Questions tagged [semi-riemannian-geometry]

It is the study of smooth manifolds equipped with a non-degenerate metric tensor, not necessarily positive-definite (and hence a generalisation of [riemannian-geometry]). Included in this are metric tensors with index 1, called "Lorentzian", which are used to model spacetimes in (general-relativity).

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Do these generalization of riemannian/semi-riemannian geometry have names?

In Riemannian geometry we require that the metric $g$ be a positive definite real matrix and in semi-Riemannian geometry we relax this requirement and instead ask that the metric be a non degenerate ...
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Integration over pseudo-Riemannian manifolds

Given a pseudo-Riemannian manifold $M$, let say for example Minkowski spacetime in $s>1$ dimensions, often we talk about integration on it. Namely the connection/metric definition gives us a ...
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Hawking and Ellis's example of an extendible manifold

In S. Hawking's and G. Ellis's book "The Large-Scale Structure of Space Time", they discuss the notion of inextendible Lorentz manifolds $(M,g)$ (see chapter 3.1). A Lorentz manifold is ...
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Convex surface in a manifold with constant curvature! [closed]

Let $S$ be a convex surface in a Riemannian manifold $M$ of constant curvature $k$. How to prove that $S$ is of sectional curvature $>k$? The lorentzian analogue: (Let $S$ be a convex surface in ...
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self intersection of two Lorentzian manifolds also Lorentzian in this case?

Consider a manifold $\zeta:=\{\varphi_S\}\cap\{\varphi_T\}$ as the non-empty transversal intersection between two Lorentzian manifolds, s.t. all three manifolds have dimension $D=3+1.$ Is $\zeta$ also ...
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Well-definedness of ADM mass, energy, linear and angular momentum

In asymptotically flat initial data sets, one can define the ADM mass, energy, linear and angular momenta via spatial limits of integrals over 2-spheres. My questions are: Is asymptotical flatness ...
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Upper and lower indices when identifying tangent space vectors with underlying vector space

Given an $n$-dimensional manifold $M$, we can construct a basis for each tangent space $T_pM$ from a local coordinate basis $\{x^i\} \subset \mathbb{R}^n$ as $$\vec{e}_i = \left( \frac{\partial}{\...
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1answer
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What kind of curvature?

In many papers of semi-Riemannian geometry, when they talk about curvature (constant or bounded) they don't precice which kind of curvature they talk about. I know the definition of: -Sectional ...
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Something like the trace of a tensor

I'm going through some Lorentzian geometry and had some questions about different ways of getting a scalar out of a $(0, 2)$-tensor. Let us denote it by $T_{ab}$. Suppose we're on a pseudo-Riemannian ...
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1answer
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Are Lorentz-tensors tensors?

It might sound like a stubid question, but I really have some problems in unterstanding it. When defining tensor fields on a given manifold $\mathcal{M}$, one find the following transformation rule: $$...
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Differential equations on manifolds

I am currently trying to understand a paper on wave maps on Lorentzian manifolds. The setup is the following: Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a ...
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Persistence of metric components and null geometry

The following setup is from Rendall's 1990 paper regarding the characteristic initial value problem for Einstein's equations, and also from material on null geometry. Given a 2-dimensional spacelike ...
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Hypersurface-orthogonal vector field and helicity

In a (3+1)-dimensional Lorentzian manifold equipped with a metric $g_{ab}$ (context: general relativity), I define a vectof field $k^a$ to be a Helical Killing Vector (HKV) if i. it is a Killing ...
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Wave maps, second energy inequality

Sorry in advance for the long setup: Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional ...
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How does calculus work on a Pseudo Riemannian manifold?

I'll first recapitulate the why calculus works for Riemannian manifolds, and then present why I believe the same does not work for Pseudo-Riemannian manifolds. I'd like to know where my current model ...
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How does topology work when taking charts on a Psuedo-Riemannian manifold?

I'll first explain why I think taking charts is sane when working with Riemannian manifolds, and then show what I believe breaks down in the Pseudo-Riemannian case with a particular choice of a Pseudo ...
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1answer
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What are Minkowski space and Lorentzian manifolds, formally speaking?

I am in general confused about what Minkowski space is. I'll write down what I know and what I believe Minkowski space is. I'd appreciate any corrections. A Riemannian manifold is a manifold (so it ...
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Intuition for curved Lorentzian manifolds through Euclidean embedding

How can one find embeddings of parts of curved Lorentzian manifolds into low dimensions ($\leq 3$) for visualization and intuition building? For example, one can embed the space $H_2$ into $\mathbb R^{...
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1answer
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Definition of Wave map on manifolds

Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. The wave equation is defined as $g. \nabla^2 u$. As far as I see, $\nabla^2 u \in \Gamma(T^{*}V \...
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Hypersurface in a Perturbation of Schwarzchild Spacetime

I have a fairly simple question about perturbations of Schwarzchild spacetime in general relativity but cannot seem to find the answer anywhere. Start with the standard Schwarzchild metric $g^{SCH}= -\...
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1answer
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second fundamental form and the mean curvature of the pseudo-sphere

I am trying to practice computing the second fundamental form and the mean curvature, and I am trying to compute them for the Pseudo-Sphere in $n+1$ Minkowski spacetime. Pseudo-Sphere in $n+1$ ...
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1answer
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Pullback bundle, connections

First of all, I know that my question is quite long. I still would really appreciate any help because I am kind of stuck with this topic, I really try to express my question as clear as possible! Also,...
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Conjugate points: circular orbits in Schwarzschild spacetime

Let $(\mathcal{M},\mathbf{g})$ be the globally hyperbolic Lorentzian manifold corresponding to the exterior region of the Schwarzschild solution, i.e. $\mathcal{M}=\mathbb{R}\times[\mathbb{R}^3\...
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Symmetry of a Perturbed Metric

If a metric has a particular symmetry, will a perturbation of that metric enjoy the same symmetry? By perturbation of a metric, I mean $g_{\mu \nu}^{perturbed} = g_{\mu \nu} + h_{\mu \nu},$ where $...
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Wave map on manifolds

Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional orientable smooth manifold. Now in a paper ...
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Show that the causal future is a subset of the closure of the chronological future

Let $(\mathcal{M},g)$ be a time-oriented Lorentzian manifold and take $p\in\mathcal{M}$. How can I show that the causal future of $p$ with respect to $\mathcal{M}$, $J^+(p,\mathcal{M})$, is a subset ...
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1answer
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Existence and Uniqueness of a Volume Element on an Oriented, Pseudo Inner Product Space?

The definition of orientation on a vector space I'm using is as an equivalence class of ordered bases, where two bases are related if and only if the determinant of the change of basis is positive. ...
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Laplace–Beltrami operator and Ricci curvature

I am taking a course on General Relativity which is more mathematically oriented than physically, and in one of the lecture notes (discussing Riemann tensor, curvature and all that) this identity is ...
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Curvature tensors and local isometries (Homework)

I am asked first to to prove that an isometry ($\Phi$) preserves the Levi-Civita connection, this is: $$\Phi_*(\nabla_XY)=\nabla^*_{\Phi_*X}\Phi_*Y$$ where $\nabla^*$ refers to the Levi-Civita ...
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1answer
20 views

Change in coordinates for a metric in a given form

With a change in coordinates, transform \begin{align} ds^2 = -z^2dt^2 + dz^2 \end{align} to \begin{align} ds^2 = -dT^2 + dX^2. \end{align} My attempt. It is clear that incoming null geodesics $\...
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1answer
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Doubt on paracompactness and lorentz signature.

While the study of Manifolds are quite interresting by itself, we know that General Relativity have on it's core the study of Manifolds. When we transport the Manifold mathematical structure to ...
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1answer
34 views

$4$-dimensional Pseudo-Riemannian manifolds with $R_{ij}=\frac 14 R g_{ij}$

Consider a $4$-dimensional Pseudo-Riemannian manifold where $R_{ij}=\frac 14 Rg_{ij}$. This would mean that $G_{ij}=-R_{ij}$, satisfying the known relation $G=g^{ij}G_{ij}=-g^{ij}R_{ij}=-R$ and the ...
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1answer
60 views

Write down what metric this transformation preserves based on animation

I made this on desmos: https://www.desmos.com/calculator/u5qpd135uc I made it because I wanted to compare and contrast it with the Lorentz boost. The transformation should move a point to ...
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1answer
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Differential of $(x,v)\mapsto (x,\exp_x(v))$

Let $M$ be a pseudo-Riemannian manifold. Let $\Omega=\{(x,v)\in TM: |v|<\epsilon\}$. I want to compute the differential of the map \begin{align*} \Omega &\to TM\\ (x,v) &\mapsto (x,\exp_xv)...
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Plotting tessarines in (the pseudo-Euclidean space) $R^{2,2}$; conventions around which axes correspond to which signs in the metric signature

Tessarines are a product of sorts of the complex numbers and the split-complex numbers. If you substitute split-complex number coefficients for real coefficients in the form of complex numbers, or ...
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1answer
26 views

Initial value problem with null surface data

Let $(M,g)$ be a $d$-dimensional Lorentzian manifold, i.e., the metric tensor $g$ has signature $(-1,+1,\dots,+1)$ where $-1$ appears only once and $+1$ appears $(d-1)$ times. A null hypersurface $\...
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Lifting class structures from a lorentzian manifold $\Bbb R^{1,1}$ to $\Bbb R^{1,2}$

Note: $\zeta^{1,1}=\Bbb M^{1,1}$ (under a $\log-\log$ change of coordinates). In other words, $\zeta^{1,1}$ with coordinates $(x,\phi)\in \zeta^{1,1}$ equals $\Bbb M^{1,1}$ with coordinates $(u,v)\in ...
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Branched manifolds in the context of general relativity

Does anyone study Branched manifolds in the context of General Relativity? I would like to define the union of two transversal spacetimes $G:=M \cup S.$ And I would also like to define a ...
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How does this Lie algebra react to this change of coordinates?

Lorentz transformations that preserve the direction of time are called "orthochronous." [Lorentz group] I want to reformulate the orthochronous transformations in $\Bbb R^{1,1}:=$ (2-dim. Minkowski ...
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1answer
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Mathematical Relativity - Topics [closed]

Doing a course on Mathematical Relativity (never had a true Relativity course before) this semester, where I have to do a project on a topic not in the programme. I already did courses on Riemannian ...
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2answers
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Geometry of transformed spacetimes?

The main question seeks to understand whether a conformal structure can be put on "transformed" Minkowski 2-space, which will be denoted as $\Bbb R^{1,1}:=\Bbb M^{1,1}.$ I will get more ...
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Do Tarski's axioms describe Lorentzian geometry if we discard the Identity of congruence axiom?

Tarski's geometry axioms describe Euclidean geometry in terms of the betweenness and congruence relations. One of the axioms states that $xy\equiv zz \rightarrow x=y$ interpreted as saying that a line ...
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1answer
67 views

Is the Levi-Civita connection on a Lorentzian manifold the same as that on a Riemannian manifold?

The Levi-Civita connection is defined by the Koszul formula to be independent of the metric. This applies to both Riemannian and pseudo-Riemannian metrics. In the latter, this is known as the miracle ...
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1answer
108 views

Exact meaning of conformally flat manifold

I would like to understand the loose statement "Every 2d manifold is conformally flat" in the context of Pseudo-Riemannian manifolds. It seems to me that there are two slightly different versions,...
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Definition of space/time orientation-preserving/reversing in the orthogonal group $O(p,q)$

It is known that elements of the pseudo-orthogonal group $O(p,q)$ can independently be space orientation-preserving/reversing and time orientation-preserving/reversing. (By convention, I call $v\in \...
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Tangent space of complex positive semidefinite cone at $E_{11}$

Take the cone of all Hermitian positive semidefinite matrices. I am interested in the tangent spaces of the boundary. Especially, what is the tangent space at the point $E_{11}$, the matrix with a $1$ ...
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Geodesic completeness and regularity of curvature invariants

Does geodesic completeness imply the absence of any divergent scalar quantity (curvature invariants and derivatives thereof) in a Lorentzian manifold? If not, is there any additional condition (for ...
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1answer
79 views

Prove that the max number of Killing fields is $N(N+1)/2$

I'm trying to prove that the maximal number of linearly independent Killing vector fields on a Riemannian manifold $(M,g)$ of dimension $N$ is $N(N+1)/2$. Here's what I got so far. By definition a ...
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Twisted $AdS_2 \times S^2$: Want a Coordinate Transformation that Makes Metric Diagonal.

I already asked this question here at mathoverflow but now that it seems to be clearer that the definition of a twisted product given below is indeed the one meant in the overview paper, the questions ...
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Affine subspaces are geodesically complete and totally geodesic

in my lecture it says that for $V \subset \mathbb{R}^n$ subspace, the submanifolds $p+V$ are totally geodesic (the second fundamental form is zero) I know that $\mathbb{R}^n$ is geodesically ...

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