In another thread, someone was talking about cosmological expansion effects on planetary orbits. (Actually it was about lunar orbits, but I think planetary orbits are more to the point). Through a somewhat round-about path, I eventually got to thinking about the following question. Suppose we look at a sphere that is 1.5*10^11 SI meters (approximately 1au) in radius around the sun. What happens to the total mass (Komar mass) enclosed inside this sphere due to the expansion of the universe? I'm assuming that the metric is quasi-static so that the Komar mass exists. I don't think this is unreasonable (though I'm willing to listen to arguments that it is). If "dark energy" is in the form of a cosmological constant, it seems to me that basically nothing should happen to this mass due to universal expansion. The only change should be due to effects that aren't being modelled by this simple model (like the radiation of the sun decreasing its mass). I suppose that the dark matter might also contribute to the mass in this sphere, but I'm not sure how plausible this really is. Ned Wright seems to think that this is at least possible in his FAQ, but I'm not quite sure how it would be possible to have the dynamics of the solar system understood if there were significant amounts of dark matter around locally - unless it perfectly mimiced the distribution of normal matter, in which case I would expect it to continue to mimic the distribution of normal matter. I don't see normal matter due to "cosmological background" (say, interstellar dust/radiation) contributing much to the mass in such a sphere, so I wouldn't expect much difference if this "background" mass disappeared. The FAQ entry I mentioned:
I do not understand the quoted part... After a quick look it seams that in the Cooperstock paper it is taken as an assumption that the scale factor may vary within the solar system. The computations are then based on this assumption: I was not able to find this Anderson paper, however. My understanding is that without an homogeneous and isotropic dark energy permeating the whole space at all scales above the Planck lenght, there should be a cut-off for the expansion of space. This should be determined as a function of some characteristic length at which matter distribution starts to be homogeneous and isotropic. At that lenght there should be then a coupling between the cosmological solution and the Schwarzschild or axially-symmetric solution at smaller scales. I would guess that the galactic distribution of matter (and therefore an axially-symmetric solution), rather than the solar system, is the solution that has to be coupled to the cosmological one. I see no reason a priori to think that this coupling would imply an influence of the external metric on the internal one at arbitrary short distances. Things are different, however, if there exists a homogeneous and isotropic distribution of dark energy at arbitrary small scales above the Planck length. I can imagine that this could imply an expansion of space at every scale, that may be however not noticeable within the solar system. This is my personal view of this, that seams not to be the "standard" one. At least not the one of Anderson, Cooperstock and Wright. But I fail to see the arguments that may invalidate it.
This is a very interesting question. The problem arises because in GR local gravitational orbits are calculated under the Schwarzschild solution to the GR field equation and cosmological expansion is a prediction of the cosmological solution of that field equation. The question is how is the Schwarzschild solution, which tends to Minkowskian space-time as r tends to infinity, embedded in a cosmologically expanding space within space-time? That all is not right with the statement that "cosmological expansion has no effect on local orbits" might be indicated by the Pioneer anomaly, (which is almost equal to cH) however the problem with a naive application of cosmic expansion to Pioneer is that it is being accelerated the wrong way - towards the Sun rather than away from it. Garth
There is a way to embed a Schwazschild solution in De-sitter space. http://arxiv.org/abs/gr-qc/0602002 for instance, takes this approach. The metric is F := 1 -2M/r - [itex]\Lambda[/itex] r^2 / 3 ds^2 = F dt^2 - 1/F dr^2 - r^2(d [itex]\theta[/itex]^2 + sin([itex]\theta[/itex])^2 d [itex]\phi[/itex]^2) The question is - is this actually the right metric to use to represent the solar system in an expanding universe? One interesting thing about the above metric is that it has an actual horizion, where F=0 and therfore g_00 = 0, as long as [itex]\Lambda[/itex]>0. The standard FRW space-time always has g_00=1. On the surface, these metrics look different, but they are supposed to be representing similar things. Perhaps there is a simple coordinate transformation that transforms the above metric into the FRW form, but so far I haven't figured out if this is true. I can two things for sure: that in an orthonormal basis of one-forms w1=sqrt(F)dt, w2=1/sqrt(F)dr, w3=r, w4=r sin([itex]\theta[/itex]) that [itex]G_{(\mu)(\nu)}[/itex] = [itex]\Lambda[/itex] Diag(1,-1,-1,-1) with respect to the basis vectors and that the above metric, if it is correct, makes the central mass have a truly static metric as long as [itex]\Lambda[/itex] is not a function of time, even though the De-sitter universe expands, because none of the metric coefficients are functions of time. It was the fact that this metric was static that made me start to think about whether the mass enclosed in a sphere around the central mass in an expanding universe should be static. As long as the universe is isotropic, I don't think it shouldn't matter what anything outside the sphere does. The net gravitational effect of a spherical shell should be zero inside the shell, regardless of whether or not the shell is expanding, by Birkhoff's theorem. (Maybe I have to read the fine print of Birkhoff's theorem to make sure it applies to this situation with a cosmological constant, though).
Why should a Schwarzschild solution embedded in an FRW solution be realistic? The solar system is embedded into the non-homogeneous, non-isotropic, axially symmetric, rotating galaxy. What is the reason to neglect this?
The galaxy is also embedded in the local group, the Virgo cluster and our local super cluster. Apart from perturbations on the spherical symmetry the main question is how iscosmological expansion handed down at each of these scales. One answer, the standard one, is that it isn't, in regions of overdensity cosmological expansion becomes replaced by halo gravitational collapse. Thank you Pervect. The Birkoff theorem only works for pressureless space, as DE is a form of negative pressure that might complicate matters a little. Garth
I think that the position of the solar system in the galaxy would indeed have an effect. I think the effect should be small, but able to be calculated by purely Newtonian means - I don't think relativistic considerations should be significant. The effects should basically be due to Newtonian tidal forces, if we neglect the motion. (I think we can neglect the effects of motion on the forces, though I can't actually prove this). For a nearby mass of mass M', I'd expect tidal forces of -2GM'/r^3 in the direction pointing to the mass, and tidal compressive forces of +GM'/r^3 in the perpendicular directions. Note that these tidal forces cancel each other out for a spherical distribution of mater as one would expect. But we aren't in a spherical distribution. Because we are on the edge of a disk, I would expect some small net tension in the plane of the galaxy, and net compression perpendicular to the galactic plane, though I haven't worked out the magnitude of the tidal forces due to the galaxy. Because of the r^3 nature of tidal forces, I would expect that nearby galaxies would not produce significant tidal forces, and the "local groups" even less. What I'm more interested in at the moment, though, is getting a handle on what the cosmological constant does (if anything) to solar system dynamics. It does seem reasonably clear from the literature that the effect is either a very very small change in orbits with time (Cooperstock) or even static (the Einstein-deSitter approach).