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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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 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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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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Comparing Honda's construction to mine
Consider a finite family $\Psi_\alpha$ where $\alpha=1,2,3,4$ of mutually transversely intersecting mixed type surfaces of $\mathbf L^3$ (Lorentz-Minkowski $3$-space) inscribed in $[-1,1]^3.$ Let the $...
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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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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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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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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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