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 ...
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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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 $\Sigma\setminus(\...
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$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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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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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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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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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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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
$$ ...
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Transformation Poincaré-coordinates to global coordinates in $\mathrm{AdS}_2$
For the two dimensional anti de-sitter space $\mathrm{AdS}_2$ one can consider the Poincaré-coordinates $\mathrm{d}s^2_P = -r^2 \mathrm{d}t^2 + \frac{1}{r^2} \mathrm{d}r^2$ which covers only half of ...
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Canonical Metrics on 3- and 4-Manifolds
From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...
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Metric with a constant Chern–Pontryagin scalar
Do there exist 3+1 dimensional spacetimes (i.e. Lorentzian manifolds with signaure (1,3)), for which the Chern–Pontryagin scalar
\begin{equation}
K_2= \epsilon^{\mu\nu\rho\sigma}R^{\alpha}{}_{\beta\mu\...
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A pseudo-Riemannian version of a theorem by Fubini
Guido Fubini, ``Sugli spazii che ammettono un gruppo continuo di movimenti,'' Annali di Mat., ser. 3, 8 (1903) 54.: Let $M$ be a Riemannian manifold of dimension $d\ge 3$. Its isometry group cannot be ...
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Automorphism group of a Lorentzian lattice
Consider the even integral lattices $L_n:=Z\times Z\times Z^{n-2}$ (where $Z$ is the set of integers) with elements $x=(x_+,x_-,x_d)$ and inner product
$$(x,y):=x_+y_-+x_-y_++2x_d\cdot y_d.$$
Its ...
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Pseudometrics on world lines
Consider the space $W$ of smooth time-like curves in $\mathbb{R}^{n,1}$ with fixed ends.
Given $\gamma\in W$, consider the space $T_\gamma$ of all smooth normal fields along $\gamma$;
one may think ...
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Harmonic maps into de Sitter Space
I am looking some references on the existence and non-existence of (space-like) harmonic maps solving the Dirichlet into the de-Sitter space.
More precisely: Let, for $n\geq 3$,
$$dS^n=\{ u\in \...
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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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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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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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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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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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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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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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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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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 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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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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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 $k$ ...
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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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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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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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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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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(...