TrickyDicky said:
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.
TrickyDicky said:
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.
TrickyDicky said:
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.
TrickyDicky said:
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.