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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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spacelike curves, in lorentzian geometry?

i have this question : Let $(M,g)$ be a lorentzian manifold, and $\gamma:[0,1]\rightarrow M$ be a spacelike curve in $M$, between two different point $A$ and $B$, so : can: $\underset{\gamma}{inf}\...
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The projection from the time-orientable double cover preserves topological properties.

In Relativity and Singularities, Natário states that A connected time-orientable Lorentzian manifold admits a nonvanishing vector field, and hence is either noncompact or has zero Euler ...
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Invariance of signature in semi-riemannian manifolds

The signature of a non-degenerate symmetric bilinear map $h:V\times V\to \mathbb{R}$ in a vector space $V$ is the number of negative numbers on the diagonal of the matrix $h_{ij} = h(e_i,e_j)$ when it'...
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Vacuum solutions of Einstein's field equations with $O(n)$ isometry group

In Relativity and Singularities, Natário states that Any vacuum solution admitting $O(n)$ as an isometry group is locally isometric to $M=\mathbb{R}^2\times S^{n-1}$ (with the Schwarzchild metric). ...
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Equivalent ways to refer to Lorentzian geometry

I'm a curious student about the Yamabe-type problem in Lorentzian geometry. I found a single article about this topic (2015). I don't know much about Lorentzian geometry and I wonder if one may ...
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The Fundamental Theorem of Geometric Calculus in a lorentzian manifold

I am trying to understand geometric calculus and apply it to physics. In this sense, I was reading Alan Macdonald's book "Vector and Geometric Calculus", and stumbled upon the Fundamental Theorem of ...
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Constant sectional curvature and unit normal vector to a hypersurface totally geodesic

I was reading about hypersurfaces totally geodesic when I found the next proposition: The sectional curvature $K$ of $M$ is constant at $p$ if and only if every unit vector in $T_{p}(M)$ is normal to ...
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For $0<\nu<n$ there are no compact semi-Riemannian hypersurfaces in $\mathbb{R}_{\nu}^{n}$

I'm reading a proof of the following proposition: For $0<\nu<n$ there are no compact semi-Riemannian hypersurfaces in $\mathbb{R}_{\nu}^{n}.$ The proof is as follows: Suppose that there is $M\...
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Intuition for warped product manifold.

I am reading about wave equations in manifold and encountered the term warped product manifold. More specifically, in my case it is defined as follows, $$N:=[0,\phi^*) \times_g \mathbb S^{k-1}$$ ...
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Connected semi-riemannian manifold is an Einstein manifold.

I'm reading the proof of the next proposition: A semi-riemannian manifold $M$ is an Einstein manifold provided $Ric=cg$ for some constant $c.$ If $M$ is connected, $n=dim(M)\geq 3$ and $Ricfg=fg,$ ...
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Showing that a regular curve is a pregeodesic

I'm trying to prove the next: To show that a regular curve $\alpha$ with $\alpha^{'}$ and $\alpha^{''}$ collinear is a pregeodesic, write $\alpha^{''}(s)=f(s)\alpha^{'}(s)$ and prove that a) $\beta=\...
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Show that Lorentzian metric restricts to Riemannian metric on hyperbolic space

Define the Lorentzian metric $\langle \ ,\ \rangle$ on $\mathbb{R}^{n+1}$ by $\langle x, x \rangle = - x_0^2 + x_1^2 + \cdots + x_n^2$. Let $$\mathbb{H}^n = \{ x \in \mathbb{R}^{n+1} | \langle x , ...
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Lorentz transformations in matrix form (Matrix Multiplication problem)

I have recently been introduced to Lorentz transformations in the form of 4-Vectors, I have been told the following: $$\Lambda^T\eta\Lambda=\eta,$$ Where $\eta$ is the matrix: \begin{bmatrix} 1 ...
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Definition of a metric tensor in Barrett O'Neil's book

I am studying Barrett O Neils's book on semi riemannian geometry with applications to relativity. In chapter 3 he states the definition of a metric tensor as follows: "A metric tensor g on a smooth ...
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How to solve a second ODE involving the Lorentzian function, possibly using analytical methods?

I'm trying to solve a second ODE involving the Lorentzian function, wherein I'm trying to solve for r(z). To see the form of the Lorentzian function (which I modified slightly for my research) and its ...
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The equivalence of two definitions of Cauchy surfaces.

I see two definitions of a Cauchy surfaces $\Sigma$ of a spacetime $M$. a closed achronal subset such that the domain of dependence is $M$ ...
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81 views

Points fixed by the isotropy group of the restricted Poincaré group in Minkowski space

What points does the isotropy group $SO(3)$ fix in the Minkowskian case? For instance, in oriented Euclidean 3-space the isotropy group $SO(3)$ of the Euclidean group $E(3)$ fixes the origin. What is ...
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Making sense of $2{\rm grad}(\langle H,H\rangle) + 4 {\rm trace}(A_{DH}) = 0$

I'm reading the paper Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its applications to biharmonic surfaces by B. Y. Chen. Then we have Lemma 2.1. Let $L\...
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Is there a coordinate free proof for this result?

Let $(M,g)$ be a smooth manifold with metric tensor $g$. For this problem, the signature of $g$ seems to be of no importance. Consider a vector bundle $\pi : E\to M$ and a smooth embedding $\phi : N\...
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Does a coordinate chart have a smooth orthonormal frame w.r.t. a semi-Riemannian metric?

If the metric is definite, then the answer is yes: the coordinate basis vectors form a smooth local frame, and each basis vector has non-zero length by definiteness, and one can run the Gram-Schmidt ...
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Can the Gram-Schmidt algorithm be extended to construct a basis for $T_p M$ of a semi-Riemannian manifold?

Prof. Lee in p. 30 of his "Riemannian Manifolds" says: "Given a pseudo-Riemannian metric $g$ and a point $p \in M$, by a simple extension of the Gram-Schmidt algorithm one can construct a basis $(E_1,...
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Do pp-wave spacetimes have a well-defined signature/index?

A pp-wave spacetime in Brinkmann coordinates has metric $$ ds^2 = H(u,x,y) \, du^2 + 2 \, du \, dv + dx^2 + dy^2 $$ and is asserted to be a Lorentzian manifold, i.e. has index 1. This is indeed true ...
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Is the dynamics of spacetime coordinate-dependent?

Consider a spacetime $(M,g)$ which admits a chart $(U, \varphi)$ with $\varphi^{-1}(p)=: x\in U$ for $p\in M$ and $\varphi^*g \equiv g_{\mu\nu}dx^{(\mu} \otimes dx^{\nu)}$ such that the spatial ...
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Variation of a differential form

Physicists sometimes get the lagrangian $$\mathcal{L}=-\frac{1}{2}\mbox dA \wedge \star \mbox dA - A \wedge \star J$$ define a functional given by $$S(A)=\int_{N_4} \mathcal{L}= \int_{N_4}-\frac{1}{2}...
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Inner product on pseudoriemannian manifold

In this topic: Scalar product on manifold. Henry said $(\omega, \eta)=\int_M \omega \wedge \star \eta=\int_M \left<\omega, \eta \right>\mathrm{d vol}$ is an inner product on forms, where $\left&...
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Chart for surface in $\Bbb L^4$ with positive relative nullity

I am reading the paper Marginally trapped surfaces in Lorentzian space forms with positive relative nullity by Chen and Van der Veken. The setup is roughly the following: we have that $M^2 \subseteq \...
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189 views

Uniqueness of connection $1$-forms and curvature $2$-forms

If $(M,\langle\cdot,\cdot\rangle)$ is a pseudo-Riemannian manifold and $\nabla$ denotes the Levi-Civita connection of the metric, we can define the connection $1$-forms and curvature $2$-forms ...
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231 views

$U$ is timelike if its orthogonal complement $U^\perp$ is spacelike

Consider the bilinear form $\left<x,y\right>_{n,1} = \sum_{j=1}^n x_j y_j - x_{n+1} y_{n+1}$ on $\mathbb{R}^{n+1}$. A vector $x \in \mathbb{R}^{n+1}$ is said to be timelike if $\left<x,x\...
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Group of positive Lorentz matrices acts transitively on hyperbolic space

We call a vector $x \in \mathbb{R}^{n+1}$ positive if $x_{n+1} > 0$. $A \in M_{n,n}(\mathbb{R})$ is called a Lorentz matrix if $J = A J A^t$ where \begin{align} J = \begin{pmatrix} 1 & & &...
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Is de Sitter space $ dS_4 = SO(1,4)/SO(1,3)$?

A $D$-dimensional de Sitter spacetime $dS_D$ is a timelike hyperbola embedded in $(D+1)$-dimensional Minkowski spacetime $\mathfrak M_{D+1}$. The metric in the embedding space is $$ ds^2 = (dX_0)^2 -...
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Existence of parametrization with dense image for an invariant surface in $\Bbb R^4_1$.

I am trying to read the paper Boost invariant marginally trapped surfaces in Minkowski $4$-space, by Haesen and Ortega. Basically, they work with metric signature $(-+++)$ in $\Bbb R^4_1$, consider ...
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Is there a natural choice of cone in Minkowski space as a real affine space?

Minkowski space is a homogeneous affine real space and under this translationally invariant perspective that doesn't have privileged points it seems easier to consider its one-sheet 3-hyperboloid ...
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What does the metric $(dt^2-dx^2)/cosh^2(x)$ describe? [closed]

I've got the metric, but I wonder what space does it describe. I think it's the hyperbolic plane, in some funny coordinates system, but I couldn't prove it. Am I right?
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Integration on manifold using the flow of a vector field

In a Physics paper the author states without proof something that seemed quite strange to me. The paper is on General Relativity, so he assumes a Lorentzian manifold $(M,g)$ is given. His hypothesis ...
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Mathematical properties of Minkowski spacetime(3+1) and its isometry group

I would like to clarify if the Minkowski subspace of points inside the null hypersurface (light-cones) at each point's tangent space is just a homogeneous subspace(transitive under G) for the Poincare ...
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74 views

Real Clifford algebra for $SO(1,2)$

Is it possible to find 3 2x2 matrices $m_i$ with real coefficients such that their anticommutator $\{m_i,m_j\}=2 \eta_{ij}*1_2$, where $\eta = diag(1,-1,-1)$ and $1_2$ is the $2\times2$ identity ...
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How does the hodge codifferential operator act over a wedge product?

We have that the exterior derivative acts over a wedge product in the following manner. Let $\alpha,\beta$ be $p,q$ forms respectively. Then we have that \begin{equation} d(\alpha\wedge \beta) = (d\...
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Lorentzian scalar product

Let $X$ be a future-directed vector and $Y$ a past-directed one in a time-oriented space-time (manifold). We want to compute $g(X,Y)$. I choose a coordinate in which $X=X^0\partial_0$ with $X^0>0$ ...
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Why these tensor fields do not depend on the hypersurfaces chosen to define them?

In the paper Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum, W.G. Dixon proposes definitions for momentum and angular momentum of a certain distribution of matter ...
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Diagonalization of $2\times 2$ Lorentz-symmetric matrix

$\newcommand\pair[1]{\left\langle #1 \right\rangle}$ Consider Minkowski plane $\Bbb L^2 = (\Bbb R^2, \pair{\cdot,\cdot}_L )$, where$\renewcommand\vec[1]{{\boldsymbol #1}}$ $$\pair{\vec{u},\vec{v}}_L\...
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Relation of commutators $[A_{\xi},A_{\eta}]$ with the normal bundle

Let $(\overline{M}^{n+m}, \newcommand\pair[1]{\left\langle #1 \right\rangle} \pair{\cdot,\cdot})$ be a pseudo-Riemannian manifold and $M^n \subseteq \overline{M}$ a non-degenerate submanifold. We have ...
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Computation of exterior derivative of an $(n-1)$ form

Suppose $(M,g)$ is a Lorentzian manifold of dimension $n$. Let $V$ be a one-form on $M$ and define the $(n-1)$ form $\omega = \ast V$ where $\ast$ is the Hodge dual. In a chart $(U,x)$, if we have $$...
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94 views

The one-parameter curves generated by Lie dragging a geodesic by a Killing field are geodesics

Let $(M,g)$ be a manifold with Riemannian or Lorentzian metric $g$ (the signature doesn't matter for our purposes). Consider $\gamma : (a,b)\to M$ a geodesic and $\phi^\xi_s : M\to M$ the flow of a ...
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113 views

Estimating the product of timelike vectors in $\Bbb R^n_\nu$

Consider pseudo-Euclidean space $\Bbb R^n_\nu$ with $\nu$ negatives first $(-,\ldots,-,+,\ldots,+)$. Split $\Bbb R^n_\nu = \Bbb R^\nu_\nu \times \Bbb R^{n-\nu}$. $\renewcommand\vec[1]{{\bf #1}}$Assume ...
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What are examples of the use of countable choice axiom and DC in (pseudo)riemannian geometry results

What are some specific results in (pseudo)riemannian geometry that require either the axiom of countable choice or the axiom of (countable) dependent choice (DC)? For instance, does the definition of ...
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Is the covariant Hessian preserved by isometries?

Let $(M_1,g_1,\nabla^1)$ and $(M_2, g_2, \nabla^2)$ be pseudo-Riemannian manifolds equipped with their Levi-Civita connections and $F\colon M_1 \to M_2$ be an isometry. $\renewcommand\vec[1]{{\bf #1}}...
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406 views

How to rigorously differentiate the square of the geodesic distance?

Let $(M,g)$ be a Lorentzian 4-dimensional manifold, in other words, $g$ has signature $(-,+,+,+)$. If $U$ is a geodesically convex set, one defines $\sigma : U\times U\to \mathbb{R}$ in the following ...
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191 views

Lorentz Transformations Vs Coordinate Transformations

I'm really confused about Lorentz transformations at the moment. In most books on QFT, Special Relativity or Electrodynamics, people talk about Lorentz transformations as some kind of special ...
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40 views

Motivation for defining space-like vector

Maybe an elementary question: What is the motivation for defining the vector $X=0$ to be space-like?
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244 views

Riemannian vs. semi-Riemannian manifolds

Some results on Riemannian manifolds are valid on semi-Riemannian manifolds and the others are not. Sometimes it takes too long to recognize between them. My question is: Is there a reference ...