# 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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### Examples of incomplete Lorentz 3-manifolds

Reading this paper where closed 3-dim. Lorentz manifolds with noncompact isometry groups are studied, I wonder if all of them are geodesically complete.
One class of 3-dim. closed Lorentz manifolds ...

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### Geodesic connectedness in static Lorentz manifold vs connectedness by trajectories with potential in Riemann manifold

What is the relationship between the study of geodesic connectedness in a standard static Lorentz manifold and the connectedness of two points by trajectories with potential (i.e. solutions to $x''(t) ...

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### Is a null hypersurface locally achronal?

Let (M,g) be a Lorentzian manifold. A subset A of M is achronal if it does not exist any timelike curve joining two points in A.
A subset A is locally achronal if for any point in A there is an open ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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?
...

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### Induced connection on null hypersurfaces

I will use a local coordinate formalism here, since this is related to research in general relativity, and my supervisor only tolerates local coordinate formalisms. Plus the research papers I base my ...

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### 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 $...

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### 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.
...

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### Smoothness of the twistor space of a lorentzian manifold, or “convexity wrt null geodesics”

Null lines in Minkowski space form a 5-dimensional manifold, represented as a (real) quadric $\mathbf{PN}\subset\mathbb{C}\mathbf{P}^3$. This is a well-known fact, on which R. Penrose’s twistor ...

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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)...

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### Number of connected components of the isometry group of a simply-connected lorentzian manifold

Let $(M,g)$ be a finite-dimensional connected lorentzian manifold. Then the group $G$ of isometries of $M$ (i.e., the group of diffeomorphisms $\varphi : M \to M$ with $\varphi^* g = g$) is a Lie ...