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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Spinor representation for indefinite signature
Recall that for a $n$-dimensional vector space with positive definite quadratic form, we have the spin representation thanks to the following inclusions
$$\Delta_n : \mathrm{Spin}(n) \rightarrow{} \...
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Local cartesian coordinates on Riemannian manifold
I'm wondering is possible for every given metric $g=g_{ij}dx^i \otimes dx^j$ on $M$ and for every given $p\in M$ to find such chart $(U, \varphi)$ around $p\in U \subset M$ that the metric $g|_U$ in ...
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Isometry group of quadric model of anti-de Sitter space
I am learning Lorentzian geometry on my own. Consider the space $\mathbb R^{p+2}$ endowed with the bilinear form $$\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 ...
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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 ...
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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 $\{E_0, E_1, E_2,...
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Geometric picture of the geodesic in a low dimensional manifold with a semi-Riemannian metric
In the context of general relativity, the simplest dimensional model consists of two-dimensions.
For a two-dimensional manifold we often use a unit sphere to depict how the physical space is curved. ...
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Is time-orientability a condition on the metric, smooth or topological structure of a manifold?
I recently asked a question on Physics Stack Exchange about orientability and time-orientability of a manifold in the language of fiber bundles. This new question is related to, but independent, of ...
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Counterexamples to show that "$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 ...
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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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