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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Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems

I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries. In pseudo-Riemannian geometry, for ...
lming2's user avatar
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N dimensional, not-locally Euclidean, non-Hausdorff topological space

Take a topological space $(M, \tau) $ where $\tau$ is the collection of open sets of $M$. Suppose: the Lebesgue covering dimension of this space is $N \geq 1$ Non-Hausdorff Not locally Euclidean The ...
Bastam Tajik's user avatar
3 votes
0 answers
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On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
G. Blaickner's user avatar
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3 votes
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What is the Lebesgue covering dimension of this topological space?

Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal. Take the induced topology defined by the Lorentzian metric called Alexandrov topology. This topology matches ...
Bastam Tajik's user avatar
13 votes
2 answers
1k views

Is the Gödel universe Wick rotatable?

Take Wick rotatability being as the way defined in the following article by Helleland and Hervik: Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
Bastam Tajik's user avatar
1 vote
0 answers
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How causal is a strongly causal purely electric spacetime?

Take a generic Lorentzian spacetime $(M, g)$ where $M$ is a time-oriented 4d manifold and $g$ is the Lorentzian metric that is strongly causal and purely electric. According to this answer: Is every ...
Bastam Tajik's user avatar
2 votes
0 answers
344 views

Is a Wick rotatable spacetime necessarily strongly causal?

There are a few viable ways to formulate Wick rotatability that preserve distinct features. One is mentioned in the post: Obtain Lorentzian manifolds from Riemannian ones by Wick rotation There's also ...
Bastam Tajik's user avatar
3 votes
1 answer
222 views

Is every strongly causal spacetime purely electric?

Take a time 4-dimensional orinted Lorentzian manifold $(M,g)$. A spacetime is called Strongly Causal at point $p$ if and only if for every neighbourhood $U$ of the point $p$ there exists a ...
Bastam Tajik's user avatar
3 votes
1 answer
306 views

reference for reading Schoen Yau positive mass theorem proof II

I am trying to read the paper by Schoen and Yau, Proof of the Positive Mass Theorem II. The notation is very different from what I am familiar with (basically Robert Wald's book on general relativity)....
Bowen Zhao's user avatar
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1 answer
170 views

Distinguishable under manifold topology but indistinguishable under the Alexandrov topology

Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal. In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
Bastam Tajik's user avatar
3 votes
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67 views

Is it always possible to find a conjugate optical function?

Optical functions (functions with null gradients) and double null foliations (foliations of a spacetime by two related optical functions) play a large roll in modern mathematical relativity research. ...
Chris's user avatar
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What is the nerve of this category?

If $\mathcal{C}$ is a thin category, we call $U \subseteq \mathrm{Ob}(\mathcal{C})$ open if for every object $X \in U$ and any morphism $X \to Y$, we also have $Y \in U$. This declares an Alexandrov ...
Bastam Tajik's user avatar
1 vote
1 answer
258 views

Temporal evolution of a globally hyperbolic spacetime

Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal). For ...
Bastam Tajik's user avatar
3 votes
0 answers
772 views

Is this set a manifold?

Take a general spacetime that is not strongly causal. Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
Bastam Tajik's user avatar
5 votes
2 answers
695 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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483 views

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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Lorentzian geometry. Comparing Honda's main theorem A construction to mine: Mixed type surfaces

This question is based on a wonderful paper by A. Honda (link below) where his main theorem A provides an incredible uniqueness result. Mixed type surfaces and type changing metrics have been ...
53Demonslayer's user avatar
40 votes
3 answers
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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
331 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
90 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
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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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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
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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
11 votes
1 answer
343 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 ...
Matha Mota's user avatar
3 votes
1 answer
230 views

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 ...
Matha Mota's user avatar
6 votes
1 answer
273 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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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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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
300 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 ${ }^{(4)} h$ ...
Matha Mota's user avatar
5 votes
1 answer
822 views

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
129 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
262 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
134 views

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
0 answers
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
111 views

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
249 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
149 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
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5 votes
1 answer
328 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
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4 votes
1 answer
179 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
206 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
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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
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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 ...
Sylvain JULIEN's user avatar
7 votes
3 answers
512 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
441 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
598 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
118 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
  • 389
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