Robertson-Walker metric and time expansion

[TrickyDicky]

The expansion is directly related to the space-time curvature (in fact, in a spatially-flat FRW universe, the expansion is the space-time curvature). You may be able to write down a metric that doesn't necessarily look like it has expansion in it, but because it is related to a real, invariant quantity, it will still have to manifest itself somehow no matter what coordinates you choose.

I still don't see the relation, forget empty universes, how is the scalar curvature or spacetime curvature related to expansion in the static Einstein universe?

 

[Chalnoth]

I still don't see the relation, forget empty universes, how is the scalar curvature or spacetime curvature related to expansion in the static Einstein universe?

Well, that universe isn't flat, so the relationship isn't direct. I *believe* that the curvature scalar is related to:

$$R \propto H_0^2 (1 - \Omega_k)$$

...though I suppose I'd have to do the calculations again to be sure (I'm a bit reluctant, because it is a bit of a lengthy calculation, and there are most likely some factors of the scale factor that I'm leaving off).

 

[TrickyDicky]

Well, that universe isn't flat, so the relationship isn't direct. I *believe* that the curvature scalar is related to:

$$R \propto H_0^2 (1 - \Omega_k)$$

...though I suppose I'd have to do the calculations again to be sure (I'm a bit reluctant, because it is a bit of a lengthy calculation, and there are most likely some factors of the scale factor that I'm leaving off).

You say that scalar curvature is intrinsically associated to expanding space, I'm just giving you examples where that is not the case, so it can't be a generalizable fact.
I don't know why that would be a feature only of spatially flat non empty universes. Scalar curvature is an invariant of different manifold modelling non-empty universes regardless of their particular configuration.

 

[Chalnoth]

You say that scalar curvature is intrinsically associated to expanding space, I'm just giving you examples where that is not the case, so it can't be a generalizable fact.
I don't know why that would be a feature only of spatially flat non empty universes. Scalar curvature is an invariant of different manifold modelling non-empty universes regardless of their particular configuration.

Well, assuming the equation I put together above is reasonably-correct (It may be a bit off, but I'm pretty sure it has to take a very similar form), then you can see how the expansion must be directly-related to the curvature in a flat universe, and how there is a relationship, though not as direct, in a non-flat one.

 

[TrickyDicky]

Well, assuming the equation I put together above is reasonably-correct (It may be a bit off, but I'm pretty sure it has to take a very similar form), then you can see how the expansion must be directly-related to the curvature in a flat universe, and how there is a relationship, though not as direct, in a non-flat one.

Aha, and when $$H_0=0$$? like in static manifolds with non zero R?

 

[Chalnoth]

Aha, and when $$H_0=0$$? like in static manifolds with non zero R?

Then we're not talking about an expanding universe, are we?

But in any event, if you're talking about a FRW universe, that situation is dynamically unstable and thus not really important.

 

[TrickyDicky]

Then we're not talking about an expanding universe, are we?

Exactly, my point is that you can't link the mere existence of the invariant scalar curvature to expansion as you did in previous posts to justify there is spatial expansion in the flat FRW metric even when it can be shown that in conformal coordinates there is no expansio wrt time. You can't assume it is a expanding universe beforehand regardless of the features of the metric or the whole discussion is meaningless.

 

[Chalnoth]

Exactly, my point is that you can't link the mere existence of the invariant scalar curvature to expansion

Um, I didn't. My point was that it goes the other way. If you have an expanding, non-empty universe, then you necessarily have non-zero curvature that is directly related to said expansion. That means that you can't just coordinate-transform the expansion away: you might hide it, but it will still appear in some form in your equations.

 

[Imax]

Can a Robertson-Walker Metric behave like a Kerr Metric near a black hole?

 

[Chalnoth]

Can a Robertson-Walker Metric behave like a Kerr Metric near a black hole?

No. The FRW metric assumes a homogeneous, isotropic universe, so it can't include black holes. To add matter that isn't perfectly smooth to the FRW metric, we make use of perturbation theory, where we start with FRW (where all matter is perfectly evenly-distributed), and add small deviations from that. This tends to work reasonably-well at large scales in the universe, but once you start to get down to the sizes of gravitationally-bound objects, such as galaxy clusters, this approximation breaks down. Black holes are way, way more dense than things like galaxy clusters, and there just isn't an easy way to place one in an FRW universe, except in very special cases (such as a universe with only a black hole and a cosmological constant).

 

[Imax]

No. The FRW metric assumes a homogeneous, isotropic universe, so it can't include black holes. To add matter that isn't perfectly smooth to the FRW metric, we make use of perturbation theory, where we start with FRW (where all matter is perfectly evenly-distributed), and add small deviations from that. This tends to work reasonably-well at large scales in the universe, but once you start to get down to the sizes of gravitationally-bound objects, such as galaxy clusters, this approximation breaks down. Black holes are way, way more dense than things like galaxy clusters, and there just isn't an easy way to place one in an FRW universe, except in very special cases (such as a universe with only a black hole and a cosmological constant).

That’s what I'm wondering about. A cosmological model based on an FRW metric can't see trees for its forest.

 

[Chalnoth]

That’s what I'm wondering about. A cosmological model based on an FRW metric can't see trees for its forest.

Well, one of the nice things is that the local details of the distribution of matter (e.g. black holes) don't have much of any impact at all on the larger-scale structure. In the regime where you can make use of perturbation theory to predict observations, observations match theory incredibly well.

At somewhat smaller scales, where perturbation theory starts to break down (this is still at a scale larger than galaxy clusters), we can make use of numerical simulations. These work quite well until you start getting into the regime where the physics of normal matter becomes important, which is around the size of galaxies. Once you get to the size of galaxies, our ability to predict what cosmological observations should look like from theory starts to break down (e.g. it's hard to predict the relative abundance of small and large galaxies). This isn't so much because we can't put black holes into the FRW metric, but instead because the physics of normal matter, and supermassive black holes in particular, are incredibly, obscenely difficult to get right.

 

[Imax]

So, a cosmological model based on an FRW metric + perturbation theory can fit with observations of our universe, but it has problems when going down to galactic scales, or around super massive black holes?

An FRW metric assumes a homogeneous and isotropic universe. Can time be different at different points in the metric, something like a twin paradox?

 

[Chalnoth]

So, a cosmological model based on an FRW metric + perturbation theory can fit with observations of our universe, but it has problems when going down to galactic scales, or around super massive black holes?

Well, the main problem here is that gravity is non-linear on smaller scales. Linearity means that if you change the system by some amount, the end result you get is always proportional to the amount you change the system. This is really important for doing calculations, because if a system is linear, you can vastly simplify those calculations. And it works really well for our universe on large scales. Linear theory predicts the CMB to tremendous accuracy, and even predicts much of the large-scale distribution of matter.

But when you start to get to smaller scales, the non-linearity of gravity becomes important. No longer is the output proportional to the input, but once you get enough matter concentrated into a small enough area, it will just keep collapsing in on itself. That kind of behavior simply cannot be modeled with a linear approximation, so what we do is make use of N-body simulations, where you imagine that matter is made up of a number of particles, and directly compute the relative gravitational attraction between all of them. This kind of calculation is relatively straightforward, but for large numbers of particles, it is extremely slow. But it works rather well at intermediate scales.

The problem with the N-body simulations, however, is that it considers matter to be made of particles, and thus this sort of simulation can't deal with gas physics. For that, we need to add another layer of complexity: hydrodynamic simulations. Here you're not only computing the gravitational interactions, but are also modeling the matter as an interacting gas, and that is where things get horribly complex, because things like supermassive black holes and supernova explosions have tremendous impacts upon the nearby gas, but we still don't know about the full behavior of these objects in the first place.

Don't get me wrong, we have made a lot of progress in understanding these amazing objects and events. It's just that figuring out what the physics we know today tells us about them is incredibly difficult to work out.

The real take-away here is just that even if we assume that the physics at work at smaller scales in the universe is completely known, we just haven't gotten to the point yet where we can say what the physics we know implies. However, at large scales, the calculations are much, much easier, so we can say with a great deal of certainty what the physics we know implies. So if you want to ask questions about whether or not experiment matches theory in terms of cosmology, your best bet is to look at the largest-scale observations you can.

An FRW metric assumes a homogeneous and isotropic universe. Can time be different at different points in the metric, something like a twin paradox?

Well, the time coordinate is arbitrary, so you can define it however you like. But there isn't much use in doing that, so typically we just do the easy thing and consider equal-time slices of the universe. Such equal-time slices, by definition, age at the same rate relative to one another. It makes the math easy, and it makes understanding the results easy.

 

[WannabeNewton]

So, a cosmological model based on an FRW metric + perturbation theory can fit with observations of our universe, but it has problems when going down to galactic scales, or around super massive black holes?

An FRW metric assumes a homogeneous and isotropic universe. Can time be different at different points in the metric, something like a twin paradox?

I'm not sure I understand you. The FRW metric is a time - dependent metric so it does evolve with time but if you take any hypersurface of time then all points on the hypersurface will be the same distance from t = 0 as defined by the metric.

 

[Imax]

Does an FRW metric allow possibilities like a twin paradox? What's a hypersurface of time?

 

[WannabeNewton]

Does an FRW metric allow possibilities like a twin paradox? What's a hypersurface of time?

Not sure on the first question sorry. A hypersurface of time is, in this case, essentially the 3D surface you get when you keep time constant. So its like if you look at the universe at some instant of time.

 

[Chalnoth]

Does an FRW metric allow possibilities like a twin paradox?

Well, yeah. The twin paradox (though not actually a paradox) is intrinsic to the nature of relativity, and since the FRW metric is based upon General Relativity, certainly it can apply.

What's a hypersurface of time?

Well, a set of points in space that all have the same time coordinate. Note that such a hypersurface depends upon what sort of time coordinate you use.

 

[Imax]

So, an FRW metric can allow for time dilation, time contraction (i.e. time expansion).

Is there any evidence that time can expand at the same rate as space?

 

[Chalnoth]

So, an FRW metric can allow for time dilation, time contraction (i.e. time expansion).

Is there any evidence that time can expand at the same rate as space?

Not in any absolute sense. With normal FRW coordinates, there is no expansion of time, just space. But you don't need to describe the system with the same coordinates. The expansion will always be there no matter what coordinates you pick, but precisely how it manifests itself depends upon those coordinates.