34
votes

Accepted

### 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) ...

Community wiki

32
votes

Accepted

### 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 ...

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 ...

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....

Community wiki

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 ...

Community wiki

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 ...

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 ...

17
votes

Accepted

### 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 ...

16
votes

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 ...

Community wiki

14
votes

Accepted

### 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 ...

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 ...

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....

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 ...

10
votes

Accepted

### 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 ...

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:...

10
votes

Accepted

### 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 ...

9
votes

Accepted

### 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 ...

9
votes

Accepted

### 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. (...

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 ...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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 + \...

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 ...

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, ...

7
votes

Accepted

### 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 ...

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 ...

7
votes

Accepted

### 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 ...

7
votes

Accepted

### 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 ...

7
votes

Accepted

### 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 ...

6
votes

Accepted

### 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}\...

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