# Tag Info

57

Short version: LIGO matches their data onto waveforms calculated in numerical relativity. The mathematical study of black hole solutions plays a significant role in this; we couldn't trust our inferences if we didn't know a priori that black holes rapidly stabilize into a handful of low-parametric stationary configurations. Classical black holes are ...

31

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) contribution of Sir Roger Penrose to mathematical physics ( not to mention his works as a geometer and his research on tilings, and so many other things). In Physics, ...

29

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 page. Finally, "The large scale structure of space-time" by S.W. Hawking and S.F.R. Ellis - another "star" with a Wikipedia page. All of them were published in the '70s, so they ...

23

The numbers 36,29,62 were obtained as the best match between the received signals and the output of computer simulations. The 90% confidence intervals on these numbers are about $\pm 4$. The details (largely beyond my understanding) are in the technical paper here.

22

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 ill-posed problems.

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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.) 1 The "black hole" theorem (incompleteness theorem) is closely related to, yet subtly different from, the Hawking-Penrose Singularity Theorems. The ...

19

Perhaps the cosmic censorship conjecture (the absence of singularities outside event horizons) is the most compelling, at least that is what is argued by Klainerman in Cosmic censorship and other great mathematical challenges of general relativity, with reference to Hilbert's requirement that a great problem in mathematics "should be clear and easy to ...

18

What is the speed of a wave in a non-linear theory? Answering before considering your question is important, because that answer will tell you where to look for your answer. A useful notion is that of domain of dependence (see for example a decent book on GR for a detailed discussion, e.g., Wald or Hawking & Ellis). If $S\subseteq \Sigma$ is a subset of ...

18

All riemannian manifolds with holonomy contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy precisely those groups. In the Lorentzian setting, you could consider a subclass of lorentzian symmetric spaces with solvable ...

16

This is a somewhat different take on Igor's answer, and I offer it just in case you are interested. First, one doesn't need to have any continuous symmetries in order to have this kind of 'Wick rotation' exist. For example, if $(M,g)$ is a real-analytic Riemannian manifold that admits a nontrivial isometric involution $\iota:M\to M$ that fixes a ...

16

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 Yamabe problem Farid Madani, Hebey-Vaugon conjecture II. (English, French summary) C. R. Math. Acad. Sci. Paris 350 (2012), no. 17-18, 849–852. Both article ...

15

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 complete. As is now well understood, this does not necessarily mean there there is a singularity (in the sense of a region of extreme curvature). A much better ...

14

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 very brief summary of Witten's achievements it should be clear that he has made a profound impact on contemporary mathematics. In his hands physics is once ...

14

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 plausible but non-rigorous physical argument. The main contribution of Penrose and Hawking and the one cited was that they showed (roughly speaking) that if one ...

13

The statement It seems that the classical programme of the PDE community, i.e., (i) existence (ii) uniqueness (iii) regularity, heavily employing concepts from functional analysis, has not found prominent application in general relativity. is just plain wrong. You are overlooking quite a lot of stuff. For a modern presentation of the mathematics I would ...

11

The consistency is proved by Cheeger, M\"uller and Schrader in 1984, "On the Curvature of Piecewise Flat Sapces". Roughly speaking, given a smooth Riemannian manifold with a smooth metric, there exists a sequence of triangulation, on which Regge's definition converges to the smooth curvature as a measure. At the linearized level, there is also a recent ...

11

Since questions on the interface between GR and QFT are admissible, here's what I consider the open problem in that direction. Fix a manifold $M$ and consider the set of globally hyperbolic solutions $\mathcal{S}(M)$ of Einstein's equations, possibly with appropriate asymptotic conditions. (a) Give $\mathcal{S}(M)$ the structure or an infinite dimensional ...

11

Curvature in Mathematics and Physics (2012), by Shlomo Sternberg, based on an earlier book Semi-Riemann Geometry and General Relativity [free download from the author's website] covers much of the same material as O'Neill but is much more recent. This original text for courses in differential geometry is geared toward advanced undergraduate and graduate ...

11

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 sketches how one might be found. The geometrical interpretation that he envisions was actually implicit in 1928 work of the French mathematician Elie Cartan [Cart ...

10

I'm going to guess that your interest in Wick rotation comes from the role it plays in the formulation of quantum field theories (QFTs) on Miknowski spacetime and some other curved spacetimes, like the examples you mentioned. Of course, it is natural to ask, like you probably have, whether Wick rotation plays a similarly important role in the formulation of ...

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This can be found in Besse "Einstein Manifolds", in chapter 4. The idea is to use Koszul formula for the Levi-Civitta connection to compute the derivative of the curvature with respect to the metric. Bianchi identities also help.

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It has found application. The main problem is that Einstein's equations have no type due to the large symmetry group (the whole diffeomorphism group). So first one has to fix a gauge; this is done by fixing a space like hypersurface $\Sigma$. Added in edit: As Deane Yang mentioned, after fixing a gauge, Einsteins equations become hyperbolic. Then the ...

10

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 $\tilde{\rho} \leq C \rho$ for some possibly different $C$ by triangle inequality. The condition is symmetric between $\rho$ and $\tilde{\rho}$, so the other inequality follows ...

8

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 classical or quantum. So the trivial answer to your question is: As stated, the singularity theorems hold for semi-classical relativity. The problem, ...

8

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 of the event horizon, would measure. Matter is falling into the black hole without ever crossing the event horizon, falling ever more slowly as it approaches ...

7

In the case of a globally hyperbolic spacetime, what you want is a smooth Cauchy temporal function (the gradient is everywhere timelike, not just causal, and each level set is a Cauchy surface that is necessarily spacelike). That global hyperbolicity is also sufficient the the existence of a smooth temporal function was also shown by Bernal and Sanchez, in ...

7

Global hyperbolicity is certainly not enough. Consider the causal diamond $$D = \{(t,x) \in \mathbb{R}^{1+1} | |t|+|x| < 1\}$$ in 1+1 dimensional Minkowski space, which is certainly globally hyperbolic. I claim that this set does not admit the so-called "global proper-time foliation". Observe that with a proper-time foliation, the total ...

7

The answer depends on the dimension. When $n=2$, Ricci-flatness of a connection implies that it is flat, so, in that case, yes, you get holonomy reduction locally. However, when $n>2$, Ricci-flatness of a torsion-free connection only implies that the (local) holonomy lies in $\mathrm{SL}(n,\mathbb{R})$. You do not generally get any further reduction ...

7

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, Shing-Tung, On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys. 65, 45-76 (1979). ZBL0405.53045.

7

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 $r = M + \sqrt{M^2 - a^2\,\cos^2\theta}$ (where, as usual, $M$ is the black hole mass and $a$ its angular momentum per unit mass). However, this $r$ coordinate is not what ...

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