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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When parallel transport isn't an isomorphism
In my research I have to deal with "connections" whose parallel transport isn't an isomorphism. Instead, the parallel transport along a curve is just a linear endomorphism that changes ...
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A global view-point on pseudo-Riemannian manifolds
A pseudo-Riemannian manifold $(M,g)$ is a generalisation of the concept of a Riemannian manifold where we relax positive-definiteness to non-degeneracy.
$\alpha$) Non-degeneracy is still enough to ...
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Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstruct a metric on $M$ from this data in a geometric way?
Let $(M,g)$ be a (semi-)Riemannian manifold, and let $(M,[g])$ be the conformal class thereof. The following two sets are naturally torsors for the collection of positive-valued functions on $M$:
The ...
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"Classifying" causally closed sets in Minkowski space
Let $M = \mathbb R^{D+1}$ be Minkowski space. Recall that the causal complement of a set $A \subseteq M$ is the set $A^\perp \subseteq M$ where $p \in A^\perp$ there is no timelike path between $p$ ...
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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 ...
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Constructing spacetimes
Consider a spacetime $(\zeta^{3,1},g)$
where $$g=\frac{du\,dv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,r,w \in (0,1)$$ Now this is just Minkowski space in different coordinates (...
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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(\...
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Decomposition of tensor field on hypersurface
Let $(\mathcal{M},g)$ be a Lorentzian manifold, which is globally of the form $\mathcal{M}\cong I\times\Sigma$, where $I\subset\mathbb{R}$ ("time") and $\Sigma$ ("space") is some $...
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Further directions in representations of surface group into a Lie group
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$.
Now I am planning to ...
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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 $h:=i^{\...
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$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 ...
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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 ...
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A question on future Cauchy developement
Let us consider the Minkowski spacetime $\mathbb R^{1+2}$ equipped with the metric
$$ \eta(t,x) = -(dt)^2+ (dx^1)^2+(dx^2)^2.$$
Let $\Omega$ be a bounded simply connected domain in $\mathbb R^2$ with ...
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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 ...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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