Why is analyticity a good mathematical assumption in general relativity?

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...?)

Great question. I once asked a GR postdoc the same thing, and he replied

Certainly I find analyticity far too restrictive an assumption, and the impression I get is that other people in my field agree, but that might be confirmation bias. For a simple example, analyticity excludes me from, say, suddenly throwing some matter into a black hole. Any matter profile must always have an infinite tail to be analytic.

For what it's worth, many important results in GR, like the no-hair theorem, require real analyticity. This fact arguably weakens the relevance of these theorems for describing the real world.