Questions tagged [lorentzian-geometry]

Lorentzian geometry is the geometry of Minkowski spacetime, hence essentially of a Euclidean space, but equipped not with the standard Euclidean Riemannian metric of signature $(+,+,+,…,+)$ (which yields Euclidean geometry) but with the pseudo-Riemannian metric of signature $(−,+,+,…,+).$

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Properties that only Lorentzian manifolds have

I wonder what are some statements that although they can be formulated for pseudoriemannian manifolds of arbitrary signature they turn out to be true only in the lorentzian case. I admit things like: &...
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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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Decomposition of a bivector of a Lorentzian manifold [closed]

In the case of the $(1, -1, -1, -1)$ metric, a bivector (or a 2-form) $F$ decomposes uniquely as the sum of two orthogonal simple bivectors if $F^2 \ne 0$. I have the impression that it is very little ...
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2 votes
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On certain umbilic surfaces

Let $(M,g)$ be a three dimensional Lorentzian manifold with signature $(-,+,+)$ and let $p\in M$ and let $U$ be a small neighborhood of $p$. Suppose there is a smooth timelike surface $S$ containing $...
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5 votes
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Complete Lorentz metric on (compact) manifolds

Let $M$ be a smooth manifold that is either compact with $\chi(M)=0$ or that is non-compact. Then we can equip $M$ with a Lorentzian metric $g$. Can we always equip $M$ with a complete Lorentzian ...
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Searching for a type of geometric flow in Lorentzian geometry

Let $(N,g)$ be a globally hyperbolic Lorentzian manifold. Given any smooth hypersurface $\Sigma$ in $(N,g)$ we define $\|\Sigma\|= \sup_{p \in N,X \in T_p\Sigma} |h(X,X)|$ where $h$ is the second ...
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3 votes
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A question on Levi-Civita connection and a fixed hyper surface

Suppose $(M,g)$ is a three dimensional smooth compact simply connected Riemannian manifold with boundary and suppose that $\Sigma$ is a smooth simply connected hypersurface in $M$ with a smooth ...
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4 votes
1 answer
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A question on null geodesics in Lorentzian geometry

Suppose that $(M,g)$ is a smooth Lorentzian manifold, that $I\subset \mathbb R$ is a closed finite interval and that $\gamma: I \to M$ is a smooth null geodesic on $M$, that is to say, it satisfies $$ ...
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2 votes
1 answer
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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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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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Learning roadmap for Lorentzian geometry

I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF. I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (...
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1 vote
0 answers
138 views

Conformal changes of metric and Ricci curvature

Let $(M,g)$ be a three dimensional smooth Lorentzian manifold and let $p$ be a fixed point in $M$ and let $S$ be a smooth symmetric tensor of rank two on $T_pM\times T_pM$. Does there exist a smooth ...
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4 votes
3 answers
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Topology on Minkowski space $\mathbb{R}^{4}$ and Lorentz invariant measure

Let us consider the Minkowski space $(\mathbb{R}^{4},\eta)$ and the mass shell $H_{m}$, $m\ge 0$, given by: \begin{eqnarray} H_{m}:=\{x=(x_{0},x_{1},x_{2},x_{3}) \in \mathbb{R}^{4}: \hspace{0.1cm} x\...
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On intersection of null geodesics

Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that $p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the ...
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1 vote
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conformal changes of Lorentzian metrics

Let $M= \mathbb R \times M_0$ with $M_0$ a smooth compact manifold with smooth boundary and let $g=-dt^2+g_0(t,x)$ be a Lorentzian metric on $M$, with $g_0$ a family of Riemannian metrics on $M_0$ ...
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8 votes
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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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2 votes
2 answers
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stability of two-sided sectional curvature bounds in Lorentzian geometry

Suppose that $(M,g)$ is a Lorentzian manifold of signature $(-,+,\ldots,+)$. Given a two plane $\Pi=\textrm{Span}\{X,Y\}$ with $X,Y \in T_pM$, we say that $\Pi$ is non-degenerate if $$ g(X,X)g(Y,Y)-g(...
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1 answer
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Arbitrary sectional curvatures at a point

Is it possible to choose a Lorentzian metric $g$ on a neighborhood of the origin in $\mathbb R^{1+n}$ so that the sectional curvature of all non-degenerate tangent timelike two planes at the origin is ...
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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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16 votes
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Geometric interpretation of the Weyl tensor?

The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops. Question: Is there a similarly direct geometric ...
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6 votes
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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
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A pseudo-Riemannian version of a theorem by Fubini

Guido Fubini, ``Sugli spazii che ammettono un gruppo continuo di movimenti,'' Annali di Mat., ser. 3, 8 (1903) 54.: Let $M$ be a Riemannian manifold of dimension $d\ge 3$. Its isometry group cannot be ...
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1 answer
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Globally hyperbolic spacetimes and future Cauchy developement

Let $(M,g)$ be a globally hyperbolic Lorentzian spacetime with a non-compact Cauchy hypersurface $S \subset M$. Let $ \Omega \Subset S$ be an open subset. Is it true that the future Cauchy development ...
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1 vote
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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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2 votes
1 answer
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conformal changes to Lorentzian curvature

Let $(M,g)$ be a Lorentzian manifold and let $R$ be the curvature tensor. We say $R\leq 0$ if $$ g(R(X,Y)Y,X) \leq 0\quad \forall \, X,Y \in TM.$$ My question is whether given a Lorentzian manifold $...
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3 votes
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Lorentzian manifolds of negative spacelike sectional curvature

Suppose $(M,g)$ is a simply connected Lorentzian manifold of signature $(-,+,\ldots,+)$ and such that the sectional curvature of any space-like surface is non-positive. Is it true that there are no ...
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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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9 votes
1 answer
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Fourier transform on Minkowski space

Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...
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A Riemannian manifold with a non-degenerate metric and an inner product $u_{\beta} u^{\beta}=1$

The question is: given a Riemannian manifold with a non-degenerate metric g and an inner product $u_{\beta}u^{\beta}=1$, is $\nabla_{\mu} (u_{\alpha}u_{\beta})=0$ without demanding the trivial ...
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Is it possible for null geodesics to suddenly "stop"?

Suppose $(M,g)$ is a smooth globally hyperbolic Lorentzian manifold of dimension $n$. Let $\beta:I \to M$ be a finite null geodesic in $M$, that is to say: $$ \nabla^g_{\dot{\beta}}\dot{\beta} = 0, \...
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16 votes
2 answers
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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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4 votes
1 answer
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Literature Request: Berger Spheres and their Construction

In a previous question by me I asked about Berger spheres and their Lorentzian analogue, squashed $AdS_3$ along Hopf fibres. It was well answered (by https://mathoverflow.net/users/13268/ben-mckay) ...
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2 votes
1 answer
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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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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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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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Example of a Lorentz-invariant probability measure?

What is an example of a Lorentz-invariant probability measure on Minkowski space other than the delta measure at the origin? Euclidean-invariance is easy to attain: the uniform measure on a ball does ...
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12 votes
1 answer
313 views

Do (3+1)-dimensional Lorentzian manifolds admit unique smoothings?

Of course, 3-dimensional topological manifolds admit unique smoothings while 4-dimensional topological manifolds generally do not. A (3+1)-dimensional topological Lorentzian manifold (definition below)...
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3 votes
1 answer
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Existence of transverse null vector bundle in a degenerate Lorentzian hypersurface

This question is cross-posted at https://math.stackexchange.com/questions/3234895/existence-of-transverse-null-vector-bundle Let $(M,g)$ be a Lorentzian manifold. Let $\dim(M)=d$. Given a null ...
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3 votes
1 answer
131 views

maximal surface, parametric approach

I am interesting in maximal surfaces: space-like surface in Minkowski $\mathbb{R}^{2,1}$ (or De Sitter $dS^3$). Of course space-like implies locally graph and almost all the literature is interested ...
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3 votes
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Harmonic maps into de Sitter Space

I am looking some references on the existence and non-existence of (space-like) harmonic maps solving the Dirichlet into the de-Sitter space. More precisely: Let, for $n\geq 3$, $$dS^n=\{ u\in \...
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2 votes
1 answer
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$(M,g)$ is complete iff $(\tilde{M},\tilde{g})$ is complete (non-Riemannian version)

I'm not sure if this question is too low level for Math Overflow (so feel free to move this to SE if you think it is). Inspired by this and this question I'm wondering if the following statement is ...
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1 vote
0 answers
133 views

Are there Lorentzian complex manifolds?

Quick and simple... Is it possible to define complex structures on Lorentzian manifolds? If so, Can you point me to some example(s)?
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Decomposition of a Jacobi field along a lightlike geodesic

Consider a Lorentzian manifold of dimension $1+n$ (with $n\geq1$) and a lightlike geodesic $\gamma(t)$ on it. One can define a Jacobi field $J(t)$ along $\gamma$ in the usual way without issues. In ...
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15 votes
0 answers
294 views

Lorentzian analogue to Thurston geometries

Is there an analogue to the eight Thurston geometries for Lorentz metrics? If so, how many "disctinct" geometries are there in the Lorentzian case? And which closed 3-manifolds admit metrics which ...
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9 votes
1 answer
439 views

Is the set of Lorentzian metrics metrizable?

Fix a differentiable non-compact manifold $M$. Denote by $\mathrm{Lor}(M) := \{\text{Lorentzian metrics on $M$}\}.$ One can define a topology on this set via: fix any open covering $\mathcal{A}$ on $M$...
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3 votes
1 answer
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Closed Semi-Riemannian manifolds with non-compact isometry group

Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group? Edit: My goal is to understand a bit better what the intuition ...
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4 votes
1 answer
336 views

Canonical Metrics on 3- and 4-Manifolds

From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...
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1 vote
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Support of functions on Minkowski space and their Fourier transform

Suppose that a tempered distribution $f(x)$ on 4-dimensional Minkowski space with signature $+---$ vanishes for $x^2<0$ and its Fourier transform $\hat f(p)$ vanishes for $p^2<0$. Are then $f$ ...
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2 votes
1 answer
87 views

Find the fixed geodesic of an orientation-preserving isometry of the $3D$ hyperboloid model

Let $\mathcal{I}^3\subset\mathbb{R}^4$ be the standard hyperboloid model for hyperbolic $3$-space and consider the usual $\mathrm{SO}(3,1)$ action of $\mathrm{PSL}(2,\mathbb{C})$ on $\mathcal{I}^3$. ...
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4 votes
0 answers
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Distributional PDE solutions as topological linear duals of PDE solutions

Let $$ P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast) $$ be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
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