Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

1
vote
0answers
15 views

Shape operator of pseudo-spheres

In the following, I will refer by O'Neil to Barrett O'Neil's book Semi-Riemannian Geometry With Applications to Relativiy. Consider the $M=\{ x \in \mathbb{R}^n_s | f(x)=r^2\}$ with $f(x)=g_s(x,x)$ ...
2
votes
0answers
24 views

On morse theory and foliations

Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: ...
2
votes
1answer
45 views

Derivative of metric along curve

Let $(M,g)$ be a semi-Riemannian manifold with Levi-Civita connection $D$. Let $\alpha : [a,b] \rightarrow M$ be a smooth curve on $M$, and let $\frac{D}{dt}$ be the induced covariant derivative on $\...
2
votes
0answers
122 views

$GL(n,G)$ General Linear group of a group $G$?

I know you can take the General linear group of some vector space $V$: $GL(V)$. For example, suppose I have a three-sphere, elements of which I might represent as $SU(2)$ since they're diffeomorphic....
0
votes
0answers
8 views

Regarding the complete reducability of the Lorentz group

I was just reading that the Lorentz group has the property of complete reducability, that is any representation can be written as the direct sum of irreducible representations. This reminds me ...
1
vote
0answers
31 views

Uniqueness of a time-orientable covering spacetime

On a standard treatment of time-orientability in Lorentzian spaces, it is usual to see a remark of the form: "If a space-time (M,g) is not time-orientable, then it has a double covering space which is....
0
votes
0answers
13 views

Zero constant mean curvature in Minkowski space versus in Euclidean space

There's a famous result in $\mathbb{R}^3$ which goes as: there are no compact minimal surfaces in $\mathbb{R}^3$.So the mean curvature cannot be zero in compact surfaces in $\mathbb{R}^3$. Now what's ...
0
votes
1answer
25 views

There are no $3$ linearly independent lightlike vectors such that $u+v+w = 0$.

Consider the Lorentz-Minkowski space $E^n_1$, also known as $\mathbb{L}^n$. I want to prove that there are not lightlike linearly independent vectors $u, v, w \in E^n_1$ such that $u + v + w = 0$. ...
0
votes
0answers
11 views

Smooth approximation of polyhedral metric!

do someone have an idea, how to prove that : For every polyhedral metric $(S,d)$ of curvature $\leq -1$, (S=surface), there exists a sequence of smooth metrics, converging uniformly to $(S,d)$. ...
0
votes
1answer
21 views

Index (or signature) of a pseudo-riemannian metric on manifold

Let $(M , g)$ a $n$-dimensional pseudo-riemannian manifold. As $g(p)$ is a bilinear mapping from $T_pM \times T_pM$ to $\mathbb{R}$, we can get a basis $$ {\left\{{\left(\frac{\partial}{\partial x_i}\...
0
votes
0answers
16 views

Anti-de Sitter space: Is the universal cover of $\text{AdS}_n$ a submanifold of $\text{AdS}_{n+1}$?

$$\text{AdS}_n = \{\vec x\in\mathbb R^{n-1,2}\mid\vec x\cdot\vec x=-1\}$$ Using an orthonormal basis $\{e_i\}$ for $\mathbb R^{n-1,2}$, with ${e_1}^2={e_2}^2=-1$ and $(e_{\geq3})^2=1$, this space can ...
0
votes
0answers
21 views

Isometries in Minkowski space

Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that: Theorem 1.7: If $\phi$ is an isometry of $E^3 $, then there exists a unique translation $T$ and a ...
1
vote
0answers
11 views

A non-orientable surface $S$ such that $T_pS$ is time-type.

I'm currently taking a course in Semi-Euclidean (Semi-Riemannian) Geometry and am given the task to give an example of a non-orientable surface $S$ in $\mathbb{R}^3_1$ which is time-type. With the ...
0
votes
1answer
36 views

Notation regarding generalized Minkowski space

In section 12 of the book Surfaces in classical geometries: A treatment by moving frames by Gary R. Jensen, Emilio Musso and Lorenzo Nicolodi (see preview here), Möbius geometry is described. They ...
0
votes
0answers
59 views

Does divergence theorem require smoothness?

Usually, one writes the divergence theorem as \begin{equation} \int_\mathcal{M} d^4x \sqrt{-g} \nabla_\mu v^\mu=\int_{\partial \mathcal{M}} d\Sigma_\mu v^\mu\\,. \end{equation} for some vector field $...
0
votes
0answers
36 views

Irreducible decomposition of Lorentz tensors with Young Tableaux

I want to understand the irreducible decomposition of Lorentz tensors by using Young tableaux. Let me start with a trivial example. Suppose we work in $n=4$ dimensions, and that we have a rank 2 ...
0
votes
1answer
47 views

Problem at proof of Cartan's theorem about the relation between metric and curvature in do Carmo's book

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 157, Cartan's theorem. My question is, why can we take a Jacobi field $J$ in such a way that $J(0)=0$ and $J(...
1
vote
0answers
40 views

Question in the appendixes A of O'Neill book

I'm reading appendixes A of O'Neill's book, "Semi-riemannian geometry" and I don't understand a something. I don't understand at the last theorem, how we construct a universal cover for $M$ and why ...
3
votes
2answers
267 views

Finding local orthonormal frame on a Pseudo-Riemannian Manifold

Suppose we have a semi-Riemannian manifold $(M^n,g)$ with metric signature $(n-k,k)$. By definition, each $p \in M$ the map $g_p : T_pM \times T_pM \to \Bbb{R}$ is a non-degenerate, symmetric, ...
2
votes
2answers
276 views

Existence of Orthonormal Basis of a Metric in a Manifold

Definition: A metric $ g$ on a manifold $ M$ is a tensor field of type $ (0,2)$ such that (1) it is symmetric, i.e. $ g(v,w)=g(w,v)$ for any $ w,v \in V_p, p\in M$, and (2) it is non-degenerate, i.e....
3
votes
2answers
48 views

Relation between lie derivative of a vector field and associated 1-form in a Lorentzian manifold

Let $(M,g)$ be a Lorentzian manifold and $X$ and $u$ represent two vector fields in $M$ such that $\mathcal{L}_X u=0$, that is, $u$ is Lie transported along the integral curve of $X$. My question is: ...
2
votes
1answer
51 views

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}\...
1
vote
1answer
17 views

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 ...
0
votes
0answers
11 views

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'...
1
vote
0answers
22 views

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). ...
1
vote
0answers
28 views

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 ...
0
votes
2answers
103 views

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 ...
2
votes
0answers
60 views

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 ...
2
votes
1answer
26 views

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\...
0
votes
0answers
37 views

Integrating along a null geodesic

I was studying the details of integration on null hypersurfaces through A Relativist's Toolkit by Eric Poisson. If we consider a null hypersurface in 4-dimensional spacetime, then the surface element,...
4
votes
1answer
90 views

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, $$N:=[0,\phi^*) \times_g \mathbb S^{k-1}$$ ...
0
votes
0answers
31 views

Gaussian curvature of $\exp_{p}(P)$

I'm reading about Gauss equation. During the reading there is a proof of a proposition that is not clear to me. It's the following: Let $\Pi$ be a tangent plane to $M$ at $p.$ If $P$ is a small ...
0
votes
0answers
33 views

Book recomendation on semi Riemannian geometry other than O'Neill?

I find very hard to follow O'Neill's book because it's not very formal on it's statements and proofs. Is there any modern treatment on semi Riemannian geometry with better statements and proofs?
0
votes
0answers
30 views

Semi-riemannian Geometry and Schur lemma

During the reading of a book I found a Schur's theorem which claims the next: If $M$ is a connected semi-riemannian manifold, with $n=dim(M)\geq 3,$ and for each $p\in M,$ $K$ is constant on the ...
2
votes
1answer
46 views

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 $Ricfg=fg,$ ...
1
vote
1answer
83 views

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=\...
0
votes
2answers
81 views

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 , ...
0
votes
1answer
25 views

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 ...
0
votes
2answers
59 views

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 ...
0
votes
1answer
39 views

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 ...
1
vote
0answers
15 views

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 $M$ ...
1
vote
1answer
71 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 ...
1
vote
1answer
150 views

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\...
3
votes
1answer
84 views

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\...
3
votes
1answer
114 views

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 ...
2
votes
1answer
107 views

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,...
1
vote
1answer
38 views

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 ...
1
vote
2answers
64 views

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 ...
6
votes
1answer
190 views

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}...
1
vote
0answers
166 views

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&...