In Cosmology, what does it mean to be 'local'?

Question

I'm trying to make a point that there is curvature of spacetime from the metric expansion that contributes to the dynamics of a galaxy. This curvature would be in addition to the curvature caused by the visible mass/energy content of the galaxy. I got back a note from an editor saying

"In the language of relativity physicists, “locally flat”, also called “locally Lorentz”, means flat at first order in separation from any chosen point. Of course, at second order one sees the influence of the Riemann curvature tensor, i.e. of the curvature."

Can someone interpret this for me? When a book says that the local geometry of spacetime is flat, how local is that? Microscopically, the size of a football field, a solar system, a galaxy? What's the cutoff for a 'local' geometry?

Answer 1

In coordinates that have the dimension of length, the dimensions of the Riemann curvature tensor are inverse length squared. Therefore at each point the components of this tensor establish length scales which you can loosely think of as radii of curvature. “Local” refers to a region whose length scales are small compared with any of these curvature-based length scales.

For example, near the horizon of a stellar-mass black hole, the radii of curvature are on the scale of kilometers. Therefore dynamics in a local region on the scale of, say, meters is barely affected by the curvature. The differences from the Minkowski metric within this local region are on the order of one part in a million.

For an even smaller region, the differences from Minkowskian are even more negligible. For a larger region, they are less negligible. There is no “cutoff”, but just “smaller and more flat” or “bigger and less flat”. Over a scale of kilometers the spacetime is not flat at all. Over no region is it perfectly flat.

To understand mathematically how spacetime is “flat at first order”, but not second order, at every point, look at the metric tensor in Riemann normal coordinates:

$$g_{\mu\nu}=\eta_{\mu\nu}-\frac13R_{\mu\sigma\nu\tau}x^\sigma x^\tau+O(|x|^3).$$

If we write

$$R\sim\left(\frac{1}{L_\text{curvature}}\right)^2$$

then the deviations from the Minkowski metric over a region of linear scale $L_\text{region}$ are of order $$\left(\frac{L_\text{region}}{L_\text{curvature}}\right)^2.$$

Answer 2

What G. Smith said is correct, but I want to also point out that

there is curvature of spacetime from the metric expansion that contributes to the dynamics of a galaxy

is incorrect. There's no such effect.

There are various ways to see this. Here's one. FLRW spacetimes have no Weyl curvature, only Ricci curvature. Ricci curvature is nonpropagating: the field equation directly equates it to the stress-energy at the exact location of the curvature. Therefore all of the curvature of FLRW spacetimes comes from locally present stress-energy. The only significant contributors to the stress-energy tensor are ordinary matter, dark matter, and dark energy. Once you've accounted for those, you're done. There is no mechanism in GR by which any extra "metric expansion" effect could enter into the dynamics.

Note that, within galaxies, none of the major contributors to the stress-energy tensor has a net outward motion, nor a tendency to dilute with the scale factor. The only reason that ordinary and dark matter dilute by $1/a^3$ at larger scales is that the voids between superclusters increase in size and the average density including those voids therefore decreases. This has no effect on the dynamics of individual galaxies. There's no mechanism in GR by which it could have an effect.

The reason for this very common misconception seems to be that people think of FLRW spacetime as a background, i.e., as what you start with before adding the galaxies. It's actually the gravitational field of a uniform matter distribution, and that matter is the same matter that makes up the galaxies. If you put a galaxy on a FLRW background, you're double-counting the matter. To get the correct dynamics, you need to put the galaxy on a de Sitter background, because that's what's actually left behind when the ordinary and dark matter clump.