All Questions
20 questions
0
votes
1
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74
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Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
- 1
4
votes
2
answers
641
views
Confusion on metric determinant derivative
Maybe it is a stupid confusion. I need to compute the derivative of the metric determinant with respect to the metric itself, i.e., $\partial g/\partial g_{\mu\nu}$, but I have an indices confusion in ...
- 378
2
votes
1
answer
58
views
Complex Tensors and Metric [closed]
It has been given that in 2-dimensions, consider the metric: $$\ ds^2 = e^{\phi(z,\overline z)}dzd\overline z .$$
Show that $$\ t_{z...z;z} = (\partial_{z} - n\partial_{z} \phi)t_{z....} .$$
I can't ...
- 23
0
votes
2
answers
194
views
Variation of the contravariant component of the metric respect to the covariant component of the metric
I am recently studying general relativity and it is a bit difficult for me to handle the rise and fall of indices in some calculations. My specific question is how could I find this variation?
$$\frac{...
0
votes
1
answer
397
views
Killing equation in coordinates
In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$
I have read that the key observation, to write the equation in ...
- 187
4
votes
3
answers
1k
views
Why is the covariant derivative of the metric tensor with UPPER indices equal to zero? [closed]
I've shown that $\nabla_{\lambda} g_{\mu\nu} = 0 $ rigorously by the following method:
$ \nabla_{\lambda} g_{\mu\nu} = \partial_{\lambda}g_{\mu\nu} - \Gamma^{\rho}_{\lambda\mu} g_{\rho\nu} - \Gamma^{\...
4
votes
1
answer
1k
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Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives
For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives.
I have tried ...
- 53
1
vote
0
answers
216
views
Lie derivative of the non-coordinate metric being 0
I'm trying to answer a question about a the Lie derivative of a metric in a non-coordinate basis.
Here, the $C^{a}_{bc}$ are from the Lie derivative of one basis vector with respect to another, or ...
- 420
1
vote
2
answers
635
views
Calculate the expression of divergence in spherical coordinates $r, \theta, \varphi$
Hi this is my first question in [Physics.SE] I saw a lot of posts and I liked them. I hope that my question will be answered too.
While I'm solving a problem in vector calculus. I recognized that I ...
user262095
0
votes
1
answer
1k
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Proof that covariant derivative of contravariant components of metric vanish for metric compatabilit
In this excercise I want to show that $\nabla_\rho g_{\mu \nu}=0$ and $\nabla_\rho g^{\mu \nu}=0$
This should probably be very easy, but excuse me I'm completly new to GR.
So to do this I used that ...
- 153
3
votes
2
answers
722
views
Differentiation of the determinant $g$
Let $g$ be the determinant of the metric tensor.
I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
- 551
0
votes
2
answers
9k
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What is the derivative of an angle? [closed]
What is the derivative of an angle? I don't understand
-1
votes
1
answer
100
views
What is $\frac{\delta (\partial_\kappa \sqrt{g})}{\delta g^{\mu\nu}}$?
Title says it all, is there a closed expression for
$$\frac{\delta (\partial_\kappa \sqrt{g})}{\delta g^{\mu\nu}}$$
where $g = \det g_{\mu\nu}$?
- 1,323
1
vote
1
answer
120
views
The dimensional analysis of the GR geodesic equation
The geodesic equation parametrized by the proper time contains two terms:
$$
{d^{2}x^{\mu } \over ds^{2}}=-\Gamma ^{\mu }{}_{{\alpha \beta }}{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\
$$
The ...
2
votes
1
answer
2k
views
Divergence of inverse of metric tensor
I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?!
I'm not so familiar with the divergence of the second ranked tensor. ...
- 119
6
votes
2
answers
12k
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Variation of square root of determinant of metric, $\delta g$ [closed]
I am trying to calculate
$$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$
where $g = \text{det} g_{\mu \nu}$.
We have
$$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 \...
- 329
1
vote
1
answer
488
views
Varying wrt metric [closed]
I saw people write $\frac{\partial( F^{ab} F_{ab})}{\partial g^{ef}}$ as
$\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}$ in a way that exposes the dependence on the metric. but ...
- 79
3
votes
1
answer
138
views
Is this covariant derivative identity true?
Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$
If this is true, I'm ...
- 3,089
2
votes
1
answer
124
views
Can these two terms cancel out?
In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$
The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with $$J^b_{\mu\lambda}J^a_\...
- 1,539
8
votes
1
answer
10k
views
How is the second-order covariant derivative of a scalar computed?
What is second-order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric takes the form $$ds^2=dr^2+g(r)d\theta^2$$ and $f$ is a ...
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