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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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 ...
JS.'s user avatar
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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 ...
Tim Campion's user avatar
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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 ...
Qfwfq's user avatar
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13 votes
2 answers
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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
12 votes
1 answer
355 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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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 ...
Tim Campion's user avatar
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11 votes
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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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9 votes
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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 $M$...
L.F. Cavenaghi's user avatar
9 votes
1 answer
730 views

On the topology induced by a Lorentzian metric

Let $(M,g)$ be a time-oriented smooth Lorentzian manifold, with Lorentzian metric $g$. In the following thread: https://physics.stackexchange.com/questions/228669/why-pseudo-riemannian-metric-cannot-...
Bilateral's user avatar
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8 votes
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On the causal structure of spacetimes: piecewise $C^1$, $C^k$ or $C^\infty$?

This is a more technical question but it seems that there is some confusion in the literature on the choice of curves used to define the causal relations in time-oriented Lorentz manifolds: the ...
Stefan Waldmann's user avatar
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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 ...
Hollis Williams's user avatar
7 votes
2 answers
263 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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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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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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6 votes
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Time-separation function on "globally hyperbolic" spacetimes with everywhere timelike boundary

It is well-known that, in globally hyperbolic spacetimes, the time separation function $\tau$ (aka Lorentzian distance function) enjoys the following property: fix a point $p$ and a point $q \in I^-(p)...
Umberto Lupo's user avatar
5 votes
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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) ...
horropie's user avatar
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Space of spacelike embeddings as infinite-dimensional manifold

Consider a four-dimensional Lorentzian manifold $(\mathcal{M},g)$ and a $3$-dimensional compact manifold $\Sigma$, such that there exists a spacelike embedding $i:\Sigma\to\mathcal{M}$ so that $h:=i^{\...
G. Blaickner's user avatar
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5 votes
1 answer
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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 ...
Joonas Ilmavirta's user avatar
5 votes
0 answers
159 views

Intersection of orbits of earthquake flow on Teichmüller space

Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...
Atlas Tasilli's user avatar
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$C^1$ isometries of pseudo-Riemannian metrics are smooth?

It is well known that $C^1$ (actually even just differentiable) isometries of Riemannian manifolds are actually $C^\infty$. The proof is based on the metric structure generated by the Riemannian ...
mitsutani's user avatar
5 votes
0 answers
137 views

Einstein metrics on spheres

We know that a closed oriented manifold $M$ carries a Lorentzian metric iff the Euler characteristic vanishes. My question concerns the existence of those Lorentzian metrics on odd-dimensional spheres ...
Julian Seipel's user avatar
5 votes
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267 views

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 ...
Adam Chalumeau's user avatar
4 votes
3 answers
356 views

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\...
IamWill's user avatar
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4 votes
1 answer
463 views

Decay estimates for wave and Klein-Gordon equation in "generic" curved backgrounds

Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $...
Pedro Lauridsen Ribeiro's user avatar
4 votes
1 answer
221 views

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 $$ ...
Ali's user avatar
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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
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4 votes
1 answer
373 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 ...
QGravity's user avatar
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1 answer
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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
4 votes
1 answer
167 views

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 ...
Thomas Schucker's user avatar
4 votes
0 answers
200 views

Automorphism group of a Lorentzian lattice

Consider the even integral lattices $L_n:=Z\times Z\times Z^{n-2}$ (where $Z$ is the set of integers) with elements $x=(x_+,x_-,x_d)$ and inner product $$(x,y):=x_+y_-+x_-y_++2x_d\cdot y_d.$$ Its ...
Arnold Neumaier's user avatar
4 votes
0 answers
80 views

Pseudometrics on world lines

Consider the space $W$ of smooth time-like curves in $\mathbb{R}^{n,1}$ with fixed ends. Given $\gamma\in W$, consider the space $T_\gamma$ of all smooth normal fields along $\gamma$; one may think ...
aglearner's user avatar
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4 votes
0 answers
239 views

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 \...
Paul's user avatar
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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 ...
Urs Schreiber's user avatar
3 votes
2 answers
403 views

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: &...
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
2 answers
353 views

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 (...
user2022's user avatar
3 votes
1 answer
176 views

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 ...
Ali's user avatar
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3 votes
1 answer
270 views

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 ...
JS.'s user avatar
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3 votes
1 answer
248 views

Reference for Bonnet Fundamental theorem of surfaces in Lorentzian spaces

I am looking for a reference for the following folklore theorem, which is the Lorentzian analogue of the Bonnet fundamental theorem of surface in Euclidean space, hyperbolic space or $3$d sphere. ...
François Fillastre's user avatar
3 votes
1 answer
168 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 ...
Paul's user avatar
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3 votes
1 answer
228 views

Minimize the area of a maximal surface in R^{2,1} with boundary on the unit sphere

Let $\mathrm{H}$ be the unit sphere in the Minkowski space $\mathbb{R}^{2,1}$ (i.e., a one-sheeted hyperboloid $x_1^2+x_2^2=x_3^2+1$). Assume that $\gamma\subset \mathrm{H}$ is a closed space-like ...
Dmitry Chelkak'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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3 votes
1 answer
140 views

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 ...
Sandesh Jr's user avatar
3 votes
0 answers
186 views

Decomposition of forms on manifolds

Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold, i.e. $M=I\times\Sigma$ with $I\subset\mathbb{R}$ being an open interval and $\Sigma$ a spacelike smooth Cauchy hypersurface. The metric is of ...
G. Blaickner's user avatar
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3 votes
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Is every finite metric space representable in a pseudo-Euclidean space?

Let $X$ be a finite set with a (true) metric $d$ and $|X| = n$. Does there exist a set $Y$ of $n$ points in $R^n$ with a pseudo-Riemannian metric with signature $(n - k, k, 0)$ for some integer $k$ ...
Steve Riley'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
3 votes
0 answers
59 views

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 ...
Ali's user avatar
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3 votes
0 answers
118 views

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 ...
Ali's user avatar
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3 votes
0 answers
98 views

Conformal Transformations that are Ricci Positive Invariant

Is there any known class of conformal transformations $\phi : M \to M$ of a riemannian/semi-riemanian manifold $(M,g)$ that have the property: $g$ is ricci-positive iff $\phi^* g$ ricci positive? ...
Zakk's user avatar
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2 votes
2 answers
152 views

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(...
Ali's user avatar
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