# 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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### 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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### 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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### 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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### $4$-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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### 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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### 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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### 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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### 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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### 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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### 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 ...