# Worldline congruence and general covariance

1. Sep 11, 2011

### 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 univers 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?

2. Sep 11, 2011

### tom.stoer

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.

3. Sep 11, 2011

### TrickyDicky

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?

4. Sep 11, 2011

### Staff: Mentor

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.

5. Sep 11, 2011

### tom.stoer

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

6. Sep 11, 2011

### TrickyDicky

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

7. Sep 11, 2011

### Staff: Mentor

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.

8. Sep 11, 2011

### Staff: Mentor

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.

9. Sep 11, 2011

### TrickyDicky

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?

10. Sep 11, 2011

### TrickyDicky

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

11. Sep 11, 2011

### Staff: Mentor

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 any one 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.

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

12. Sep 12, 2011

### TrickyDicky

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 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.

Last edited: Sep 12, 2011
13. Sep 12, 2011

### Staff: Mentor

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.

Last edited: Sep 12, 2011
14. Sep 12, 2011

### Staff: Mentor

I meant this one:

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

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).

15. Sep 12, 2011

### TrickyDicky

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.

16. Sep 12, 2011

### PAllen

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. ]

Last edited: Sep 12, 2011
17. Sep 12, 2011

### Staff: Mentor

No problem here.

Agreed.

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:

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.

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

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.

18. Sep 12, 2011

### TrickyDicky

I'm fine with this uses.

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?

19. Sep 12, 2011

### PAllen

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.

20. Sep 12, 2011

### TrickyDicky

I'm always referring to timelike geodesics hypersurface orthogonal.

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.

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.

21. Sep 12, 2011

### TrickyDicky

Thanks Pallen, you got it.

22. Sep 12, 2011

### PAllen

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.

23. Sep 12, 2011

### Staff: Mentor

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).

Ah, I see; that makes sense.

24. Sep 12, 2011

### Staff: Mentor

Ok.

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

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).

25. Sep 13, 2011

### TrickyDicky

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.

Last edited: Sep 13, 2011