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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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 ...
Hollis Williams's user avatar
8 votes
0 answers
304 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 ...
Hollis Williams's user avatar
49 votes
4 answers
8k 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 ...
30 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 ...
ThiKu's user avatar
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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 ...
Tim Campion's user avatar
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5 votes
1 answer
335 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 \...
gustavo's user avatar
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7 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-...
Tim Campion's user avatar
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3 votes
0 answers
69 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 ...
Quarto Bendir's user avatar
20 votes
0 answers
2k 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 ...
Quarto Bendir's user avatar
1 vote
1 answer
130 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 ...
Earl Jones's user avatar
1 vote
0 answers
130 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 ...
Hollis Williams's user avatar
0 votes
0 answers
80 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 ...
Sun's user avatar
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7 votes
1 answer
240 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 ...
José Navarro's user avatar
2 votes
2 answers
210 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, ...
user143410's user avatar
7 votes
1 answer
323 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 ...
Nicolò Cavalleri's user avatar
6 votes
0 answers
236 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 ...
Zurab Silagadze's user avatar
5 votes
1 answer
327 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/...
Sylvain JULIEN's user avatar
3 votes
1 answer
367 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?
Nothing's user avatar
  • 159
5 votes
1 answer
219 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 ...
user avatar
3 votes
0 answers
318 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 (...
Thomas Schucker's user avatar
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 ...
Qfwfq's user avatar
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2 votes
0 answers
139 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 ...
annie marie cœur's user avatar
2 votes
1 answer
716 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 ...
horropie's user avatar
  • 649
2 votes
1 answer
355 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 "...
horropie's user avatar
  • 649
2 votes
0 answers
105 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-...
Hollis Williams's user avatar
4 votes
2 answers
2k 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 ...
horropie's user avatar
  • 649
1 vote
0 answers
124 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 ...
AmorFati's user avatar
  • 1,349
1 vote
1 answer
132 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 ...
horropie's user avatar
  • 649
11 votes
1 answer
594 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 ...
Hollis Williams's user avatar
2 votes
1 answer
451 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 ...
horropie's user avatar
  • 649
2 votes
0 answers
138 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 ...
horropie's user avatar
  • 649
7 votes
1 answer
218 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 ...
Bence Racskó's user avatar
3 votes
0 answers
294 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 ...
Bence Racskó's user avatar
22 votes
2 answers
3k 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 ...
Hollis Williams's user avatar
0 votes
0 answers
119 views

Stably causal spacetimes

Consider a stably causal Lorentzian spacetime $(\mathcal{M},g)$, so that a non-zero timelike vector field $t^a$ exists such that $$ \tilde{g}_{ab}=g_{ab}-t_a t_b $$ and $(\mathcal{M},\tilde{g})$ has ...
user12588's user avatar
  • 101
0 votes
1 answer
115 views

Analytic approach to geodesic connectedness in Semi-Riemannian manifolds

Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
user avatar
0 votes
0 answers
363 views

Mathematical Problems of General Relativity II

In the introduction of D. Christodoulou's book "Mathematical Problems of General Relativity I", he refers a few times to the second volume. My question is does it exists? Has it been (or will it be) ...
MBN's user avatar
  • 133
1 vote
0 answers
151 views

The converse to the positive mass theorem

Let $(M^n,g)$ be an asymptotically flat manifold of decaying-order $\tau>\frac{n-2}{2}$, the positive mass theorem states that if the scalar curvature $S_g$ is non-negative, then the ADM mass $m_g$...
Overflowian's user avatar
  • 2,523
2 votes
2 answers
346 views

Einstein warped product manifold Ricci flat

Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold Ricci flat (i.e. $Ric=\lambda g$ with $\lambda=0$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and ...
MathDG's user avatar
  • 242
6 votes
1 answer
815 views

Physical (GR) Differential Geometry?

I am looking for problem lists or books which contain open problems in the area of mathematics motivated by physics. Ideally, I am looking for questions asking about which reduce to some calculation ...
Giulia S-A.'s user avatar
6 votes
2 answers
1k views

Manifolds with negative dimension – Definition, References

Does the concept of differential manifold with negative dimension make sense, in differential geometry? If yes, how is it defined? Do you have any reference to recommend? My problem was born in ...
MathDG's user avatar
  • 242
1 vote
0 answers
179 views

Timelike geodesic congruences covering the whole Schwarzschild spacetime

For some reason, I would need to know what are the timelike geodesic congruences which cover the entire exterior region of the Schwarschild spacetime. In fact the only thing I really need is the ...
Fabien Besnard's user avatar
2 votes
1 answer
288 views

Schwartz distributions, Colombeau algebra and applications

I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book: "Geometric Theory of Generalized Functions with ...
Andrew's user avatar
  • 559
36 votes
7 answers
5k views

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

I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free. The idea is to use ...
Leo's user avatar
  • 395
2 votes
0 answers
469 views

Scalar curvature and warped-product manifolds - intuition

Let $(M, g) = (N_1, g_1) \times_f(N_2, g_2)$ be an Einstein warped-product manifold, with metric $g=g_1+f^2g_2$. What does it mean if the scalar curvature of its base-manifold $(N_1, g_1)$, equal to ...
MathDG's user avatar
  • 242
3 votes
1 answer
213 views

A problem about closed 2-forms on Minkowski space

The problem is: For any closed 2-form in the Minkowski space $\mathbb{R}^{3,1}$ satisfiying $dF=0$ and $\delta F \ne 0$ (with $\delta$ denoting the codifferential), does there exist a Lorentz ...
jacktang1996's user avatar
3 votes
0 answers
161 views

Parallel frame for marginally trapped bi-harmonic surfaces in $\Bbb R^4_2$

I'm reading the paper Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its applications to biharmonic surfaces by B. Y. Chen. Summarizing it quickly: he first ...
Ivo Terek's user avatar
  • 1,061
1 vote
1 answer
694 views

Invariance of a vector under parallel transport along an infinitesimal orthogonal loop

I'm not very familiar with differential geometry and am coming from a general relativity background, so would appreciate help with a question from that context. If this question could be posed in a ...
Aegon's user avatar
  • 173
6 votes
2 answers
190 views

Why are they called "screen" distributions?

If $V$ is a vector space and $g$ is a symmetric degenerate bilinear form on $V$, every complementary subspace to the radical ${\rm rad}(V)$ is called a "screen subspace" of $V$: we have an orthogonal ...
Ivo Terek's user avatar
  • 1,061
9 votes
2 answers
543 views

Some Mathematical Questions on Gravitational Waves and Numerical Relativity

Due to the recent spate of detections of gravitational waves by LIGO, my amateurish interest in the mathematics of general relativity has been revived. The wave-forms of the detected gravitational ...
Transcendental's user avatar