Questions tagged [general-relativity]

For questions about mathematical problems arising from general relativity, the branch of physics which provides and studies the currently accepted geometric description of gravity.

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Convergence of spheres in nonlinear stability of Minkowski space

My question is about Lemma 3.3.1 in Christodoulou and Klainerman's proof of nonlinear stability of Minkowski space. This lemma says the following: Consider a family of metrics $m_u$ on $S^2$ defined ...
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7 votes
2 answers
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In which dimensions is a strongly causal Lorentzian manifold determined conformally by its causal structure?

Let $M$ be a strongly causal Lorentzian manifold. If $M$ has dimension 4, a theorem of Hawking, King, and McCarthy (see Thm 5) says that $M$ is determined up to conformal isomorphism by its class of ...
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4 votes
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Example of a bicontinuous poset which is not jointly bicontinuous?

Recall that a poset $P$ is said to be continuous if, for every $p \in P$, the set $\{q \in P \mid q \ll p \}$ is directed with supremum $p$. Here $q \ll p$ is the "way below" relation (see ...
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Completeness of infinitely intersecting causal geodesics in strongly causal spacetimes

Let $(M,g)$ be a connected, smooth, strongly causal Lorentzian manifold, and consider an inextendible causal geodesic $\sigma : [0,b) \to M$ (a priori, $b$ may be $\infty$) with the following property:...
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Two identical objects circling the center of mass periodically in general relativity

In Newton's gravity we can have two identical objects circle the center of mass periodically (assuming the surroundings are vacuum). Is something like this possible in general relativity? Is there an ...
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3 votes
0 answers
203 views

What are quantum extremal surfaces from a mathematical viewpoint?

It is said that they are surfaces which locally maximize area and bulk entanglement entropy. It would be great if I could receive some introductory material on it and some prerequisites to understand ...
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2 votes
1 answer
99 views

Hyperboloids in Minkowski geometry

Let $(\mathbb R^{1+2},\eta)$ be Minkowski with the metric $\eta= -dt^2+(dx^1)^2+(dx^2)^2$. Suppose $\Sigma$ is a smooth timelike hypersurface and denote by $h$ the second fundamental form on $\Sigma$. ...
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4 votes
1 answer
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Spin connection in the tetradic Palatini-formalism of general relativity

$\DeclareMathOperator\SO{SO}$I am trying to understand the tetradic Palatini-formalism of general relativity from a mathematical point of view. I am graduate student and quite new to mathematical ...
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3 votes
1 answer
95 views

Compactly supported transverse traceless tensors

Let $(M, g)$ be a Riemanian manifold (or $\mathbb{R}^n$ if you prefer). A TT-tensor is a symmetric 2-tensor $\sigma_{ab}$ satisfying $g^{ab} \sigma_{ab} \equiv 0$ ($\sigma$ is trace free), $\nabla^a ...
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3 votes
1 answer
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A question on light cones in Lorentzian manifolds with timelike boundary

Suppose $M= \mathbb R \times M_0$ with a Lorentzian metric $g(t,x)=-dt^2+ g_0(t,x)$ where $M_0$ is a compact manifold with a smooth boundary and $g_0$ is a family of smooth Riemannian metrics on $M_0$ ...
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4 votes
2 answers
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Reference for mathematical Palatini formalism of general relativity

I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community. I am looking for a reference ...
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Morphism of non-commutative algebras

Disclaimer: this question is a "big picture" one that comes from my personal thoughts in physics. If it doesn't fit this site, please tell me. While having a walk, I thought a bit about what ...
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7 votes
3 answers
393 views

Preservation of metric signature in Cauchy problem for the Einstein equations

In Choquet-Bruhat's solution to the Cauchy problem for Einstein's equation, one reduces the Einstein equations to a quasidiagonal quasilinear hyperbolic system on $ M := [0, T] \times \bar M$ where $T ...
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1 vote
3 answers
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Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths

While going through this paper by Witten and seeing a discussion about different aspects of Raychaudhari Equation and Einstein Field Equation. I want to ask if Raychaudhari Equation find any ...
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2 answers
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In what sense exactly are the Einstein metrics distinguished?

EDIT: In general relativity given a manifold $M$ one can consider a functional on (pseudo-) Riemannian metrics $g$ $$\int_M R\,\, dvol_g,$$ where $R$ is the scalar curvature and $vol_g$ is the (pseudo-...
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6 votes
0 answers
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Deriving (Gaussian) curvature bounds from bounds on the metric

I am trying to understand a bound in Christodoulou's 2008 paper on black hole formation. The paper considers a spacelike surface $S$ diffeomorphic to a sphere, with two metrics: the induced metric $\...
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15 votes
2 answers
925 views

Counterexamples to the Penrose Conjecture

I have noticed that in the literature on causality in general relativity one sees apparent counterexamples to the cosmic censorship hypothesis (somehow you have models for gravitational collapse which ...
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8 votes
0 answers
277 views

Proving the Hawking Area Theorem without Cosmic Censorship

I notice that some of the classic results and theorems in black hole physics from the 1960s like the Hawking area theorem use the cosmic censorship hypothesis at some point in the proofs of the ...
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50 votes
4 answers
7k views

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

I received an email today about the award of the 2020 Nobel Prize in Physics to Roger Penrose, Reinhard Genzel and Andrea Ghez. Roger Penrose receives one-half of the prize "for the discovery ...
29 votes
3 answers
2k views

Penrose’s singularity theorem

Roger Penrose won today the Nobel Prize in Physics for the singularity theorem, which at first glance seems to be a result in pure mathematics. Questions about the theorem: What kind of mathematical ...
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7 votes
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Homotopy types of causal / chronological pathspaces in Lorentzian manifolds?

Let $M$ be a Lorentzian manifold, and let $p,q \in M$. Let $\Pi^J(p,q)$ be the space of causal paths from $p$ to $q$ (in the compact-open topology). Question 1: Is it reasonable to expect that the ...
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5 votes
1 answer
289 views

Proving an identity used in general relativity

I need to prove the following identity for scalar field ($\phi:M\rightarrow R$) in curved spacetime without torsion called $M$ $\nabla_{\mu}[\Box \phi \nabla^{\mu}\phi-\frac{1}{2}\nabla^{\mu}(\nabla \...
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6 votes
0 answers
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On thinking of spacetime as a local Scott domain

An observation of Martin and Panangaden links the study of Lorentzian manifolds and the semantics of programming languages via the theory of Scott domains. Background: Recall that if $M$ is a time-...
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3 votes
0 answers
55 views

Lorentzian cobordism through the dominant energy condition

Is the answer to the following problem, or some close variant thereof, known? Briefly: Given two initial data sets $I_1=(M,g_1,k_1)$ and $I_2=(M,g_2,k_2)$, is there a time-oriented spacetime ...
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20 votes
0 answers
1k views

Schoen and Yau's proof of the higher dimensional positive mass theorem

In April 2017 Schoen and Yau posted on the arxiv their solution of the time-symmetric positive mass theorem in all dimensions, which has been a significant conjecture since the 70s. As of now, July ...
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1 vote
1 answer
91 views

Conjugate point to spacelike hypersurface

Suppose you have a smooth spacelike hypersurface $\Sigma$ in some spacetime (four-dimensional Lorentzian manifold). Let $\gamma$ be a timelike geodesic meeting $\Sigma$ orthogonally and let $p$ be a ...
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1 vote
0 answers
108 views

Perturbation of a spacetime in general relativity

In general relativity one has the Schwarzchild metric for a non-rotating black hole $g_{SC} = -\phi^2 \: dt^2 + \Bigg(1 + \frac{m_0}{2r} \Bigg)^4 \delta $ and from this one has the spacelike ...
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0 answers
73 views

Prerequisites/Preparation for understanding a research paper - global solutions to Einstein field in Bondi Coordinates

I would like to read this paper: João L. Costa, Filipe C. Mena, Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi ...
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1 answer
220 views

Electromagnetic energy in Lovelock gravities

To fix ideas, let us recall that General Relativity describes gravitational phenomena on a 4-dimensional pseudo-Riemannian manifold $(X,g_{ab})$ with field equations that relate the energy-momentum ...
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2 votes
1 answer
113 views

Lower bound for domain of exponential map on Lorentzian manifolds

Let $M$ denote a manifold admitting a Lorentzian metric $g_{ab}$. Essentially, I would like to know the "minimum domain" on which the exponential map is defined at $p\in M$. To make this concrete, ...
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7 votes
1 answer
257 views

Completeness hypothesis in the positive mass theorem

I am trying to understand and further formalize Witten's proof of the positive mass theorem. Dan Lee, in his book "Geometric relativity" did a wonderful job with formalizing and carrying out the ...
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5 votes
0 answers
164 views

Gravity, connection, and curvature

Starting with Synge and Fock, many modern authors identify gravity with curvature. On the other hand, Einstein always emphasized that gravity should be equated with a connection, but not with ...
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5 votes
1 answer
308 views

Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval?

Disclaimer: this a cross post from MSE, where this question was asked on November 4th 2019 and has so far received no upvote, no comment and no answer whatsoever. Glancing at https://en.wikipedia.org/...
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3 votes
1 answer
313 views

Gauss-Bonnet-Chern Theorem [closed]

I am currently doing an undergraduate project about Gauss-Bonnet-Chern Theorem. Is there any particular books/papers regarding the application of the theorem in the theory of general relativity?
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5 votes
1 answer
195 views

Are quadrics the cones of maximal symmetry?

A paper by Ehlers, Pirani, and Schild axiomatizes the geometry of general relativity in what seems like a nice way. However, Jacobson criticizes one aspect of the system as not natural: One deep ...
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3 votes
0 answers
306 views

Maximal symmetry at the speed of light

Are there examples of 1 + 3 dimensional pseudo-Riemannian manifolds with 6 dimensional isometry group whose orbits are light-like (i.e., the metric restricted to each orbit is degenerate)? Here is a (...
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16 votes
2 answers
1k views

What's the "actual" shape of a black hole accretion disk?

[Warning: I have no expertise in general relativity, so this question might not be very rigorous] More and more often we come across science popularization articles like this one which show beautiful ...
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2 votes
0 answers
122 views

Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
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2 votes
1 answer
434 views

Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$

In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given: A (singly) warped product $B \times_b F$ of two pseudo-Riemannian ...
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2 votes
1 answer
243 views

Explanation for "Squashing" and "Stretching" (Lorentzian Analogue of Berger Spheres)

In the paper Anti-de Sitter space, squashed and stretched Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up Berger spheres, it seems what is meant with "...
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2 votes
0 answers
101 views

Marginal surfaces in spacetimes

Is there some result on existence of marginally trapped surfaces in spacetime 4-manifolds? Am I right in saying that a marginal surface (like a trapped surface in general) is a compact spacelike 2-...
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4 votes
2 answers
1k views

Transformation Poincaré-coordinates to global coordinates in $\mathrm{AdS}_2$

For the two dimensional anti de-sitter space $\mathrm{AdS}_2$ one can consider the Poincaré-coordinates $\mathrm{d}s^2_P = -r^2 \mathrm{d}t^2 + \frac{1}{r^2} \mathrm{d}r^2$ which covers only half of ...
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1 vote
0 answers
107 views

Condition for Integrability of an Almost Complex Structure

The following question concerns a remark made in the paper: Lebrun, C., Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 52 ...
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1 vote
1 answer
86 views

Connection of the existence of Killing-Yano tensor and Killing tensor

Stephani states that in 4 dimensions a spacetime admits a non-reducible Killing-Yano tensor only if the Weyl tensor either is of Petrov type D or vanishes. Does this imply that the spacetime also ...
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11 votes
1 answer
542 views

Are there currently any plausible approaches to proving the Penrose сonjecture?

I have recently been reading some of the literature on the Penrose inequality, especially the papers by Bray and by Huisken and Ilmanen. One notices immediately that the existing proofs for the ...
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2 votes
1 answer
276 views

Gaussian null coordinates

I find it hard to find information on the so-called "Gaussian null coordinates", which Wikipedia says is used to describe "near horizon geometries". Can someone provide a reference where I can read ...
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2 votes
0 answers
114 views

Example Petrov Classification

I would like to calculate the Petrov type for a specific spacetime, unfortunately I am not able to find a step by step algorithm or example for the process, either using null-tetrad, nor calculating ...
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6 votes
1 answer
154 views

What exactly is a Cartan radius vector (and its role in Poincaré gauge theories)

I am studying approaches to gravity where the Poincaré group is "gauged". The original motivation of this is to understand what is meant on the statement that "Teleparallel gravity is a gauge theory ...
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3 votes
0 answers
231 views

Gauge structure of teleparallel gravity

I am interested in references that treat teleparallel gravity in a mathematically rigorous manner, especially in regards to it being a "gauge theory of the translation group". The standard reference ...
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22 votes
2 answers
2k views

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

I noticed that the only official reason given for awarding Edward Witten the Fields medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully ...
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