# Why isn't everything expanding in an expanding universe?

Viewpoint 1:
Because gravity or other forces that are holding the thing concerned (be it a galaxy, a ruler or an atom) together are way stronger than the "force" caused by the expansion of space. So strictly speaking, space does in fact expand everywhere, including the space inside an atom between its nucleus and its electrons. But gravity or other forces keep the size of the object constant.

Viewpoint 2:
The space within the thing concerned (such as the space inside an atom) is of a different nature from the space between two galaxies (the space that "occupies" the vast region of space). Let's call the former type-1 space and the latter type-2 space. Only type-2 space expands for some reason; eg., type-2 space is filled with (or littered with) dark energy, which drives the expansion, but type-1 space is void of dark energy.

I think the correct or the widely accepted viewpoint is the first one. But I don't understand how exactly the size of an object is being kept constant.

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phinds
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Viewpoint 2 sounds like nonsense to me. Where did you hear this? Do you have reputable citations?

Chalnoth
Everything isn't expanding because the universe doesn't have perfectly uniform density. When you work through the equations of what gravity predicts in the face of differences in density from place to place, you get an average, large-scale expansion, but small-scale bound systems that are reasonably stable over time (e.g. galaxy clusters and smaller).

• PeterDonis
PeterDonis
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the "force" caused by the expansion of space
There is no such thing; the expansion of the universe does not give rise to any force.

There is no such thing; the expansion of the universe does not give rise to any force.
Yes, that's why it was in quotation marks. iirc, Prof Susskind mentioned in one of his lectures that the expansion can be modelled using a small force.

PeterDonis
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Prof Susskind mentioned in one of his lectures that the expansion can be modelled using a small force.
Do you have a reference? I suspect the use of the word "force" in this connection was ill-advised, since it leads to the erroneous inference you have made. Expansion does not "try" to push bound objects apart, which is what the word "force" implies.

wabbit
Gold Member
Yes, that's why it was in quotation marks. iirc, Prof Susskind mentioned in one of his lectures that the expansion can be modelled using a small force.
I suspect he may have been discussing accelerated expansion in the presence of a cosmological constant, where something like that is possible (within limits) but I don't think it makes sense at all for unaccelerated expansion.

Chronos
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We can safely conclude the historical earth sun recession rate is much smaller than that predicted by the expansion rate of the universe based on a variety of unrelated lines of evidence as discussed in this paper;http://arxiv.org/abs/1306.3166, A Closer Earth and the Faint Young Sun Paradox: Modification of the Laws of Gravitation, or Sun/Earth Mass Losses? Given the existing expansion rate measured at 1% per 140,000,000 years the earth would have been about 27% closer to the sun at the beginning of the Archean era 3.8 billion years ago. This is inconsistent with a vast body of climate and paleontology evidence.

As far as I know, the universe at the smallest level does expand. A possible fate of the universe is called the big rip, where the expansion of space becomes so fast that atoms aren't able to keep themselves together.

Chronos, that's interesting, I never thought of that before, so why has the solar system not expanded? A gravitational orbit in an expanding space shouldn't remain stable should it?

wabbit
Gold Member
I don't understand how expansion per se can affect orbits at all. The earth is not comoving and the geometry is locally Schwarzchild. Can it be affected by a distant symmetric distribution of galaxies, whether these are receding or approaching?
The cosmological constant should have an effect though, I recall seeing a paper analysing that, will try to dig it up.

Chalnoth
Do you have a reference? I suspect the use of the word "force" in this connection was ill-advised, since it leads to the erroneous inference you have made. Expansion does not "try" to push bound objects apart, which is what the word "force" implies.
In some equations, the expansion acts as a sort of friction, tending to damp differences in velocity between things in the universe over time. That *might* be what he was talking about, but I'm not sure of the context.

Chalnoth
I don't understand how expansion per se can affect orbits at all. The earth is not comoving and the geometry is locally Schwarzchild. Can it be affected by a distant symmetric distribution of galaxies, whether these are receding or approaching?
The cosmological constant should have an effect though, I recall seeing a paper analysing that, will try to dig it up.
No, it doesn't. Bound objects are stable in an expanding universe without a cosmological constant. This drops right out of the linearized equations using perturbation theory to describe a universe that isn't homogeneous (linearized equations only describe really big objects accurately....smaller objects are going to be even less impacted by the expansion).

wabbit
Gold Member
I was saying there might be an effect with a CC, not without it - not sure though.
Couldn't find the paper I had in mind but this one seems relevant:
http://arxiv.org/abs/0810.2712
Influence of global cosmological expansion on local dynamics and kinematics
Matteo Carrera, Domenico Giulini
We review attempts to estimate the influence of global cosmological expansion on local systems. Here `local' is taken to mean that the sizes of the considered systems are much smaller than cosmologically relevant scales. For example, such influences can affect orbital motions as well as configurations of compact objects, like black holes. We also discuss how measurements based on the exchange of electromagnetic signals of distances, velocities, etc. of moving objects are influenced. As an application we compare orders of magnitudes of such effects with the scale set by the apparently anomalous acceleration of the Pioneer 10 and 11 spacecrafts, which is 10^-9 m/s^2. We find no reason to believe that the latter is of cosmological origin. However, the general problem of gaining a qualitative and quantitative understanding of how the cosmological dynamics influences local systems remains challenging, with only partial clues being so far provided by exact solutions to the field equations of General Relativity.

Chalnoth
Right. I was a little ambiguous in my reply. I was agreeing with you (my first sentence responds to your first sentence).

wabbit
Gold Member
Ah OK, got it now thanks:)

For a spherically symmetric distribution, distant stars should have zero effect - this is correct in Newtonian gravity but is it always true in GR? Looking at these papers the issue doesn't seem so simple so presumably this may be only correct up to a Newtonian approximation, or otherwise up to second order effects?

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PeterDonis
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For a spherically symmetric distribution, distant stars should have zero effect - this is correct in Newtonian gravity but is it always true in GR?
Yes; the "shell theorem" holds in GR as well as Newtonian gravity.

wabbit
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Great. Can we also say that if we add one mass at the center of this spherical vacuum, we get a Schwarzchild geometry in place of flat spacetime?
Newton-wise we would just add forces, but in GR unless the sphere is large enough there should some correction?
Otherwise the orbits around a comoving mass surrounded by a spherically symmetric distribution of matter would be unaffected even by a cosmological constant - but this isn't quite true, in the vacuum case we get Schwarzchild-de Sitter geometry with some corrections to orbits?

Edit: but how does the shell theorem work with a CC? It must be saying something like, not the vacuum is flat, but it has de Sitter geometry? Otherwise a sphere cut out in de Sitter space would seem to provide a counterexample.

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Chalnoth
Great. Can we also say that if we add one mass at the center of this spherical vacuum, we get a Schwarzchild geometry in place of flat spacetime?
Newton-wise we would just add forces, but in GR unless the sphere is large enough there should some correction?
Otherwise the orbits around a comoving mass surrounded by a spherically symmetric distribution of matter would be unaffected even by a cosmological constant - but this isn't quite true, in the vacuum case we get Schwarzchild-de Sitter geometry with some corrections to orbits?

Edit: but how does the shell theorem work with a CC? It must be saying something like, not the vacuum is flat, but it has de Sitter geometry? Otherwise a sphere cut out in de Sitter space would seem to provide a counterexample.
Found a proof here:
http://arxiv.org/abs/0908.4110

They say that locally, the vacuum solution in a spherically-symmetric space-time in the presence of a cosmological constant is necessarily equivalent to either de Sitter space-time, or Schwarzschild-de Sitter space-time (Theorem 1 in the paper). However, they also say that the predicted space-time is not necessarily static, and I don't entirely understand what that means.

• wabbit
PeterDonis
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Found a proof here:
http://arxiv.org/abs/0908.4110
This is a proof of Birkhoff's Theorem, not the shell theorem; they're two different things.

Birkhoff's Theorem, in the generalized form proven in this paper, says that any vacuum, spherically symmetric solution to the EFE with cosmological constant must be the Schwarzschild-de Sitter geometry.

The shell theorem says that, if we have a distribution of stress-energy that is spherically symmetric outside some region, the geometry inside that region is unaffected by the stress-energy distribution outside it.

We can combine the two results to say that, for example, if the stress-energy distribution is spherically symmetric outside some region, and inside the region we have a spherically symmetric vacuum, then the geometry inside the region must be Schwarzschild-de Sitter.

they also say that the predicted space-time is not necessarily static, and I don't entirely understand what that means.
It means that the Schwarzschild-de Sitter geometry is only static between the two horizons--outside the black hole horizon and inside the cosmological horizon. In the other regions it is not static; there is still an extra Killing vector field (in addition to the ones implied by spherical symmetry), but it is not timelike (it is null on the two horizons and spacelike inside the black hole or outside the cosmological horizon).

• wabbit
Chalnoth
This is a proof of Birkhoff's Theorem, not the shell theorem; they're two different things.
I don't think so. One consequence of Birkhoff's theorem is that the spacetime inside a spherical shell is Minkowski.

Birkhoff's Theorem, in the generalized form proven in this paper, says that any vacuum, spherically symmetric solution to the EFE with cosmological constant must be the Schwarzschild-de Sitter geometry.
Right, which reduces to de Sitter geometry when m=0. And as with Birkhoff's original theorem, this is also valid for vacuum inside a spherical shell.

wabbit
Gold Member
Thanks @Chalnoth and @PeterDonis. So from this theorem, if we have a central mass in a spherically symmetric universe,
- if ##\Lambda=0## the geometry is locally exactly Schwarzchild;
- if ##\Lambda>0## but ##\Lambda\ll M^{-2}## it is locally exactly SdS.
The "locally" still describe a "large" region, spherically symmetric, which is if I understand it correctly is sufficient to describe orbits around the central mass.
In this case - say, carving out a hollow comoving sphere in a FRW spacetime and putting a mass at the center, the orbital dynamics inside that sphere are entirely unaffected by expansion, and a small cosmological constant changes them a bit but leaves them bounded.

A more realistic situation might be to look at a scale where homogeneity holds and carve out a comoving sphere where the outside is still symmetric, but then replace the homogenous inside with some random distribution of matter (with the constraint that its center of mass is at the center of the sphere?)
It would be nice if we could to conclude that dynamics inside are still unaffected by (spherically symmetric) dynamics outside, so that again expansion has exactly zero effect on local gravitational dynamics (and the cosmological constant effect is that of the constant local curvature it adds, not an effect of outside expansion), which seems to be the conclusion suggested by the shell theorem.

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PeterDonis
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One consequence of Birkhoff's theorem is that the spacetime inside a spherical shell is Minkowski.
More precisely, one consequence of Birkhoff's theorem is that a vacuum spherically symmetric spacetime region with zero cosmological constant and zero mass is Minkowski. But the shell theorem is much more general: the region inside the shell does not have to be vacuum (or zero mass or zero cosmological constant). No matter what is inside the shell, if the shell is spherically symmetric, the geometry inside it is unaffected by the geometry outside it.

PeterDonis
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carving out a hollow comoving sphere in a FRW spacetime and putting a mass at the center, the orbital dynamics inside that sphere are entirely unaffected by expansion
Yes.

A more realistic situation might be to look at a scale where homogeneity holds and carve out a comoving sphere where the outside is still symmetric, but then replace the homogenous inside with some random distribution of matter (with the constraint that its center of mass is at the center of the sphere?)
It would be nice if we could to conclude that dynamics inside are still unaffected by (spherically symmetric) dynamics outside, so that again expansion has exactly zero effect on local gravitational dynamics (and the cosmological constant effect is that of the constant local curvature it adds, not an effect of outside expansion), which seems to be the conclusion suggested by the shell theorem.
Yes.

wabbit
Gold Member
Thanks, this seems clear then.

But now I am having doubts about the approach of the (Newtonian approximation to the) two body problem in an expanding background described in http://arxiv.org/abs/0810.2712 (p.7) and elsewhere, which takes the acceleration of comoving bodies and applies it to orbiting bodies (by itself this step seems arbitrary).

The clearest case is that of a significant comoving mass and a small orbiting one, both added to an FRW background. The only reason I can see for a correction, is that the background fluid is escaping a fixed sphere around the comoving mass, so that the effective mass within a fixed sphere decreases over time (and the solution seems to be a TLB spacetime) - but this doesn't seem to be a realistic prescription, when the comoving fluid represents galaxies and such and we're looking at a smaller scale : here, the empty ball embedded in a symmetric spacetime seems much more relevant, with the conclusion that the two body probem is the same as with no expansion.

Is this correct ?

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Do you have a reference? I suspect the use of the word "force" in this connection was ill-advised, since it leads to the erroneous inference you have made. Expansion does not "try" to push bound objects apart, which is what the word "force" implies.
I've wondered about this... I interpreted, perhaps wildly incorrectly, the information content of Verlinde's holographic screen, p.2 sec3

"Thus we are going to assume that information is stored on surfaces, or screens. Screens separate points, and in this way are the natural place to store information about particles that move from one side to the other. Thus we imagine that this information about the location particles is stored in discrete bits on the screens. The dynamics on each screen is given by some unknown rules, which can be thought of as a way of processing the information that is stored on it. Hence, it does not have to be given by a local field theory, or anything familiar. The microscopic details are irrelevant for us."

to be a set of as yet un-identified dynamics well-covered by the term "expansion", like curvature distribution in LGQ, or MERA renormalization through disentanglers.

http://arxiv.org/pdf/1001.0785v1.pdf

On the Origin of Gravity and the Laws of Newton
Erik P. Verlinde
(Submitted on 6 Jan 2010)
Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic.