system that is "not rotating" with respect to the asymptotic spacetime at infinity will agree with observers who are at rest in a coordinate system that is "rotating with the black hole" on where the light cones are, which direction of time is the "future" (assuming both coordinate systems make the same choice of which half of of the light cones is the "future" half), that the second law of thermodynamics is valid, etc.

The two coordinate systems I mention for Kerr spacetime are described in the Wikipedia article here:

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

Btw, I realize on reading back through previous posts that I may have caused some confusion in my description of the "Solar System centric" coordinates, in relation to the standard "comoving" FRW coordinates for our universe. I did not mean to imply that the "Solar System centric" coordinates are hypersurface orthogonal; I am not sure they would be.

 

[TrickyDicky]

Maybe we're having a terminology problem again. Can you give an example of "non-timelike orthogonal hypersurface coordinates"? As far as I know, *any* valid set of coordinates has to agree on the interior and exterior of light cones, since those are invariant. Valid coordinate systems could disagree about the direction of time (which half of the light cone was "future" and which was "past"), and therefore about whether the universe was expanding or whether the second law of thermodynamics was valid, but that has nothing to do with whether the coordinates are hypersurface orthogonal; it's just a binary choice that is made when setting up coordinates. Also, being able to find a set of hypersurface orthogonal coordinates is a property of the spacetime; not all spacetimes admit such a set of coordinates.

Maybe I should give an example to make it clearer where I'm coming from. Kerr spacetime does not admit a hypersurface orthogonal coordinate system; no matter how you set up coordinates, there will be at least one $dt d\phi$ "cross term" in the metric that can't be eliminated. But, for example, observers who are at rest in a coordinate system that is "not rotating" with respect to the asymptotic spacetime at infinity will agree with observers who are at rest in a coordinate system that is "rotating with the black hole" on where the light cones are, which direction of time is the "future" (assuming both coordinate systems make the same choice of which half of of the light cones is the "future" half), that the second law of thermodynamics is valid, etc.

Well, the Kerr spacetime is not the best example here given the fact that being stationary its time derivatives all vanish and therefore besides not having a worldline congruence (cross term that can't be eliminated) it is time traslation invariant. So it is not possible for them to agree on which direction of time is the "future" that the second law of thermodynamics is valid, etc. because first of all time doesn't flow in such manifolds and second we can't assume they make the same choice of which half of of the light cones is the "future" half because it is not timelike hypersurface orthogonal.
In any case, Kerr spacetime as we know has nothing to do with our manifold and I'm asking about what happens in our non-stationary universe.

 

[Mentz114]

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

 

[TrickyDicky]

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

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

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

 

[PeterDonis]

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[TrickyDicky]

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

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

 

[PeterDonis]

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

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

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

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

 

[Mentz114]

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

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

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

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

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

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

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