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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Is this set a manifold?

Take a general solution of the Einstein's Field Equation that is not strongly causal. Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric ...
Bastam Tajik's user avatar
5 votes
2 answers
580 views

What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?

In the same spirit of this question: How much of mathematical General Relativity depends on the Axiom of Choice? I want to go radically further ahead and ask for what remains of mathematical general ...
Bastam Tajik's user avatar
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1 answer
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Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds

$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
Bastam Tajik's user avatar
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0 answers
65 views

Comparing Honda's construction to mine

Consider a finite family $\Psi_\alpha$ where $\alpha=1,2,3,4$ of mutually transversely intersecting mixed type surfaces of $\mathbf L^3$ (Lorentz-Minkowski $3$-space) inscribed in $[-1,1]^3.$ Let the $...
53Demonslayer's user avatar
40 votes
3 answers
4k views

How much of mathematical General Relativity depends on the Axiom of Choice?

One of the cornerstones of the mathematical formulation of General Relativity (GR) is the result (due to Choquet-Bruhat and others) that the initial value problem for the Einstein field equations is ...
Pelota's user avatar
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4 votes
1 answer
314 views

Metric with a constant Chern–Pontryagin scalar

Do there exist 3+1 dimensional spacetimes (i.e. Lorentzian manifolds with signaure (1,3)), for which the Chern–Pontryagin scalar \begin{equation} K_2= \epsilon^{\mu\nu\rho\sigma}R^{\alpha}{}_{\beta\mu\...
Michał Jan's user avatar
2 votes
0 answers
84 views

Non-compactness on Penrose singularity

I've been studying singularities in GR, and (obviously), came across PST. Let us state it as the following: Let $(M, g)$ be a connected globally hyperbolic spacetime with a noncompact Cauchy ...
Johann Wagner's user avatar
0 votes
1 answer
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Curvature of an affine system

I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It reminded me that many textbooks on Riemannian geometry only tell us about metrics, ...
lumw's user avatar
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86 views

Behavior of lapse function at infinity: stability of Minkowski

In the Stability of Minkowski Spacetime, Christodoulou and Klainerman prove a local existence proof for a particular class of quasilinear wave equation for a symmetric, traceless, covariant 2-tensor $...
Chris's user avatar
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2 votes
0 answers
108 views

Naked curvature singularity vs Cauchy horizon in stably causal space-time

There is a result that says (theorem 2.11) that any stably causal space-time $M$ is either a product $\Sigma\times \mathbb{R}$ or the time-like gradient $\nabla f$ of a time function $f:M\rightarrow \...
Grothendieck's Ox's user avatar
10 votes
1 answer
315 views

Synthetic differential / conformal geometry of Lorentzian manifolds?

Let $M$ be a sufficiently nice Lorentzian manifold of dimension $\geq 3$. It's known [1] (see also [2]) that the differential and even conformal structure of $M$ is completely encoded in the causal ...
Tim Campion's user avatar
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Number of divergence free symmetric two tensor in dimension 4 [duplicate]

In a $4$ dimensional (semi)-Riemannian manifold $(M^{4}, g)$, both Einstein tensor $G= \operatorname{Ric}(g)- \frac{R(g)}{2}g$ and stress-energy tensor $T$ symmetric and divergence-free. Is there any ...
Boka Peer's user avatar
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3 votes
1 answer
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Definitions fundamental forms and their geometric Intuition

Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many ...
Boka Peer's user avatar
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6 votes
1 answer
235 views

The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds

(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.) Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...
Hrhm's user avatar
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1 answer
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Numerical reconstruction of Einstein's field equations

A few analytic solutions are known to the Einstein field equations: $$ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R - kT_{\mu\nu} = 0$$ Taking a preexisting analytic solution such as Schwarzchild's solution: $$...
James's user avatar
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0 answers
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Understanding the boundary condition of spherical waves in the flat spacetime

I am trying to understand one of the two boundary conditions one has to impose to find the solutions of the wave equation in the flat space-time inside a collapsing null shell. For the spherical wave, ...
Hadamard1's user avatar
1 vote
0 answers
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Invariance signature in infinite dimension

Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that: $g_0$ is positive-definite $g_t$ is non-degenerate for ...
John117's user avatar
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0 answers
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What spacetime symmetry groups does U(2,2) contain?

The group U(2,2) has 16 real parameters. It contains the Lorentz group. What other spacetime groups does U(2,2) contain? How is it related to GL(4,R) which also has 16 real parameters?
Kevin Cahill's user avatar
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1 answer
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Mathematical characterization of gravitational geons as reference request, and their properties as main question

I've edited (ten days ago) a question on Physics Stack Exchange, this Mathematical characterization of gravitational geons, post with identifier 726281 the users of the site were kind adding in the ...
user142929's user avatar
2 votes
1 answer
246 views

Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper

The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1 1.1. Lemma. If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ ...
Boka Peer's user avatar
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5 votes
1 answer
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On imaginary time

I've heard a few times that "the time was imaginary before the Big Bang". I am guessing it means that at this stage, the space-time was a Riemannian $4$-manifold, but I am not sure this ...
aglearner's user avatar
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1 answer
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How to choose a set of non-orthonormal basis vectors for the absolute space of a stationary and axisymmetric space-time in General Relativity?

In General Relativity, the space-time is described by the metric tensor $g_{\mu\nu}$, where $\mu,\nu=0,1,2,3$ and the interval is written as $$ds^2=g_{\mu\nu}dx^\mu dx^\nu$$. A 3+1 split allows to ...
Richard's user avatar
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3 votes
1 answer
126 views

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 ...
Chris's user avatar
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7 votes
2 answers
243 views

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 ...
Tim Campion's user avatar
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4 votes
1 answer
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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 ...
Tim Campion's user avatar
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1 vote
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70 views

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:...
jawheele's user avatar
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2 votes
0 answers
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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 ...
soft-drinks's user avatar
3 votes
0 answers
240 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 ...
Siddharth Panigrahi's user avatar
2 votes
1 answer
131 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 $\Sigma$. ...
Ali's user avatar
  • 3,883
5 votes
1 answer
277 views

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 ...
G. Blaickner's user avatar
4 votes
1 answer
156 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 ...
Romain Gicquaud's user avatar
3 votes
1 answer
176 views

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 $M_0$ ...
Ali's user avatar
  • 3,883
4 votes
2 answers
1k 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 am looking for a reference ...
B.Hueber's user avatar
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0 answers
166 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 ...
Sylvain JULIEN's user avatar
7 votes
3 answers
481 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 ...
Chris's user avatar
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1 vote
3 answers
369 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 ...
aitfel's user avatar
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6 votes
2 answers
581 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-...
asv's user avatar
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6 votes
0 answers
109 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 $\...
Chris's user avatar
  • 367
15 votes
2 answers
1k 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 ...
Hollis Williams's user avatar
8 votes
0 answers
298 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
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 ...
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
0 answers
143 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 ...
Tim Campion's user avatar
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5 votes
1 answer
314 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
163 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-...
Tim Campion's user avatar
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3 votes
0 answers
68 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
120 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
124 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
78 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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