# Asymptotical flatness for galaxies - Why?

During the construction of solutions for General Relativity (when leaving out Lambda), asymptotical flatness is always assumed. Why?

I fully understand that it fits the solar system, as it results in Newtons law here.

However, why is it assumed to be the boundary condition for galaxies as well?

The solar system is moving in free fall - condition around the centre of the galaxy and therefore, asymptotical flatness would be the boundary condition of the solar system regardless of the gravitational law and the boundary condition in galaxies. Galaxies aren't moving in free fall around bigger objects, are they?

Spacetime is curved at the centres of gravity, so why should it be flat in the infinity?

There is an analogy between curvature of spacetime and electromagnetic fields. And electromagnetic fields are not flat in the infinity, instead they are vanishing in the infinity.

Is there an explanation why galaxies have to be asymptotically flat? Is there an experiment which confirmed asymptotical flatness for galaxies?

• Surely the analogy is between curvature and electromagnetic potential (not fields). It is charges that determine potential and masses (and energy) that determine curvature. Commented Mar 11, 2022 at 7:55

## 2 Answers

Asymptotical flatness is not assumed in GR in the sense of assumed to hold in fact. Rather, various interesting and useful results hold if the asymptotic regime is flat, and it is good to find those facts (e.g. conservation laws and things like that). Meanwhile what the condition really is at long distances is an empirical issue. It is studied in cosmology, and as it happens the evidence suggests spatial flatness or near-flatness at large scales. But this was not assumed, it was tested by observation of background radiation, galaxy surveys, supernovae and things like that (and it is thought to be implied by inflationary scenarios of early-universe physics).

• Thank you for your answer! You say "it's an empirical issue" and refer to cosmology scale studies. Are there also galactic-scale studies? Commented Mar 10, 2022 at 21:49
• Solar-system-scale, galaxy-scale and cosmological-scale may not lead to the same boundaries, do they? Only, there is no other alternative now then the asymptotical flatness, isn't it? Commented Mar 22, 2022 at 20:47

Asymptotic flatness is not an assumption of GR, but an assumption of the solutions that are constructed. In a lot of circumstances (as you have pointed out, like the solar system) it make sense to assume that the universe above a certain scale is flat -- after all in a flat universe geodesics are just straight lines. And since people are often interested in localised matter distributions, where we know from Newton that far away they produce a negligible gravitational force, this seems reasonable to assume.

However, there are solutions to the Einstein Equations that are not asymptotically flat, for example a de Sitter or an anti de Sitter universe. In this type of universe, the Ricci scalar is a (non-zero) constant over the entire space time. Also the $$\Lambda$$CDM universe (which we think describes our universe, although there are some problems like the Hubble tension) with a curvature term that is non-zero is another example for a solution where we do not impose asymptotic flatness.

There even seems to be evidence that we are living in a non-flat universe from observations of the cosmic microwave background.

• Thank you for the remark on asymptotical flatness being "not an assumption of GR, but of the solutions that are constructed". I changed that in the original question. Commented Mar 25, 2022 at 8:49
• You mention the solar system, there, Newtons' law is proven and therefore, asymptotical flatness is reasonable there. Furthermore, you mention the whole universe and discuss whether it might be flat as a whole or not. However, my question is explicitly meant to ask for the validity of the asymptotical flatness of galaxies (And I changed the title to be more precise). It's also assumed there and I didn't see yet why it is assumed there. Commented Mar 25, 2022 at 8:53
• You state "During the construction of solutions for General Relativity (when leaving out Lambda), asymptotical flatness is always assumed.". That is not true. The Friedmann equations are solutions of GR and they allow for a non-flat space. To answer why solutions for galaxy assume asymptotic flatness, I think it is mostly out of simplicity, although people have built solutions of black holes for space that is not asymptotically flat. Commented Mar 26, 2022 at 12:00
• The question is: Do you expect the boundary conditions of the boundary being non-flat to have a measurable impact on the observables, such as the rotational curves of the galaxies? If not, there is no need to include such complicated boundary conditions. Commented Mar 26, 2022 at 12:03
• No, the question simply is: Do the rotational curves of the galaxies support the assumption of asymptotical flatness? And how do they do that? For example, in a far away distance from the galactic centre (outside the assumed dark matter halo), do the rotation curves finally become Newtonian there again? Commented Mar 27, 2022 at 5:58