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.
139
questions
3
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2answers
183 views
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 already asked this ...
0
votes
0answers
126 views
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 ...
7
votes
3answers
306 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 ...
1
vote
3answers
224 views
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 ...
4
votes
2answers
430 views
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-...
6
votes
0answers
76 views
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 $\...
15
votes
2answers
874 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 ...
6
votes
0answers
162 views
Proving 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 ...
48
votes
4answers
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 ...
28
votes
3answers
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 ...
6
votes
0answers
112 views
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 ...
5
votes
1answer
273 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 \...
6
votes
0answers
123 views
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-...
0
votes
0answers
25 views
Outer Minimizing Horizon for a Perturbed Metric
I am reading this paper which proves the Riemannian Penrose inequality in general relativity. On pages 10 - 11, it is stated that the Riemannian $3$-manifold $(M^3, g_t)$ has a strictly outer ...
3
votes
0answers
52 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 ...
16
votes
0answers
602 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 ...
1
vote
1answer
69 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 ...
1
vote
0answers
86 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 ...
0
votes
0answers
67 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 ...
7
votes
1answer
210 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 ...
0
votes
0answers
65 views
Mathematical construction: ADM formulation in general relativity
I'm doing my undergraduate thesis and now I'm looking for references that presents ADM Formulation in general relativity mathematically.
I studied the basics of general relativity theory by O'Neill ...
2
votes
1answer
80 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, ...
6
votes
1answer
221 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 ...
5
votes
0answers
132 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 ...
4
votes
1answer
298 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/...
3
votes
1answer
295 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?
4
votes
1answer
147 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 ...
3
votes
0answers
304 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 (...
16
votes
2answers
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 ...
2
votes
0answers
104 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 ...
2
votes
1answer
357 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 ...
2
votes
1answer
212 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 "...
2
votes
0answers
91 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-...
3
votes
2answers
774 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 ...
1
vote
0answers
101 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 ...
1
vote
1answer
69 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 ...
11
votes
1answer
525 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 ...
2
votes
1answer
196 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 ...
2
votes
0answers
94 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 ...
5
votes
0answers
69 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 ...
3
votes
0answers
196 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 ...
20
votes
1answer
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 ...
0
votes
0answers
83 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 ...
0
votes
1answer
111 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?
0
votes
0answers
241 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) ...
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0answers
140 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 $m_g$...
asked Nov 14 '18 at 8:46
Warlock of Firetop Mountain
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3
votes
2answers
243 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 ...
6
votes
1answer
624 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 ...
6
votes
1answer
453 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 ...
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0answers
138 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 ...
