All Questions
Tagged with differentiation curvature
45
questions
1
vote
3
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86
views
Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
- 366
1
vote
0
answers
45
views
A covariant derivative computation in General Relativity [duplicate]
I am trying to compute $\nabla^\mu\nabla^\nu R_{\mu\nu}$.
I proceed as follows:
\begin{align}
\nabla^\mu\nabla^\nu R_{\mu\nu}&=g^{\mu\rho}g^{\nu\lambda}\nabla_\rho\nabla_\lambda R_{\mu\nu} \\
&...
- 392
1
vote
4
answers
230
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How to find the double covariant derivative of a general vector?
I have been reading through Carroll's GR textbook and there is a line in the derivation of the Riemann tensor that I do not understand.
$$\nabla_\mu \nabla_\nu V^\rho=\partial_\mu(\nabla_\nu V^\rho) - ...
- 51
8
votes
2
answers
825
views
How does the covariant derivative satisfy the Leibniz rule?
In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
- 103
1
vote
0
answers
79
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Scalar curvature from Riemannian metric
I want to compute the scalar curvature for points on an empirical manifold (sampled data).
I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
0
votes
1
answer
122
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Einstein field equations from covariant derivative of a general linear gauge transformation
A general linear transformation is given by
\begin{align}
\psi'(x) \to g \psi(x) g^{-1},
\end{align}
The gauge-covariant derivative associated with this transformation is
\begin{align}
D_\mu \psi=\...
- 1,548
1
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0
answers
160
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What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?
I know the covariant derivative of a tensor is
$$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$
Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
- 67
0
votes
1
answer
1k
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Covariant derivative, directional derivative, and curvature tensor
I'm confused about how to connect those three things together, so hopefully the question doesn't end up vague. The main problem is understanding how the curvature tensor is a commutator of covariant ...
- 860
1
vote
1
answer
250
views
Are Riemann curvature tensors defined up to a total derivative?
Consider a case where the Riemann Tensor is given by
$$R^{\mu}_{~~~\nu\rho\sigma} = P^{\mu}_{~~~\nu\rho\sigma} +\nabla_{\rho}A^{\mu}_{~~~\nu\sigma}-\nabla_{\sigma}A^{\mu}_{~~~\nu\rho}$$
It seems to me ...
- 1,200
0
votes
0
answers
99
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Derivatives of the metric in the local Lorentz frame
In the local Lorentz frame (local flatness at point P) we have:
$$
g_{\alpha \beta} (P) = \eta_{\alpha \beta}, \quad \Gamma^{\rho}_{\alpha \beta}(P) = 0.
$$
In the reference that I am following (The ...
- 297
1
vote
1
answer
350
views
Covariant derivative of spherical harmonics
Given is the metric $\gamma_{jk}$ for the surface of a Sphere $S^2$ with $\gamma_{22}=1,\gamma_{23}=\gamma_{32}=0$ and $\gamma_{33}=\sin^2(\theta)$. The coordinates are $x=$($t,r,\theta,\phi$) and $j$ ...
- 83
4
votes
1
answer
620
views
Derivation of Gauss-Codazzi type equation (Ricci relation)
I am following Padmanabhan's book Gravitation for the particular derivation. The derivation goes as follows,
\begin{align}
R_{abst}n^t&=\nabla_a\nabla_b n_s-\nabla_b\nabla_a n_s=\nabla_a(-K_{...
- 682
1
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0
answers
98
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Question on differentiation, Sakharov equation (related to a previous post) [closed]
I'll keep this as short and sweet as possible, the other day I was going through some old physics notes I had wrote. I'm quick to admit, my knowledge on physics is good but I'm in no way a pure ...
3
votes
1
answer
2k
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Physical meaning of the vector Laplace operator
I have seen here a question asking for the physical interpretation of the Laplace operator for a scalar field. However, there is also a vectorial version of this operator, the vector laplace operator, ...
- 974
3
votes
2
answers
171
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Does covariant derivative include magnitude change of a vector as well as direction change of the same vector?
Does covariant derivative include magnitude change of a vector as well as direction change of the vector? In some explanations I followed I have not noticed mentioning of magnitude change along with ...
- 1,815