Why is analyticity a good 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...
I recommend you sit down and take a long hard look at page 848 of MTW. That's where you can see these two diagrams:
Image copyright W H Freeman and company, publishers of Gravitation
Look carefully at the Schwarschild image on the left. See how it's truncated at the top? The infalling body somehow crosses the event horizon at time t = infinity. Then it comes back down the chart, tracing out the curve to the left of the vertical dashed line. It ends up in the central point singularity at r = 0 at proper time tau τ = 35.1 M. Yes, according to MTW an infalling body goes to the end of time and back. But that’s not all. If you look horizontally across the Schwarzschild chart at time t = 45, you will notice that the infalling body is at two locations at the same time t. It’s outside the event horizon with a proper time τ = 33.3 M, and at the same time it’s inside the event horizon with a proper time of circa τ = 34.3 M. That’s why you read about the elephant and the event horizon, where the elephant is in two places at once. Analyticity isn't going to fix that. All it does is make the situation worse by throwing in tortoise seconds that last forever. And then that the story goes that the infalling body reaches the point singularity in finite proper time. Finite proper time that hasn't happened yet and never ever will.
To paraphrase Newton, that gravity should lead to such things is to me "so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it". But there again, I'm the only one around here who's read the Einstein digital papers.
and also maximally extend the Kerr and Reissner-Nordstrom spacetimes.
Ah, the Kerr and Reissner-Nordstrom spacetimes. Let's take a look at another question on that, namely What is unstable about the wormhole in the Reissner-Nordström solution? That's where we find this Penrose diagram:
It features a parallel universe, an antiverse, a parallel antiverse, a wormhole, and a white hole. And more. How anybody could take this seriously for a moment absolutely beats me. At this juncture I really would like to be quite forceful about this. But you will forgive me if I hold my tongue.
However, I don't understand why this is a good assumption.
Hopefully you should by now understand why it isn't.
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.
You could also say the metric is an abstract thing, and that the map is not the territory. And that if you think gravity can open some door to some parallel antiverse, you've maybe dropped a stitch somewhere.
By contrast the predictions of analyticity, such as the 'other universes' at the other end of a Reissner-Nordstrom black hole have never been verified, and to the best of my knowledge can never be.
Knzhou, they're a fantasy. On a par with thinking a furnace door is a gateway to paradise.
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".
Quantum field theory has its own problems. And I would venture to say that they are much worse than you think. One for another day.
Analyticity feels unphysical to me because it implies that the entirety of spacetime is determined by an arbitrarily small piece of it. Is there a way to motivate why the analytic extensions in general relativity should be viewed as physical?