All Questions
11 questions
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How to solve nonlinear diff eq for a general Schwarzschild metric?
So I have a general form of a spherically symmetric metric:
$$ds^2 = -g(r)_t \, dt^2 + g(r)_r \, dr^2 + g(r)_s (d\theta^2 + \sin^2 \theta \, d\Phi^2)$$
$$ R_{\theta\theta} = \frac{-g'_s g'_t g_r + ...
- 11
2
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1
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543
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How to know which the independent components of the Riemann tensor are before calculating them?
I want to know how to figure out the non-vanishing and independent components of the Riemann Tensor before computation so I don't need to calculate all of them.
For example, a $3x3$ metric would give ...
2
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2
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572
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The Christoffel symbol of a maximally symmetric space
What is the form of the Christoffel symbol associated with the curvature tensor of a maximally symmetric space given by $$R_{\mu\nu\alpha\beta}=\frac{R}{d(d-1)}\left(g_{\mu\alpha}g_{\nu\beta}-g_{\mu\...
- 262
1
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0
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191
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Carroll's GR: The Schwarzschild Metric
On page 195 of Sean M Carroll's An Introduction to Special Relativity: Spacetime and Geometry, he calculates the Christoffels and Riemann tensors for a generic metric that is static and has spherical ...
- 589
2
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1
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101
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Determining the curvature by symmetries of the metric
Given the Kahn-Penrose metric:
$$
ds^2=2dudv-(1-u)^2dx^2-(1+u)^2dy^2
$$
I calculated the Riemann Tensor and found that all elements equal 0.
Is there some symmetry principle by which I could have ...
- 21
1
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1
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261
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Common misunderstanding of Birkhoff Theorem
I just found a paper "On a common misunderstanding of the Birkhoff theorem". This means that inside a spherically symmetric thin shell there is no gravitational force, BUT there is time ...
- 782
0
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1
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122
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maximally symmetric spacetime
An empty spacetime has zero or constant Ricci Scalar (depending on the cosmological constant). Is there a theorem which guarantees that such a spacetime should be Minkowski or dS/AdS? In other words, ...
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2
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1
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521
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A priori knowledge of the components of the Ricci tensor
Source: Thomas Moore's A General Relativity Workbook
In Moore's "diagonal metric worksheet" he doesn't explain his process of determining the "only possible non zero components" of the Ricco tensor, ...
- 1,081
1
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1
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214
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Proof of Schur's Theorem
On Pg. 123 of Schaum's Tensor Calculus:
At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$
for any ...
- 1,047
0
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1
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Proof of first Bianchi identity
The proof is often simplified by using the following theorem:
"If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g_{...
- 1,047
5
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159
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Assuming a fixed total mass, will the spacetime geometry outside a spherical mass distribution depend on the shape (of the distribution)?
Consider two independent spheres of equal masses but of different radius and in different spacetimes. The first sphere is less dense than the second one, i.e., it has a larger radius. For example, if ...
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