Why do people care about vacuum solutions in GR?

[ohwilleke]

TL;DR Summary

People study vacuum solutions in general relativity, but I don't really understand why.

I understand that there are vacuum solutions of the equations of general relativity (GR) (i.e. equations with no mass-energy content contributing to the stress-energy tensor), that are studied by physicists interested in GR.

I don't understand why these are studied or what purpose they serve. Why aren't these solutions simply ignored as non-physical?

Could someone explain this at a very basic level?

 

[Ibix]

They're often mathematically tractable.

You don't need to write down an equation of state for matter that you may not entirely believe under extreme circumstances anyway. (Contributes to the previous paragraph.)

Most of space on interstellar scales is vacuum to a very good approximation, so you need vacuum solutions. Because of this, realistic and semi-realistic astrophysical models are often vacuum solutions around non-vacuum regions. And things like gravitational waves are effectively propagating in vacuum.

Analysing mathematically tractable models can give you insight into how to approach numerical simulations of more general cases. And elephant traps to be aware of.

 

[Orodruin]

TL;DR Summary: People study vacuum solutions in general relativity, but I don't really understand why.

Could someone explain this at a very basic level?

Vacuum does not mean uninteresting. Most of the universe is very well approximated by vacuum - making Minkowski space a pretty good approximation locally. In addition you have things like Birkhoff’s theorem ensuring that the solution outside a spherically symmetric streas-energy distribution is the exterior Schwarzschild solution - which is a vacuum solution.

 

[ohwilleke]

And elephant traps to be aware of.

Who knew that there were elephants in space? Life always brings surprises I guess.

 

[ohwilleke]

Vacuum does not mean uninteresting. Most of the universe is very well approximated by vacuum - making Minkowski space a pretty good approximation locally. In addition you have things like Birkhoff’s theorem ensuring that the solution outside a spherically symmetric streas-energy distribution is the exterior Schwarzschild solution - which is a vacuum solution.

Does this mean that a vacuum solution requires boundary conditions, or for that matter, just boundaries?

 

[renormalize]

Could someone explain this at a very basic level?

Consider the analogous situation in electrostatics. The field equation in vacuum for the electrostatic potential $\varphi$ is $\nabla^{2}\varphi\left(r\right)=0$. The spherically-symmetric solution to this is $\varphi\left(r\right)=\frac{k}{r}+\varphi\left(\infty\right)$, i.e., it's the potential outside of an isolated charge. This is a physical result. The gravitational analog is the vacuum Schwarzschild solution of general relativity, i.e., it's the metric tensor outside of an isolated mass. This is also a physical result.

 

[Ibix]

Does this mean that a vacuum solution requires boundary conditions, or for that matter, just boundaries?

No, but the universe has matter in it, so you will often be interested in solutions that join a matter-filled solution to a vacuum solution. For example, the Oppenheimer-Snyder black hole is a toy model of stellar collapse that consists of a small part of a closed FLRW solution started from the instantaneous rest point (this is the star) stitched to a Schwarzschild vacuum (the space around the star).

Separately, there is the notion of "domain of applicability". Minkowski spacetime is a fine model for studying even something like Fizeau's experiments with light in water - clearly not a vacuum, but the failure to be vacuum is a tiny tiny correction to the result. SR is applicable in domains where the curvature is negligible.

 

[PeterDonis]

Why aren't these solutions simply ignored as non-physical?

Um, because they're not?

Our universe has plenty of significant regions which are, to a very good approximation, vacuum. Vacuum solutions describe these regions to a very good approximation, certainly good enough for practical use. A specific solution in GR does not have to describe an entire spacetime. It can perfectly well describe a region of spacetime that is joined to another non-vacuum region.

 

[PeterDonis]

Does this mean that a vacuum solution requires boundary conditions, or for that matter, just boundaries?

Generally speaking, a vacuum region will have a boundary with a non-vacuum region, yes.

 

[ohwilleke]

A specific solution in GR does not have to describe an entire spacetime.

This is what I hadn't realized about these solutions.