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34 votes
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What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

It seems (as mentioned by Sam Hopkins above) that the Singularity Theorem is the official reason for the Nobel Award. But that is by no means the only (and perhaps not even the most important) ...
32 votes
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How much of mathematical General Relativity depends on the Axiom of Choice?

The dependence on AC through the use of Zorn's lemma in the proof of the Choquet-Bruhat–Geroch theorem on the existence of a maximal globally hyperbolic development for solutions of the Einstein ...
Igor Khavkine's user avatar
31 votes

Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations?

Try "General Relativity for mathematicians" by R. Sachs and H. Wu. Also, "Gravitation" by C.W. Misner, K.S. Thorne, J.A. Wheeler - it's so famous that it's got its own Wikipedia ...
Alex M.'s user avatar
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25 votes

What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

I answered about the incompleteness theorem in the other thread. Let's talk about some of his other contributions here. (This list is definitely incomplete*, but just some stuff off the top of my head....
22 votes

What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

A very interesting contribution (not directly related to relativity) is joint with Moore on the so-called Moore-Penrose inverse or generalized inverse, which is crucial in inverse problems theory and ...
19 votes

Is Witten's proof of the positive mass theorem rigorous?

You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3). 6. Conclusion From this ...
Francesco Polizzi's user avatar
19 votes

Penrose’s singularity theorem

1 Penrose's singularity theorem is a bit of a misnomer. Penrose never showed that there is a singularity in the spacetime. What he proved is that the spacetime cannot be timelike or null geodesically ...
Willie Wong's user avatar
17 votes
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What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

I know two applications: a) The convergence of the Yamabe flow in dimension 6 and higher Simon Brendle, Invent. math. 170, 541–576 (2007) DOI: 10.1007/s00222-007-0074-x b) Solution of the equivariant ...
Bernd Ammann's user avatar
16 votes

What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

I would say Penrose is a mathematical physicist and I don't think he can be considered (at least not primarily) to be a pure mathematician. For example, his argument for the Penrose inequality is a ...
14 votes
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Manifolds with negative dimension – Definition, References

Smooth manifolds of negative dimension are defined in derived geometry. Recall that if A→M and B→M are two transversal submanifolds of codimension a and b respectively, then their intersection C is ...
Dmitri Pavlov's user avatar
12 votes

General Relativity and Differential Geometry intuitions of Second Bianchi Identity

From The "Foreword to Feynman Lectures on Gravitation" by John Preskill and Kip S. Thorne: In §9.3, Feynman comments that he knows no geometrical interpretation of the Bianchi identity, and he ...
Zurab Silagadze's user avatar
12 votes

On imaginary time

The introduction of imaginary time as a way to resolve the Big Bang singularity is a proposal by Hawking and others. I don't think it plays a role in modern cosmology, see Emerging from imaginary time....
Carlo Beenakker's user avatar
12 votes

What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?

This is perhaps more of an extended comment than a real answer, but I do think it goes a long way towards answering these kinds of questions. The set-theoretic result referred to as Shoenfield ...
James E Hanson's user avatar
10 votes
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A step in the proof on the uniqueness of mass

Yes it follows from the previous sentence. Since $\tau > (n-2)/2 \geq 0$ by assumption you have that $|\varphi^i| \leq C \rho$ from the definition of the norm. This implies that $\tilde{\rho} \leq ...
Willie Wong's user avatar
10 votes

How much of mathematical General Relativity depends on the Axiom of Choice?

It's not the headline theorem you wanted, but this gives at least a lower bound on what is possible in much weaker foundations than usually assumed, and also analyses a serious theorem in that setting:...
David Roberts's user avatar
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10 votes
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Is the Gödel universe Wick rotatable?

$\DeclareMathOperator\SL{SL}$Clearly, as Robert Bryant indicates, it is Wick-rotatable to a different Lorentzian space. However, it is also Wick-rotatable to a Riemannian space, albeit negative ...
Sigbjørn Hervik's user avatar
9 votes
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Is Witten's proof of the positive mass theorem rigorous?

The positive mass theorem is more or less to do with the geometry of a type of positive scalar curvature condition. Witten's work considers harmonic spinors, which are solutions to a certain linear ...
Quarto Bendir's user avatar
9 votes
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In what sense exactly are the Einstein metrics distinguished?

If I understood your question correctly, the answer indeed is due to Lovelock. I think it's important to state all the hypotheses clearly, because they are not always reported accurately. Theorem. (...
Igor Khavkine's user avatar
9 votes

Is the Gödel universe Wick rotatable?

I may be misreading the sources that you list for the definition of Wick-rotatable, but, I believe that the following construction does fit that definition: According to the Wikipedia page that the ...
Robert Bryant's user avatar
8 votes
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Singularity theorems for semiclassical gravity

In terms of singularity theorems: The Hawking-Penrose singularity theorems require certain energy conditions be satisfied; the theorems are in particular not-sensitive to the underlying fields being ...
Willie Wong's user avatar
8 votes
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What's the "actual" shape of a black hole accretion disk?

The "shape" of an accretion disc is the spatial profile of the gas density $\rho$ at a given time $t$. Here is a recent calculation, arXiv:1810.0083. This image shows what a distant observer, outside ...
Carlo Beenakker's user avatar
8 votes
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Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval?

It's not a Cassini oval. To see this, recall that we're talking about the outer static limit of the black hole, whose (Boyer-Lindquist) $r$ coordinate in function of $\theta$ is given by $r = M + \...
Gro-Tsen's user avatar
  • 30.3k
8 votes

How much of mathematical General Relativity depends on the Axiom of Choice?

My (attempt at an) answer goes in the direction of "2. natural restrictions". First of all, as noted in the comments, provable in ZF are restrictions of AC to the language of second-order ...
Sam Sanders's user avatar
  • 3,999
7 votes

Gauss-Bonnet-Chern Theorem

Gauss-Bonnet is used extensively in the proofs of “no-hair” and “positive mass”: Israel, Werner, Event Horizons in Static Vacuum Space-Times, Phys. Rev. 164, 1776-1779 (1967). Schoen, Richard; Yau, ...
Francois Ziegler's user avatar
7 votes
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Completeness hypothesis in the positive mass theorem

Completeness is necessary. Otherwise you can just take a maximal spatial slice of the negative-mass Schwarzschild solution (i.e. a constant $t$ slice in Boyer-Lindquist coordinates) and it has ...
Willie Wong's user avatar
7 votes

Penrose’s singularity theorem

Wille's answer is technically true, but he doesn't talk about the historical context of the result. I think that is important for understanding why such a "simple" result is deserving of a ...
Ben Whale's user avatar
  • 480
7 votes
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Preservation of metric signature in Cauchy problem for the Einstein equations

The dynamic metric is the metric for the quasidiagonal system; and so for self consistency the solution can only be proven to exist (and be unique) when the metric (i.e. the solution itself) is ...
Willie Wong's user avatar
7 votes
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Is every strongly causal spacetime purely electric?

These are quite orthogonal conditions. To start, one is a global condition, while the other is a local one. Every point has a small enough neighborhood that is strongly causal (even globally ...
Igor Khavkine's user avatar
7 votes
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Topology and local isometry, spinning cosmic string

I think in your question, as currently formulated, the whole rotating cosmic string is a red herring. If I interpret your notation correctly, $a$ and $\kappa$ are constants. And hence locally you can ...
Willie Wong's user avatar
6 votes
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Invariance of a vector under parallel transport along an infinitesimal orthogonal loop

In dimension $3$, the curvature tensor can always be written with respect to an orthonormal frame as something like $$ R_{ijkl} = R_{ik}\delta_{jl} + R_{jl}\delta_{ik} - R_{il}\delta_{jk} - R_{jk}\...
Deane Yang's user avatar

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