Questions tagged [general-relativity]
Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).
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Clarification regarding the transformation law of the Christoffel symbols
I'm learning about general relativity from Sean M. Carroll's textbook. I recently encountered the transformation law for the Christoffel symbols, and I'm confused, as it seems like I'm seeing two ...
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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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Double Trace of the tensor product of the metric tensor with vector fields.
So I am currently preparing for an exam on General Relativity and while reading the notes I stumbled upon this:
$$
tr[tr[g \otimes X \otimes Y]]= g(X,Y)
$$
Where
$$
g=g_{ij} dx^{i}\otimes dx^{j}
$$
is ...
- 3
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Conditions on positivity of Riemann squared products in Euclidean space?
I wonder if it is known whether there are simple conditions on $u,v$ such that:
$$R_{abcd}R^{abcd}+ u R_{ab}R^{ab} + vR^2 = A_{abcd}A^{abcd}$$
and thus proving that the combination is always positive ...
- 4,425
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A question about the definition of the normal variation of a tensor field
My question comes from the proof of Prop. 7.32 in Dan Lee's book Geometric relativity.
The setting is we have a hypersurface $\Sigma$ with unit normal $\nu$ in an initial data set $(M, g, k)$. We now ...
- 294
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General relativity problem from Wald's book
I'm self-studying General Relativity using Carroll's and Wald's book, and I'm confused about this problem from Wald (Chapter 3, Problem 2).
Suppose that $M$ is a manifold with metric $g_{ab}$ and ...
- 1,222
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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 ...
- 589
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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_{\...
- 589
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Are there different types of metric?
I don't know if this is the right place to ask this question. I'm reading Bernard Schutz's First Course In General Relativity (2nd edition) and in page 61 under the title 'Picture of a one-form', ...
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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 $\{E_0, E_1, E_2,...
- 3,162
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Derivation of a spin connection in general relativity
On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle.
The ...
- 191
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Is time-orientability a condition on the metric, smooth or topological structure of a manifold?
I recently asked a question on Physics Stack Exchange about orientability and time-orientability of a manifold in the language of fiber bundles. This new question is related to, but independent, of ...
- 530
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Counterexamples to show that "$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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