# 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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### Question on Cauchy problem on manifolds

Let $(M,g)$ be a globally-hyperbolic manifold, i.e. $M=\mathbb{R}\times\Sigma$ and $g=-\beta^2 \, dt^2+h_{t}$. Furthermore, let $E$ be a vector bundle over $M$ and $\square$ a normally-hyperbolic ...
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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 ...
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1 vote
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### How causal is a strongly causal purely electric spacetime?

Take a generic Lorentzian spacetime $(M, g)$ where $M$ is a time-oriented 4d manifold and $g$ is the Lorentzian metric that is strongly causal and purely electric. According to this answer: Is every ...
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### Is a Wick rotatable spacetime necessarily strongly causal?

There are a few viable ways to formulate Wick rotatability that preserve distinct features. One is mentioned in the post: Obtain Lorentzian manifolds from Riemannian ones by Wick rotation There's also ...
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### 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 ...
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### 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 ...
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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 $k$ ...
1 vote
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### Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?

Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that $$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$ where $\Sigma$...
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### Distinguishable under manifold topology but indistinguishable under the Alexandrov topology

Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal. In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
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### 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 ...
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### What is the nerve of this category?

If $\mathcal{C}$ is a thin category, we call $U \subseteq \mathrm{Ob}(\mathcal{C})$ open if for every object $X \in U$ and any morphism $X \to Y$, we also have $Y \in U$. This declares an Alexandrov ...
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1 vote
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### Temporal evolution of a globally hyperbolic spacetime

Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal). For ...
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### 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 ...
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### Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds

$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
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### Lorentzian geometry. Comparing Honda's main theorem A construction to mine: Mixed type surfaces

This question is based on a wonderful paper by A. Honda (link below) where his main theorem A provides an incredible uniqueness result. Mixed type surfaces and type changing metrics have been ...
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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 K_2= \epsilon^{\mu\nu\rho\sigma}R^{\alpha}{}_{\beta\mu\...
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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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1 vote
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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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1 vote
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### Transporting a Cauchy foliation of Minkowski space

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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### 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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### 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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### 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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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-...