# 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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### 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 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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### 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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### 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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### 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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### 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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### 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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### Continuous dependence on initial data for (null) geodesics in Lorentzian manifolds

I'm looking for a reference for the following theorem: Theorem Let $\mathcal M$ be a smooth manifold with smooth Lorentzian metric $g$. Let $p$ and $q$ be two points on the same (causal) geodesic, ...
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### 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-...
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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 ...
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### Tensor Field Decomposition in Space time

For vector fields in $R^3$ one knows that there exists a unique decomposition of vector fields in to solenoidal (divergence free) and potential parts. A generalization of this Theorem for symmetric ...
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### 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? ...