black holes - Why is analyticity a good mathematical assumption in general relativity?

In general relativity, real-variable analytic continuation is commonly used to understand spacetimes. For example, we use it to extend the Schwarzschild spacetime to the Kruskal spacetime, and also maximally extend the Kerr and Reissner-Nordstrom spacetimes. It is also used as an essential condition to prove theorems, for example the theorem (see p. 92 of these lecture notes):

If $(M, g)$ is a stationary, non-static, asymptotically flat, analytic solution to the Einstein-Maxwell equations that is suitably regular on, and outside an event horizon, then $(M, g)$ is stationary and axisymmetric.

Note that this is a nontrivial hypothesis; it is necessary in the proof, and plenty of theorems do not use the analyticity assumption, i.e. it's not something you just automatically put in.

I don't see why analyticity is a good assumption. Mathematically, you could say the metric is just defined to be analytic, like it's defined to be smooth, but physically there's a big difference. Smoothness directly reflects observation -- a violation of smoothness would require infinite energy as argued here. Analyticity is much stronger: it implies that the entirety of any spacetime is determined by an arbitrarily small piece of it. While I think there's plenty of evidence that the real world is smooth, I don't see why we should treat it as analytic.

Contrast this with another use of analyticity, in quantum field theory. We can analytically continue to imaginary time by Wick rotation and perform the computation there, then continue back to real time. In this case analyticity is used purely as a calculational device; we never view the imaginary time solutions as physically "real".

Is there a way to physically motivate the assumption of analyticity in general relativity?

Answer

Why is analyticity a good mathematical assumption in general relativity? While I think there's plenty of evidence that the real world is smooth, I don't see why we should treat it as analytic.

I don't think analyticity is a good assumption in GR, for exactly the reason you give.

In my experience, discussion of analyticity comes up most often because we're talking about the maximal analytic extension of a spacetime. The point of considering the maximal extension is that we want to rule out unphysical examples that look geodesically incomplete, but are in fact just a geodesically complete spacetime with a piece cut out. The reason for making it analytic is probably just the desire to be able to talk about "the" maximal extension.

For example, suppose I have a spacetime that is the portion of Minkowski space with $t<0$. (Wald has a nice example on p. 148 in which this is initially represented as a certain singular metric so that it's not immediately obvious what it is.) We want to be able to talk about "the" maximal extension of this spacetime and say that it's Minkowski space. But uniqueness may not hold or may be harder to prove if we don't demand analyticity. (It's pretty difficult to prove that Minkowski space is even stable, and I think Choquet-Bruhat only proved local, not global, existence and uniqueness of solutions of Cauchy problems in vacuum spacetimes.)

This is probably analogous to wanting to extend the function $e^x$ from the real line to the complex plane. If you only demand smoothness but not analyticity, you don't have uniqueness.

I don't know how well this analogy holds in detail, and it seems to be true that in many cases you can just require some kind of regularity, but not analyticity. For example, Hawking and Ellis prove uniqueness of maximal developments for vacuum spacetimes (p. 251) using only the assumption that it's a Sobolev space with the metric in $W^4$ (i.e., roughly speaking, that it's four times differentiable). (This is probably their presentation of Choquet-Bruhat's work...?)