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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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Local cartesian coordinates on Riemannian manifold

I'm wondering is possible for every given metric $g=g_{ij}dx^i \otimes dx^j$ on $M$ and for every given $p\in M$ to find such chart $(U, \varphi)$ around $p\in U \subset M$ that the metric $g|_U$ in ...
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Isometry group of quadric model of anti-de Sitter space

I am learning Lorentzian geometry on my own. Consider the space $\mathbb R^{p+2}$ endowed with the bilinear form $$\langle x, y \rangle_{p,2} = \sum_{i = 1}^{p+2} x_iy_i -x_{p+1}y_{p+1}-x_{p+2}y_{p+2}$...
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Adjoint of Cartan's magic formula

It is well-known that if $X$ is a vector field and $\omega$ is a form, then we have Cartan's "magic" formula $$ L_X \omega = d\iota_X \omega + \iota_X d\omega. $$ Assuming that we are on a (...
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Show that the conformally complete Schwarzschild spacetime is asymptotically flat at null infinity

I am trying to show that the conformal factor used to conformally complete the Schwarzschild spacetime renders it asymptotically flat at null infinity (according to the mathematical definition given ...
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Sign error in the computations of $R_{\theta\phi\theta\phi}$ for the Schwarzschild metric

I am computing the components of the Riemann tensor for the Schwarzschild metric using the following formula $R_{\alpha\beta\mu\nu}$=$(\partial_\alpha\Gamma^l_{\beta\mu}-\partial_\beta\Gamma^l_{\...
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Displacement map $\Delta: C\times C\to TM$ has open image in semi-Riemannian manifold

Hi I’m following o’neill’s book and have the following question regarding the displacement function. Let $M$ be a semi-Riemannian manifold and $C\subset M$ a convex open set (so it is a normal ...
user1325753's user avatar
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Diagonal metric with induced metric being diagonal as well

A nondegenerate cyclic surface in $\Bbb L^3$ (Lorentz-Minkowski space) with constant (Gaussian curvature) $K \ne 0$ is a surface of revolution. Take a surface of revolution $S\subset \Bbb L^3$ with ...
zeta space's user avatar
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What condition of the Hawking singularity theorem fails for de Sitter and Minkowski spacetimes?

I am trying to figure out what condition of the Hawking singularity theorem fails for de Sitter and Minkowski spacetimes. Some has to fails otherwise we would had geodesic incompleteness which its ...
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Contradictions in "A diagonalizable energy-momentum tensor $T$ satisfies the SEC iff $\rho + p_1+p_2+p_3 \ge 0$ and $\rho + p_i ≥ 0 (i = 1, 2, 3)$

I am trying to understand the prove of this proposition : Let $T$ be a diagonalizable energy-momentum tensor, that is, (T_{µν}) = diag$(\rho, p_1, p_2, p_3)$ on some orthonormal frame $\{E_0, E_1, E_2,...
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Geometric picture of the geodesic in a low dimensional manifold with a semi-Riemannian metric

In the context of general relativity, the simplest dimensional model consists of two-dimensions. For a two-dimensional manifold we often use a unit sphere to depict how the physical space is curved. ...
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Is time-orientability a condition on the metric, smooth or topological structure of a manifold?

I recently asked a question on Physics Stack Exchange about orientability and time-orientability of a manifold in the language of fiber bundles. This new question is related to, but independent, of ...
Níckolas Alves's user avatar
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Counterexamples to show that "$p < q ⇒ t^{±}(p) < t^{±}(q)$", and "$t^{±}$ are continuous" are not true

About this proposition For a general spacetime $(M, g)$ the volume functions $t^{±}$ (a) $p < q ⇒ t^{±}(p) \le t^{±}(q)$, (b) $t^{±}$ are upper/lower semicontinuous. What counterexamples could I ...
some_math_guy's user avatar
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How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic?

How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic? I am just starting with this so I don't really know how to lay out this arguments. I consulted Beem's ...
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Is causality preserved between two points on an immersed hyperboloid in Minkowski $R^3$?

Consider $\mathbb{R}^{1+2}$ with coordinates $\{t, x, y\}$ and Minkowski metric $g = diag(-1,1,1)$. Suppose we have a hyperboloid $x^2 + y^2 - t^2 = 1$ inside the previous Minkowski spacetime. My ...
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Is causal convexity equivalent to causal completeness?

Let $M$ be a spacetime (i.e., a differentiable manifold equipped with a Lorentzian metric) that admits a choice of time orientation. For any region $R\subset M$, we define the causal complement of $R$ ...
Bruno De Souza Leão's user avatar
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How to make sense of $\int _{\gamma }f(z)\,\left|dz\right|$?

Context I understand from [1] that to define the contour integral, let $f:\mathbb {C} \to \mathbb {C} $ be a continuous function on the directed smooth curve $\gamma$. Then the integral along $\...
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What changes in the 1st and 2nd variation formulas and the information contained in the index form for a Lorentzian instead of a riemannian manifold?

In Lee's Intro to Riemannian manifolds I learned that for a Riemannian manifold (M,g) (1)The first variation of the arc-lenght functional is given by equation 6.1 (see below) (2)The second variation ...
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Equality characterization of the reverse Cauchy-Schwarz inequality in a Lorentzian manifold

Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone(ie, both future-directed or past-...
some_math_guy's user avatar
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Bijection from (proper) Lorentz group to PSL(2,C)

It is well known that $SL_2(\mathbb{C})$ is the universal cover of $SO^+(1,3)$, see for example the Wikipedia page on the Lorentz group 1. The map goes like this (some of this is not standard ...
TheEmptyFunction's user avatar
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Does every n-torus admit a semi-Riemannian metric?

I know every smooth manifold admits a Riemannian metric, and I think I can state that a Riemannian metric is a semi-Riemannian metric,right? The former being just a special case of the later, so the ...
some_math_guy's user avatar
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How do I find a cont. non-vanishing vector field on $\Bbb S^n$, $n=$odd, such that there exists a Lorentz metric?

I know that the Hairy ball theorem plus the following proposition Proposition For a smooth manifold $M$. There exists a Lorentz metric on $M$ $\iff$ there is a continuous non-vanishing vector field on ...
some_math_guy's user avatar
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Pade approximation

I am trying to model the Pade approximation of a Lorentzian graph from the taylor series. I am trying to model PA[2/2] from taylor series expansion of order N+M=4th order of derivatives taken at the ...
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Isometries of a submanifold with induced metric

I need help to find isometries of a submanifold of a semi-Riemannian manifold. To be crystal clear, let me start with what I mean by an isometry. $\textbf{Definition:}$ Let $(M,g)$ be a semi-...
lolabol's user avatar
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3 votes
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Derivative of a function with respect to a function

This is Definition 10 from Semi-Riemannian Geometry With Applications to Relativity by Barrett O'Neill, page 7: On both sides of Definition 10, he is differentiating with respect to functions, not ...
John's user avatar
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3 votes
2 answers
121 views

Proof check that a given metric on $\mathbb{S}^1\times\mathbb{R}$ is not time-orientable

I want to prove that there exist a non time-orientable Lorentzian metric on the manifold $M=\mathbb{S}^1\times\mathbb{R}.$ I have an idea on how to do this but since it's the first time I approach ...
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Metric expression of the hypertorsion of (the image of) a timelike curve in 3+1 dimensional spacetime

From mostly recent papers such as these, and their common reference to papers by J. R. Letaw and J. D. Pfausch from the early 1980s, we can learn that [...] a trajectory in (3 + 1)-dimensional ...
user12262's user avatar
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How to derive Lorentz (general) transformation in Special Relativity?

Let $M$ be a 4-vector space over $\mathbb R$. Since $M$ is linearly isomorphic to $\mathbb R^4$, the linear isomorphism is also a diffeomorphism between $M$ and $\mathbb R^4$. For each $p\in M$, we ...
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1 answer
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How to show that the gradient is normal everywhere on a regular level set?

I'm trying to solve Problem 2.9 in John Lee's book Introduction to Riemannian Manifolds. It says Suppose that $(M,g)$ is a Riemannian manifold, $f\in C^{\infty}(M)$, and $\mathscr{R}\subseteq M$ is ...
Nicholas James's user avatar
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Characterization of the Levi-Civita connection through length-extremization property of geodesics

I know that geodesics on a Riemannian manifold equipped with the Levi-Civita connection $\nabla$ have two equivalent characterizations: They are energy-extremizing curves (which is equivalent to ...
Francesco Paronetto's user avatar
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Converse of "a parallel form is closed and co-closed" on Lorentzian manifold

Let's work on $(\mathcal{M},\texttt{g},\nabla)$, an $n$-dimensional Lorentzian manifold $\mathcal{M}$ with metric tensor field $\texttt{g}$ and Levi-Civita connection $\nabla$. Let $\{e^{a}\}$ denote ...
AloneAndConfused's user avatar
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Does $T$ define an isometry?

Consider a Lorentzian manifold $(\zeta,g)$ with metric: $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$$ For $u,v,r,w \in(0,1)$. It can be derived that: $$\Omega_s(x,y,z)=\varphi_s(x)\varphi_s(...
zeta space's user avatar
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Decomposition of differential form on a globally-hyperbolic manifold

Let $(M,g)$ be Lorentzian manifold of the form $$M=I\times \Sigma,\quad\quad g=-dt^{2}+h_{t}$$ where $\Sigma$ is a spacelike hypersurface and $h_{t}$ is a Riemannian metric on $\Sigma$ for every $t\in ...
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1 vote
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Restricted covariant derivative

Consider a globally-hyperbolic manifold $(M,g)$, i.e. $M\cong\mathbb{R}\times\Sigma$ with metric $g=-\beta\mathrm{d}t^{2}+h_{t}$ where $\beta\in C^{\infty}(M)$ and $h_{t}$ is a time-dependent ...
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Trouble with proving certain equality of the covariant derivative of a pseudo-riemannian manifold

I am trying to fill the details from this paper (it's free available) and I get stuck proving the equation (3.16) from the mentioned article. Let me give some context. Part 1. Setting the notation. ...
rowcol's user avatar
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On the calculation of the gradient of the squared distance function on a Riemannian manifold

I recently gave an answer to this question, that I'm requesting you to check the correctness of. "Let $f(x)=\text{dist}_M^2(p,x)$, and $p \in M$ (Riemannian manifold) is fixed. Show that $\text{...
Learning Math's user avatar
1 vote
0 answers
25 views

Proof on inmersions involving the Laplacian

I'm reading "Finite type submanifolds in pseudo-euclidean spaces and applications", by B.Y. Chen. On Lemma $4$ I have a couple of doubts on the proof, the Lemma says: Let $\psi\,\colon M\...
Carlos Cabezas's user avatar
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On the definition of pregeodesics as stationary points of the length functional in (pseudo-)Riemannian geometry

On a Riemannian manifold $(M,g)$, pregeodesics can be defined either of the following two equivalent ways: curves for which the (Levi-Civita) acceleration $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ is ...
Inzinity's user avatar
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1 vote
0 answers
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Proof of commutativity of trace and covariant derivative in orthonormal frame

Suppose we have a symmetric (0,2) tensor $k$. (The original setting is that $k$ is the scalar second fundamental form for a Riemannian hypersurface, but I don't think it matters in my specific ...
Ning's user avatar
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1 vote
1 answer
83 views

Proving $\phi$ is a smooth map and constructing an explicit isometry

Consider a Lorentzian manifold $(\zeta,g)$ with metric: $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$$ For $u,v,w,r,>0$. Suppose we take a Cauchy foliation of $\zeta,$ called $\mathcal F$,...
zeta space's user avatar
2 votes
0 answers
91 views

Application of the spectral theorem to shape-operator.

This was a question brought up by a classmate of mine; In the general case (i.e. when not neccessarily considering $\mathbb{R}^n$ but just a smooth manifold $\overline{M}$ of dimension $m \in \mathbb{...
Ben123's user avatar
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0 votes
1 answer
102 views

Boundary points of a smooth manifold with boundary independent of chart.

Setup: Let $$\mathbb{R}^n\_ := \{(x^1,\ldots,x^n) \subset \mathbb{R}^n: x^1 \leq 0\}$$ and $$\partial \mathbb{R}^n\_ := \{0\} \times \mathbb{R}^{n-1}$$ i.e. $$x \in \partial \mathbb{R}^n\_$$ are on ...
Ben123's user avatar
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0 votes
1 answer
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Integral of an $n$-form with compact support.

Setup: Let $M^n$ be an oriented manifold and let $$\mathcal{A} := \{(U_i,\varphi_i):i \in I\}$$ be a positively oriented atlas ($\varphi_i:U_i \to \varphi_i(U_i)$ preserves orientation). Furthermore, ...
Ben123's user avatar
  • 1,252
3 votes
1 answer
82 views

Comparison of terms in local formulation of Christoffel-symbols in relation to isometries.

Let $(M,g),(N,h)$ be semi-riemannian manifolds, and $\varphi \in C^{\infty}(M,N)$ an isometry (hence a diffeomorphism). Let $(U,\psi = (x^1,\ldots,x^n))$ be a coordinate chart of a point $p \in U \...
Ben123's user avatar
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3 votes
2 answers
166 views

Sectional curvature independent of basis.

Given a semi-Riemannian manifold $(M,g)$; for $p \in M$, we define the sectional curvature of a non-degenerate $2$-plane $\sigma$ with basis $\{u,v\}$ as $$K(\sigma) := K(u,v) = \frac{R(u,v,v,u)}{Q(u,...
Ben123's user avatar
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1 vote
1 answer
65 views

Equation for Ricci tensor induced on a metric hypersurface derived from first Gauss-Codazzi equation

The first Gauss-Codazzi equation for a metric hypersurface $(S, h_{ab})$ of $(M, g_{ab})$ is: $$\mathcal{R}^{a}_{\ bcd} = -2 \pi^{a}_{\ [c}\pi_{d]b} + h^a_{\ m}h^n_{\ b}h^p_{\ c}h^r_{\ d}R^m_{\ \ npr}$...
fr_'s user avatar
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0 answers
54 views

How to define addition law in hyperboloid model(lorentz space) of hyperbolic space

I know mobius addition and Einstein addition are well defined in Poincaré ball model . But how to define addition in hyperboloid model(lorentz space) of hyperbolic space,and can we define the exact ...
Zoe.peace's user avatar
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1 answer
147 views

Existence of geodesically convex neighborhoods in semi-Riemannian manifolds.

I am studying the text by Barrett O’Neill referenced below. On page 130, O'Neill states, as Proposition 7, that every point in a semi-Riemannian Manifold has a convex neighborhood. Convex is defined ...
fisher_martin's user avatar
3 votes
0 answers
72 views

Reference request: Lorentzian Ricci flow

I have been studying some aspects of Ricci flow, namely existence, uniqueness, finite time extinction, the preservation of curvature bounds via the maximum principle, and the modifications of Ricci ...
QuantumFieldMedalist's user avatar
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0 answers
52 views

Why is $\text{Ric}(g)=0$ a quasi-linear PDE in harmonic coordinates?

While studying the dynamics of the Einstein vacuum equations $$ \text{Ric}(g)=0 $$ for $(M,g)$ unknwon, I've come across the statement that in harmonic coordinates $x^\lambda$ defined by $\Box_g x^\...
Gandalf The Gray's user avatar
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0 answers
69 views

Laplace-Beltrami operator in coordinates

I know that on a Lorentzian manifold the Laplace-Beltrami operator of a function $\phi\in C^\infty(M)$ is defined as $$ \triangle_g \phi = \text{div}(\text{grad}\phi). $$ Now I've come across the ...
Gandalf The Gray's user avatar

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