I About the meaning of "expanding universe"

[cianfa72]

TL;DR Summary

About the meaning of expanding universe from the point of view of FRW models

As required I start this thread on the meaning of "expanding universe" in the context of GR.

FRW standard models have a special timelike congruence named "comoving" congruence. One can pick an adapted global chart in which the comoving congruence's worldlines are "at rest". Such a chart defines the cosmological time $t$. In general FRW models are not stationary (i.e. they have not a timelike KVF), yet the timelike comoving congruence is hypersurface orthogonal (irrotational).

That said, in general the geometry of spacelike hypersurfaces of constant coordinate time $t$ isn't the same.

In any of such spacelike hypersurfaces, one can take the spacelike geodesic (geodesic curve w.r.t. the metric induced on the hypersurface from the spacetime metric) connecting the timelike worldlines of the galaxies' CoM.

"Expanding universe" just means the "proper length" of the above "restricted spacelike geodesic curves" keeps increasing with the coordinate time.

 

[Ibix]

I think you are overcomplicating it rather, while missing some important points.

Expanding universe means that you can pick a coordinate system such that $ds^2=dt^2-a(t)d\Sigma^2$, where $d\Sigma^2$ is a spatial metric that is homogeneous and independent of time, and where $da/dt>0$. If you want a coordinate-free definition, it would be that the expansion scalar of the congruence you mention is a function solely of the proper time along the worldlines of the congruence.

The important point you missed is that the distances don't just increase, they scale. If the distance from A to B is twice the distance from B to C, this will always be so whatever the actual value of the distances in question. If they were increasing in different ratios I wouldn't call that "expansion" - or at least I think it would need qualifying.

Note also that, while it does mean that the geometry of spatial surfaces is not the same, the only difference is the scale factor. With that factored out, the rest of the spatial metric is independent of time.

 

[cianfa72]

Expanding universe means that you can pick a coordinate system such that $ds^2=dt^2-a(t)d\Sigma^2$, where $d\Sigma^2$ is a spatial metric that is homogeneous and independent of time, and where $da/dt>0$.

Ok, the above metric form means that the timelike congruence "at rest" in such global chart/coordinate system is hypersurface orthogonal (there aren't mixed terms involving $t$ and some other "spatial" coordinate).

If you want a coordinate-free definition, it would be that the expansion scalar of the congruence you mention is a function solely of the proper time along the worldlines of the congruence.

Ok.

The important point you missed is that the distances don't just increase, they scale. If the distance from A to B is twice the distance from B to C, this will always be so whatever the actual value of the distances in question. If they were increasing in different ratios I wouldn't call that "expansion" - or at least I think it would need qualifying.

Ok, the distance you were talking about is the length of the shortest (spacelike) "restricted geodesic curve" connecting for instance A and B on any spacelike hypersurface of constant cosmological time.

Note also that, while it does mean that the geometry of spatial surfaces is not the same, the only difference is the scale factor. With that factored out, the rest of the spatial metric is independent of time.

Ok, so the scale factor on any spacelike hypersurface of constant cosmological time $t$ is actually $a(t)$.

 

[PeterDonis]

"Expanding universe" just means the "proper length" of the above "restricted geodesic curves" keeps increasing with the coordinate time.

This is one way of stating it, but a more compact way, which does not depend on picking out any particular set of spacelike hypersurfaces, is to say that the expansion scalar of the comoving congruence is positive.

 

[PeterDonis]

If you want a coordinate-free definition, it would be that the expansion scalar of the congruence you mention is a function solely of the proper time along the worldlines of the congruence.

I'm pretty sure this is true for any timelike congruence.

The coordinate-free definition of "expanding universe" for the specific case of FRW models is that the expansion scalar of the comoving congruence is everywhere positive.

 

[cianfa72]

I'm pretty sure this is true for any timelike congruence.

Do you mean that holds true for any timelike congruence in any spacetime ?

Btw, when it comes to the question in the other thread, do FRW models have any CTC ?

 

[Ibix]

I'm pretty sure this is true for any timelike congruence.

But it's the same function of time for all worldlines, which is the relevant thing here (and I note I didn't say). Otherwise couldn't you have a case where the expansion scalar increased in some spatial direction at fixed cosmological time?

 

[martinbn]

But it's the same function of time for all worldlines, which is the relevant thing here (and I note I didn't say). Otherwise couldn't you have a case where the expansion scalar increased in some spatial direction at fixed cosmological time?

That would violate isotropy. This diriection would be diffetent from the others.

 

[Ibix]

That would violate isotropy. This diriection would be diffetent from the others.

Yes, that's another way of stating the criticism I'm making. If we're trying to define "an expanding universe" and we want it to be isotropic we either need to say "it's isotropic" or imply that somehow, and I think Peter's "expansion scalar is everywhere positive" doesn't do that. I accept his criticism of my original statement, and I think "the expansion scalar is the same function of proper time along every worldline in the congruence" does do the job.

Or, on second thought, perhaps Peter is taking the more general view that "an expanding universe" is just one in which the expansion scalar is everywhere positive, which is a more general class of spacetimes than FLRW and includes anisotropic ones.

 

[cianfa72]

Yes, that's another way of stating the criticism I'm making. If we're trying to define "an expanding universe" and we want it to be isotropic we either need to say "it's isotropic" or imply that somehow, and I think Peter's "expansion scalar is everywhere positive" doesn't do that.

By isotropic do you mean spatially isotropic? In other words the induced metric on any spacelike hypersurface of constant cosmological time has the 3 rotational symmetries (i.e. the Lie algebra of KVFs of the induced metric is 3-dimensional and a basis is given by 3 independent rotational KVFs).

 

[Ibix]

By isotropic do you mean spatially isotropic?

I'm not really sure how a Lorentzian metric could be isotropic in any sense other than spatially. A plane that contains a spacelike and a timelike vector is necessarily anisotropic, surely.

 

[PeterDonis]

Do you mean that holds true for any timelike congruence in any spacetime ?

I believe so, yes.

when it comes to the question in the other thread

Stop hijacking your own threads. If you want to ask a different question, start a different thread.

 

[PeterDonis]

If we're trying to define "an expanding universe" and we want it to be isotropic

These are two different things. Defining "expanding" is done in terms of the expansion scalar, as I said. Adding isotropy as another requirement imposes further constraints on the expansion scalar besides it just being positive everywhere, yes.

 

[Ibix]

These are two different things. Defining "expanding" is done in terms of the expansion scalar, as I said. Adding isotropy as another requirement imposes further constraints on the expansion scalar besides it just being positive everywhere, yes.

Right, so it's terminology. "Expanding universe" usually refers to an FLRW (or nearly FLRW on a large scale) spacetime, but strictly doesn't have to. Any spacetime you can fill with a timelike congruence withan everywhere positive expansion scalar is expanding in that broader sense, as you say. In the usual FLRW sense you also need to say something like "isotropic space" or "same dependence on time along a worldline for all worldlines in the congruence" or something.

It's not clear to me which one the OP means.

 

[cianfa72]

Any spacetime you can fill with a timelike congruence with an everywhere positive expansion scalar is expanding in that broader sense, as you say.

Ah ok, so one can define for any timelike congruence an expansion scalar.

Btw, S. Carroll in his book section 8.1 defines a manifold $M$ isotropic around the point $p$ if, for any two vectors $V$ and $W$ in $T_pM$, there is an isometry of $M$ such that the pushforward of $W$ under the isometry is parallel with $V$ (not pushed forward).

To me isn't clear what that actually means: $V$ and $W$ are both elements of the same $T_pM$, whereas an isometry associated pushforward maps vectors belonging to different tangent spaces.

 

[PAllen]

Going beyond FLRW, I would say a spacetime that admits a spacetime filling congruence with everywhere positive expansion scalar can reasonably be considered to be 'expanding'. Further, this is an uncommon property of GR solutions. For example, though I am fond or using the Milne coordinates on flat spacetime to refute many false claims of what features of cosmology require curvature, the Milne congruence does not really make Minkowski space expanding in this sense - the congruence fails to fill all of spacetime.

 

[PeterDonis]

so one can define for any timelike congruence an expansion scalar.

Yes. The magic phrase you need to look up is "kinematic decomposition of a timelike congruence". (You can also do it for a null congruence--that case plays a significant role in proving the Hawking-Penrose singularity theorems.)

 

[cianfa72]

Thinking again about it, Carroll defines the notion of isometry as a particular class of diffeomorphisms $\theta$ w.r.t. the metric tensor $g$ defined on the manifold $M$.

The definition of manifold isotropic around point $p$ from Carroll then is: for any $V,W \in T_pM$ there exists an isometry $\theta_p$ in the subgroup of isometries that leave fixed $p$ and $k \in \mathbb R$ such that $$\theta_p(W)=kV$$
In any case, I believe the relevant manifold $M$ to which the notion of isotropy applies isn't the spacetime as whole but its spacelike hypersurfaces.