Why doesn't the universe expand on small scales?

[Fredrik]

I got a PM from "Old Smuggler" that contained links to a couple of interesting articles. One of them listed this one in the references. It analyzes some of these questions quantitatively. In particular it calculates the effect of the cosmological expansion on Earth's orbit around the sun. The result they found is an outward acceleration of about 3*10^-47 m/s². (Compare this with the centripetal acceleration of about 6*10^-3 m/s² in the opposite direction).

 

[MeJennifer]

As has been said previously, that's a very good question. (Although I don't know how accurate it is to say that the metric is partly FRW and partly Schwarzschild.) The FRW metric is valid only if the matter distribution is homogeneous and isotropic. Thus, it only applies on the >~100 Mpc scale.

Thus, the metric of the universe is almost FRW at distances of 10⁸ pc.

I am not agreeing with this notion, since the FRW metric does not treat the distribution of matter in the universe as a dust.

The FRW metric is obtained by smoothing out all matter in the universe and treating it like a perfect fluid. The "expansion" is nothing more than the separation of neighboring flow lines with or without a cosmological constant.

But one can't have one's cake and eat it too by first treating all matter in the universe as some kind of smoothed out fluid and then make definitive statements about gravitationally bound systems. The FRW metric simply does not model these systems.

 

[Fredrik]

The FRW metric is obtained by smoothing out all matter in the universe and treating it like a perfect fluid.

Yes, a homogeneous and isotropic perfect fluid in a homogeneous and isotropic space. The additional assumption that matter is a perfect fluid is just there to make sure that there are no internal stresses that contribute to the stress-energy tensor (i.e. to make sure that density and pressure are the only things that contribute to it).

But one can't have one's cake and eat it too by first treating all matter in the universe as some kind of smoothed out fluid and then make definitive statements about gravitationally bound systems. The FRW metric simply does not model these systems.

Agreed. But we are still near a spherical distribution of mass in a universe that's homogeneous and isotropic on large scales. Hence I find it more than reasonable to expect the metric on Earth to be "a little bit like FRW and a lot like Schwarzschild". The first correction we'd have to make to our approximately Schwarzschild metric here on Earth is of course not due to the large-scale stuff, but due to the influence of the sun. Then there's the other planets in our solar system, the large concentration of mass near the center of the galaxy, other nearby stars, etc. Corrections due to the large-scale homogeneity and isotropy of space are way down on the list, but they must be present in some form.

It's annoying that the non-linearity of Einstein's equation prevents us from just calculating a bunch of metrics for various ideal situations and just adding them together to get our actual metric. This makes it hard to see how the large-scale distribution of matter contributes to the small-scale structure of space-time.

 

[MeJennifer]

It's annoying that the non-linearity of Einstein's equation prevents us from just calculating a bunch of metrics for various ideal situations and just adding them together to get our actual metric. This makes it hard to see how the large-scale distribution of matter contributes to the small-scale structure of space-time.

Correct.

Unfortunately many cosmologists claim that expansion is zero in gravitationally bound systems without demonstrating it mathematically using GR. A typical case of "trust us, we know" without backing it up.

 

[Count Iblis]

Huh? Isn't there an exact solution available for an FWR universe with a single mass in it?. It looks spherical symmetric to me, so what's the problem in writing down the exact solution?

 

[Ich]

There is the AdS Black Hole, and it's not too hard to find some articles (like http://arxiv.org/abs/gr-qc/0612146" ) with calculations. No need to lament.

 

[CJames]

I think the issue that is making this confusing is thinking of the universe as expanding meaning that space is expanding. That's true in a way. But really what it means is that on average two geodesics in the universe will diverge from one another. This is not true of a solar system or a galaxy. Those geodesics converge.

As for the meter sticks, I think I have a satisfactory answer. You've heard it before, but I want to reiterate it so it doesn't sound so complicated. Remember, we're talking about diverging geodesics. All that means in the real world is you have two objects in free fall that are separating from one another. So think of dropping two balls on the moon. They start one meter apart, but they will diverge with time. Now, drop a meter stick next to those two balls. You don't expect the meter stick to expand, right? You expect to measure a change in the distance between the two balls.

 

[Fredrik]

That's an easy way to see that the endpoints of a meter stick don't move on geodesics, but the question then is what do they move on and why? I think a pretty good answer is that when we consider a small enough region, a non-gravitational interaction in a local inertial frame of the curved space-time can be approximated by the theory describing those interactions in an inertial frame of Minkowski space. (How small the region must be depends on the curvature of course). So if the theory of those interactions in Minkowski space says that two particles will stay 1 unit of length apart, they will stay almost exactly 1 unit of length apart in a local inertial frame, not in a FRW frame.

 

[Ich]

So if the theory of those interactions in Minkowski space says that two particles will stay 1 unit of length apart, they will stay almost exactly 1 unit of length apart in a local inertial frame, not in a FRW frame.

But you can figure out what they would do in FRW spacetime. For a start, assume a meterstick to be rigid, i.e. with constant proper lenth. Applying the laws of motion you find that, in a FRW metric, each point is differently accelerated. In the meterstick's (flat tangent) space these accelerations appear as fictious forces that will stretch the meterstick, depending on its material properties. It's easy to see that it will not follow the FRW expansion, but rather be constantly a little bit longer than in flat spactetime.

 

[Fredrik]

But you can figure out what they would do in FRW spacetime.

Yes, that was the point of what I just said. It seems to be a very reasonable assumption that physics on small scales in local inertial frames on a FRW space-time will be approximately the same as physics in inertial frames in Minkowski space-time. This implies that two nearby points on the meter stick will stay the same coordinate distance apart in the coordinates of the local inertial frame that's co-moving with one of the points, which means that their coordinate distance in FRW coordinates will decrease. It also implies that the proper length of a meter stick, along the shortest possible path between the sticks endpoints in a space-like hypersurface of constant FRW time, is going to be constant.

For a start, assume a meterstick to be rigid, i.e. with constant proper lenth. Applying the laws of motion you find that, in a FRW metric, each point is differently accelerated. In the meterstick's (flat tangent) space these accelerations appear as fictious forces that will stretch the meterstick, depending on its material properties. It's easy to see that it will not follow the FRW expansion, but rather be constantly a little bit longer than in flat spactetime.

You're not wrong, but I think you're oversimplifying. "Proper length" is only defined along a curve, and it's not obvious what curves we're talking about. Why should they even be paths in the hypersurface of constant FRW time? It's obvious that those are the paths we're interested in, but it's not obvious that those are the paths along which the proper length of the meter stick is always the same. I think we need all that stuff I said about local inertial frames to motivate that.

 

[Ich]

You're not wrong, but I think you're oversimplifying.

I'm not so pessimistic. We are in a weak field regime, where nonlinearities can safely be ignored. We can parametrize a bundle of worldlines that all keep constant distance to one another when compared at a certain cosmological time t. We can calculate the four-acceleration of all worldlines, which is constant for quite long time intervals. And I'm sure that we can split the acceleration in 3+1 d without big ambiguity.
"we can" of course means "you can", as I am overqualified and underchallenged. Or the other way round.