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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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 ...
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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 ...
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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 $\mathcal F$,...
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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{...
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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 ...
- 447
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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, ...
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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 \...
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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 $$K(\sigma) := K(u,v) = \frac{R(u,v,v,u)}{Q(u,...
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Equation for Ricci tensor induced on a metric hypersurface derived from first Gauss-Codazzi equation
The first Gauss-Codazzi equation for a metric hypersurface $(S, h_{ab})$ of $(M, g_{ab})$ is:
$$\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}$...
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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 ...
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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 ...
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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 ...
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Physical implications of turbulization of the leaves of a foliation of a spacetime?
There is a technique in differential topology called turbulization that essentially twists the leaves of the foliation along the ideal boundary of a manifold, say the open 3-ball, rendering the ...
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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^\...
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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 ...
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Pullback of a Lorentzian metric is non-degenerate
Background: Recently I've read this post of one person trying to prove that the pullback $F^*g$ of a riemannian metric $g$ is a Riemannian metric iff $F: N \to (M,g)$ is a smooth immersion.
Question: ...
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Pullback of lorentzian metric is degenerate at the boundary of embedded submanifold
Let $\Sigma = \mathbb{R} \times [0,1]$ be a smooth manifold with boundary $\partial \Sigma = \mathbb{R} \times \{0,1\}$.Let $M = R^{1,D-1}$ be the $D$-dimensional Minkowski space with scalar product (...
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A global view-point on pseudo-Riemannian manifolds [closed]
A pseudo-Riemannian manifold $(M,g)$ is a generalisation of the concept of a Riemannian manifold where we relax positive-definiteness to non-degeneracy.
$\alpha$) Non-degeneracy is still enough to ...
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What is Higher Teichmuller Theory?
I am interested in representation of surface group. So, I am studying the book Lectures on Representations of Surface Groups by Francois Labourie. The main goal of representation of surface groups is ...
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Constructing certain Lorentz-Minkowski isometries of dimension 3
I need to pick a basis of elements $u$, $v$ and $w$ such that $u$ and $w$ are lightlike (i.e., $g(u,u) = 0 = g(w,w)$, where $g(x,y) = x_1y_1+x_2y_2-x_3y_3$), $g(v,v) = 1 = -g(u,w)$, $\langle u \rangle^...
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What is the metric for this product manifold?
Consider a spacetime $(\zeta^{3,1},g)$
where $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,w,r \in (0,1)$$ Now this is just Minkowski space in different coordinates (related ...
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$\mathrm{Iso}(M)$ and the metric that's preserved
Consider a semi-Riemannian manifold $(M,g)$ with the metric $g$ being a modified Bessel function, $K_1.$
However, $K_1$ was not randomly chosen. It comes from the Fisher information metric of a ...
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How to show that the special orthochronus pseudo orthogonal group is the connected component of the identiy?
Let $\mathbb{R}^{t,s}$ be the the vector space $\mathbb{R}^{t+s}$ equipped with the non degenerate indefinite scalar product $\eta$ that satisfies:
$$\eta(e_i,e_i)=-1\text{ for $1\leq i\leq t$}\qquad \...
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Misunderstanding with indices in covariance and contravariance
I am getting confused with naming things covariant or contravariant. I am reading Barret O Neil's book on semi riemannian geometry with applications to relativity, and I need some help with the ...
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Why is Weingarten mapping well-defined?
Let $\Sigma$ be a three dimensional submanifold of the lorentzian manifold $(M,g)$ and let $q$ the riemannian metric on $\Sigma$ induced by $g$. We define the Weingarten mapping as
$$ W:T\Sigma \...
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Manifolds where $\Delta t$ is conserved for any trajectory.
If we consider a medium where perturbations always take the same time to reach every point in space when measured from an arbitrary observer's frame of reference then how would length have to contract ...
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Total scalar curvature of a compact space-like manifold in De Sitter space
Consider the De Sitter space $S^{5,1} = \{ x\in \mathbb{R}^7 : (x^1)^2 + ... + (x^6)^2 - (x^7)^2 = 1 \}$, with its lorentz norm $\eta = (dx^1)^2 + ...+(dx^6)^2 - (dx^7)^2$.
Consider a compact 4-...
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What is the relationship between a submanifolds' normal vectors and its induced signature?
I'm following an example about how to determine the signature of a metric induced on a submanifold of some space using the normal vectors to the submanifold but don't understand how the full process ...
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Are $\mathrm{Pin}(3, 1)$ and $\mathrm{GL}(2, \mathbb{C})$ isomorphic?
In physics, it’s very common to utilize the group (exceptional) isomorphism
$$
\mathrm{Spin}^{+}(1, 3) \approx \mathrm{SL}(2, \mathbb{C})
$$
in problem solving. I’m working with $\mathrm{Pin}$ ...
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What does it mean for covariant derivative of metric tensor is zero in general relativity?
Let $(M,g)$ be a Riemannian manifold and $\nabla$ be the Levi-Civita connection. I understand that the total covariant derivative $\nabla g$ of the metric tensor is zero. However, In general ...
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Reducing vacuum Einstein field equations to wave equations using wave coordinates
In a lot of research papers, authors claim that it is well know that in wave coordinates, Einstein field equations can be reduced to wave equations. But I did not find any reference. Can anybody ...
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A computation for null vector fields in general relativity
Denote $g_{ij}$ to be the metric and $(g^{-1})^{ij}$ to be its inverse. Let $u$ and $\underline{u}$ satisfy eikonal equations, i.e. $$(g^{-1})^{\mu\nu}\partial_{\mu}u\partial_{\nu}u=0,\quad(g^{-1})^{\...
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Is $w$ Killing for some semi-Riemannian metric?
Consider the vector field $w=(x\log x,-y\log y,-z\log z)$ for $0<x,y,z<1.$ I'm wondering if $w$ is Killing for some semi-Riemannian metric.
If we consider a lower dimensional version of $w$ i.e. ...
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Deformation of causal curves
In Chapter 14 of the Barrett O'Neill's book Semiriemannian geometry, in the proof of theorem 55B, there is a claim that, even that it is intuitively obvious, I don't know how to prove it rigorously.
...
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Geodesic distance from inner product in an embedding space
Let $\mathbb{R}^{2,d}$ be the $(d+2)$-dimensional space with semi-Riemannian metric $\eta$ with signature $(-,-,+,\dots,+)$. We shall enumerate the Cartesian coordinates on this space as $X = (X^0,X^1,...
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bound on the distance of an isometry to the identity
Consider the space
$$
\mathbb{S}^{3,1} := \{ x \in \mathbb{R}^5 : (x_1)^2 + (x_2)^2 + (x_3)^2 +(x_4)^2=1+(x_5)^2 \}
$$
with the lorentz metric on $\mathbb{R}^5$ :
$$
\langle x,y\rangle _{lor} := x_1 ...
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Plotting tessarines in (pseudo-Euclidean) $R^{2,2}$ or $C^{1,1}$; conventions around which axes correspond to which signs in the metric signature
[This is a slightly tweaked re-posting of a question I asked in April 2020, that subsequently got auto-deleted as an "abandoned question." Hopefully launching a second attempt now (over two ...
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Showing a symmetric bilinear form is nondegenerate
I wan to show that the symmetric bilinear form given by
\begin{align}
\begin{split}
g: T_pSL_2(\mathbb{R}) \times T_pSL_2(\mathbb{R}) & \to \mathbb{R}, \\
\bigg{(}
\...
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For space-like linear independent vectors $x,y$, $V:=\text{span}\{x,y\}$ is space-like iff the Lorentz-orthogonals of $x$ and $y$ intersect
A vector $x \in \mathbb{R}^{n+1}$ is called space-like if $\lVert x \rVert^2 > 0$ and time-like if $\lVert x \rVert^2 < 0$ with respect to the norm induced by the Lorentzian scalar product.
A ...
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general form of a Lorentz transformation in $\mathbb{R}^2$
Is it true that every Lorentz transformation acting on $\mathbb{R}^2$ is of the form
\begin{pmatrix}
\cosh(s) & \sinh(s) \\
\sinh(s) & \cosh(s)
\end{pmatrix}
for some $s \in \mathbb{R}$ ?
If ...
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Conjugate points and expansion of the geodesic congruence
I am working in a Lorentzian manifold $(M, g)$ (but I think the problem would be quite similar in a Riemannian manifold) and I am considering a timelike geodesic whose tangent vector field is denoted ...
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Pullback Connection/Riccati-Equation
I'am currently trying to understand the paper:
https://epub.uni-regensburg.de/23578/1/MP171.pdf
The point where I'am stucked at is in the proof of Theorem 4.6.
You dont need to read the whole article, ...
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Extendibility of a curve in a compact subset (Lemma 14.2 in O'Neill's book)
Lemma $14.2(5)$ of O'Neill's book Semi-Riemannian geometry with applications to relativity states the following:
If $\mathcal{C}$ is a convex open set in $M$, then, a causal curve $\alpha$ contained ...
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Shape operator as a tensor field defined in a submanifold
Definition $4.18$ in O'Neill's book Semi-Riemannian geometry, with Applications to Relativity states that given a unit vector field $U$ normal to a hypersurface $M\subset \overline M$, then, there is ...
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Contraction of a lightlike vector with another vector (free cordinate)
I'm trying to prove that there exist always a vector w whose contraction with a lightlike vector u ($g(u,u)= 0$) it's always different from zero:
$$g(u,v)\ne 0$$
I know how to do this with coordinates,...
- 111
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Non-degenarate subspace with indefinite inner product.
I'm little overshadowed with this problem.
Let $S = (v_1, · · · , v_m) ⊂ V$ be a linearly independent set which spans a non-degenerate
subspace W = L(S) and put $W_k = L(v_1, . . . , v_k)$ for every $...
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Gram-Schmidt process to construct orthonormal base in a finite vector space with indefinite scalar product.
Im choking with this exercise because of the indefinite scalar product. I know the process for the definite one.
The first thing I'm asked to do is to check GS is still valid for indefinite scalar ...
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Completeness of projection of vector fields
Let $X$ and $Y$ be complete commuting vector fields on a (semi-) Riemannian manifold.
Denote by $\pi X$ the component of $X$ orthogonal to $Y$, i.e.
$\pi X = X - \frac{\langle X,Y\rangle}{\langle Y,Y\...
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Doubt in the Second fundamental form of a immersed manifold
Let $f \colon M^{n} \rightarrow \overline{M}^{n+m}$ be an immersion. Identifying $TpM$ with $Im(df_{p})$, we have the following decomposition
$$T_{p}\overline{M} = T_{p}M \oplus (T_{p}M)^{\perp}$$
...
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Integrals over a space with Lorentz signature metric
$$\int_{spacetime}\frac{d^4x}{(x^2)^2}=\int_{-\infty}^{\infty} dt\int_{-\infty}^{\infty} dx\int_{-\infty}^{\infty} dy\int_{-\infty}^{\infty} dz\frac{1}{(t^2-x^2-y^2-z^2)^2}$$
To show that $\int_{...
