All Questions
Tagged with tensor-calculus curvature
189
questions
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Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion
I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding.
The starting form of action (3-Dimensional) is:
$$
S_{...
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1
answer
68
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Selecting Indices for the Riemann Tensor
How do I know when computing the Riemann Tensor (in two dimensional) which indices to select? Consider the Riemann Tensor $R^a_{bcd}$ how do I know what values to take for $a$?
As an example, consider ...
1
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0
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35
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Timelike normal vector becomes null
I have a metric given by
\begin{equation}
ds^2 = \frac{e^{2 A(z)}}{z^2} \left(-g(z) dt^2 + \frac{dz^2}{g(z)} + dx^2 + dx^2_1 + dx^2_2 \right)
\end{equation}
where $A(z) = -a \ln(b z^2 + 1)$ and $g(z)$ ...
- 404
1
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28
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Why are the zero modes of the below operator Killing vectors? (2+1 dimensional gravity)
I'm trying to understand the eigenmodes of the following operator:
$$(\Delta_{(1)}^{L L}-\frac{2}{3} R)V_\nu \equiv -\nabla^\mu \nabla_\mu V_\nu+R_{\nu \mu} V^\mu -\frac{2}{3} RV_\nu $$
Where $R_{\mu\...
- 11
2
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1
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95
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Independent Components of the Riemann Curvature Tensor
I am struggling to understand a general method to calculate the independent components of the Riemann Curvature Tensor (RCT).
Firstly, as far as I am aware the number of independent components of the ...
- 155
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2
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92
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No torsion with calculating the commutator of the covariant derivatives
For simplicity, I only calculated half of the commutator. I didn't leave everything in components because I'm uncomfortable considering
(I previously messed up the indices. The following is the ...
0
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0
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38
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Extrinsic curvature of constant time hypersurfaces in Minkowski
Along the geodesic of a stationary observer in Minkowski spacetime we have the following tangent vector
$$t^\mu = (1,0,0,0)$$
We have that hypersurfaces of constant time along this are just 3D ...
- 49
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1
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71
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Does the variation of $I$ yield Bach tensor？
For $$I_1=\int \sqrt{-g}C_{abcd}C^{abcd}d^4x,$$ where $C_{abcd}$ is the Weyl tensor. If we neglect the Gauss-Bonnet term this can be reduced to $$I_2=2\int \sqrt{-g}(R^{ab}R_{ab}-\frac {1}{3} R^2)d^4x....
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Verifying whether the vanishing term in the Quasi-minkowskian metric is indeed a tensor
In Weinberg's Gravitation and Cosmology, on page 165 he notes that $h_{\mu \nu}$ is lowered and raised with the $\eta$'s since unlike $R_{\mu\kappa}$ it is not a true tensor (or at least implies it). ...
- 11
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2
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177
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Is space-time curvature same for all observers?
Since tensors are invariant under coordinate transformations, and a moving observer is just in another coordinate system, he should measure the same Riemann curvature tensor (with different components)...
- 1,161
3
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100
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Curvature expression using Levi-Civita symbol
I've seen the following expression to curvature tensor
$$R^j_{ab}=2 \partial_{[a}\Gamma^j_{b]}+ \epsilon_{jkl}\Gamma^k_a \Gamma^l_b $$
both on Thiemann (equation $4.2.31$) and Rovelli's (equation $3....
- 403
5
votes
1
answer
268
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Curvature tensor and parallel transport around an infinitesimal loop (quadratic terms)
Given a manifold endowed with a connection $(M, \nabla)$, I want to see how the curvature tensor appears parallel-transporting a vector around a closed loop. To avoid complications with holonomies, I'...
- 1,989
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164
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Ricci tensor proportional to Ricci scalar in 2D [duplicate]
How can we prove that in 2D the Ricci scalar is proportional to the Ricci tensor? I started with the second Bianchi identity, set 2 axes equal - and it led me to the fact that in 2D the covariant ...
- 1,161
1
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1
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220
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How do I conceptualize the difference between the Weyl tensor and Riemann Curvature tensor?
Currently, I am studying General Relativity from Sean Carroll's An Introduction to General Relativity: Spacetime and Geometry. In chapter 3 of this book, he develops the Riemann Curvature Tensor. I ...
- 1,261
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Ricci Scalar Curvature under conformal transformation
Consider the Klein-Gordon equation in curved spacetime with metric $g$
$$\square_g \phi - \xi R \phi = 0$$
and consider a conformal transformation
$$g \longmapsto \tilde{g} = \Omega^{2} g \quad , \...
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