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).

219 questions
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Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'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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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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Failure of geodesic uniqueness - what does it say about the manifold?

I am more of a physicist than a mathematician, but this question is properly mathematical rather than physical, even though it is motivated by a physical application; please assume mathematical ...
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No Lorentzian metric on $S^{2}$

In the book Riemannian Geometry by Gallot et al there is a remark at the beginning that there is no Lorentzian metric on $S^{2}$. Is it a difficult theorem? Or there is an easy solution? Any hint/idea ...
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Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a semi-...
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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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On geodesics in Schwarzschild spacetime

I am required to show that a circular lightlike geodesic exists in the Schwarzschild spacetime, and to find its radius. What's the best way to start this?
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Definition of gradient of a function $f$ in Riemannian manifold

I'm reading Semi Riemannian Geometry with applications to relativity by Barret Oneill and I'm trying understand the definition of gradient of a function $f$ in Riemannian Manifold. I know that ...
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How to visualize rotation on a hyperbola?

I am studying Lorentz transform and I do not quite get what it means to use the hyperbolic matrix to rotation a point on a hyperbola, mainly it is because the hyperbola consists of two divergent ...
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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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Conformal Killing vectors fields on Minkowski spacetime

As is well known Minkowski spacetime (which is four dimensional vector space with scalar product $\eta _{\mu \nu}$ of signature $-+++$) is maximally symmetric, which manifests itself in presence of ...
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Calculating Christoffel symbols using variational geodesic equation

Given the line element $$ds^2 = e^v dt^2 - e^{\lambda} dr^2 - r^2 d \theta^2 - r^2 \sin^2 \theta d \phi^2$$ we wish to compute the Christoffel symbols $\Gamma^{a}_{bc}$ using the geodesic equation. ...
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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 ...