All Questions
Tagged with differentiation metric-tensor
125
questions
3
votes
1
answer
65
views
Covariant and controvariant bases derivative
How to show that
$\overrightarrow{\textbf{e}}_\sigma\cdot\partial_\mu \overrightarrow{\textbf{e}}_\nu = \overrightarrow{\textbf{e}}_\sigma\cdot\partial^\mu \overrightarrow{\textbf{e}^\nu}$
where $\...
- 133
4
votes
3
answers
2k
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Laplace operator and tensor calculus:
I'm studying Tensor calculus and I found this interesting problem:
Show that:
$$ \Delta F=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_kF\right)$$
Here's some ...
user263457
1
vote
4
answers
2k
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Deriving the Covariant Derivative of the Metric Tensor
First off, I did look through some other questions:
Covariant Derivative of Metric Tensor
Why is the covariant derivative of the metric tensor zero?
https://math.stackexchange.com/q/2174588/
But they ...
- 81
0
votes
2
answers
3k
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Covariant Derivative of Metric Tensor
I'm an amateur studying General Relativity. I'm reading some notes of lectures by Susskind. In them, it is written that
"we know that [the covariant derivative of the metric tensor] is zero. ...
- 113
0
votes
1
answer
75
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Why $\Gamma^\mu_{\beta\delta;\gamma} =\Gamma^\mu_{\beta\delta,\gamma}$?
Gravitation Page 276 Exercise 11.3 solution indicated that
$$\nabla_\gamma \nabla _\delta e_\beta
=e_{\mu}\Gamma^\mu_{\beta\delta,\gamma} +(e_\nu\Gamma^\nu_{\mu\gamma}) \Gamma^\mu_{\beta\delta}$$
...
- 3,256
0
votes
1
answer
354
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Is Ricci's theorem can be simply deduced using covariant derivatives of fundamental tensors? [duplicate]
Well Ricci's theorem is given by:
$$\mathrm{D}g_{ij}=\mathrm{D}g^{ij}=0$$
I was wondering that if the theorem can be proved using covariant derivatives of $\delta_i^k$, $g_{ij}$ and $g^{ik}$.
I ...
user262095
1
vote
0
answers
208
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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
519
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Which derivative to use in the change of metric tensor due to a gauge transformation?
I'm used to calculating the change in the metric due to a gauge transformation in the following way:
The gauge transformation up to linear order is
\begin{equation}
x^\mu \rightarrow x' ^\mu =x^\mu ...
- 4,827
1
vote
2
answers
559
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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
0
answers
239
views
Covariant derivative of a metric determinant
The covariant derivative of a metric is zero $g_{\alpha\beta;\sigma}=0$. Is the covariant derivative of a metric determinant zero following the assumption($g_{\alpha\beta;\sigma}=0$):
$$
g_{;\sigma}=...
- 626
0
votes
2
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827
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Covariant derivative of four-position
May someone confirm or deny that covariant derivative of four-position is just metric tensor?
I mean:
$\nabla_{\gamma}X_{\alpha} = g_{\gamma \alpha}$
When I try to rewrite it with base vectors it ...
- 9
0
votes
1
answer
2k
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Derivative of a metric tensor
I would like to ask you a question - maybe simple - but bothering me.
We have two four-position vectors product in curvilinear coordinates given by
$(1) \quad X^{\alpha}g_{\alpha \beta}X^{\beta} = \...
- 9
0
votes
1
answer
1k
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Proof that covariant derivative of contravariant components of metric vanish for metric compatabilit
In this excercise I want to show that $\nabla_\rho g_{\mu \nu}=0$ and $\nabla_\rho g^{\mu \nu}=0$
This should probably be very easy, but excuse me I'm completly new to GR.
So to do this I used that ...
- 153
1
vote
0
answers
65
views
Differentiating the four-velocity contracted with itself
Let us denote $u^\mu$ as the contravarient component of a four velocity at a point in some coordinate system for a pseudo-Riemannian manifold. I want to examine the following equation.
$$\partial_\nu(...
- 541
0
votes
1
answer
173
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4-velocity lowering index question
The 4-velocity in contravariant form is given by
$$V^\mu=\frac{dx^\mu}{d\tau}$$
for some general co-ordinates $x^\mu$ and proper time $\tau$.
Is the 4-velocity in covariant form given by
$$V_\nu=V^\...
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