All Questions
19 questions
0
votes
2
answers
329
views
Transformation of Lie derivative of one-form
In the textbook Supergravity ( by Freedman and Proeyen, 2012), they have defined the Lie derivative of a covariant vector with respect to a vector field V on page 139:
$$ \mathcal{L}_V \omega_\mu = V^\...
- 542
4
votes
3
answers
354
views
Ricci Identity with Torsion Proof
In the notes I'm using for General Relativity, the author begins their proof of the Ricci identity with torsion by writing
$$\nabla_{[\mu}\nabla_{\nu]}Z^{\sigma}=\partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}...
0
votes
1
answer
159
views
Relationship between derivatives of tensors in different Cartesian coordinate systems
I'm new to tensor calculus: I'm reading a little introductory book whose title is "Quick Introduction to Tensor Analysis", written by R.A Sharipov. I've reached the section called ...
- 540
2
votes
1
answer
215
views
Tensor Differentiation
In the book "Tensors, Relativity and Cosmology" the author derived Maxwell's Equation in covariant form using the EM field tensor Lagrangian $L=-\frac{1}{4}F^{jl}F_{jl}$ (source=0). One of the steps ...
- 1,047
3
votes
1
answer
420
views
Exterior and Covariant Derivatives
Is the following guaranteed to be true for any covariant vector $f_\mu$ (1-form $\boldsymbol{f}$) in the absence of torsion?
$$\nabla_{[\alpha}\nabla_{\beta}f_{\mu]}=\partial_{[\alpha}\partial_{\beta}...
- 307
1
vote
0
answers
373
views
Second-order covariant derivative in index notation [closed]
So I'm having problems finding the second order covariant derivitive in index notation. My teacher said to just find the covariant derivative of a covariant derivative, so I first started with finding ...
0
votes
2
answers
4k
views
Why is covariant derivative a tensor?
I am trying to prove that the covariant derivative is a tensor (ie it transforms well under a change of coordinates) but I can't succeed to it.
Here is the definition of the covariant derivative :
$$...
- 1,560
0
votes
2
answers
388
views
Covariant derivatives of null tetrads
I am trying to understand the Newman Penrose null tetrads and facing some problems. Given $\ell_k$ is a null tetrad in Newman-Penrose formalism, what is $\ell_{k;i}=?$
0
votes
1
answer
506
views
Divergence on tensor product [closed]
Can someone explain how first equation can be expanded as third equation?
I'm familiar with vector calculus, but not so familiar with tensor calculus, though I know all the definitions.
I don't have ...
- 11
0
votes
3
answers
174
views
Question about differentiation of tensors
According to Arnab Rai Choudhuri, Astrophysics for physicists Page 363:
$$\frac{\partial \overline A^i}{\partial \overline x^l}=\frac{\partial A^k}{\partial x^m}\frac{\partial x^m}{\partial \overline ...
- 859
1
vote
1
answer
483
views
Tensors and derivatives
I am a maths student taking a module in (the mathematics of) Relativity so I get quite confused when looking for stuff that may help me understand where I go wrong in certain questions as I'm not ...
- 11
6
votes
5
answers
8k
views
Covariant Derivative of Kronecker Delta
I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
- 147
7
votes
2
answers
3k
views
Covariant derivative of a covariant derivative
I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$.
This is something I've taken for granted a lot in calculations, namely I though that by the ...
- 623
0
votes
1
answer
72
views
Calculating motion of equation in tensor form
for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$
how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$
...
- 337
0
votes
1
answer
756
views
Variation of a tensor
Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity.
Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means $...
- 221
6
votes
2
answers
12k
views
Variation of square root of determinant of metric, $\delta g$ [closed]
I am trying to calculate
$$ \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
1k
views
Covariant derivative ordering
I was working on a problem involving Bianchi identities, in a particular case I have to take the covariant derivative of the following, which indeed is the Ricci tensor in linearised limit
$$r^{\mu}_{\...
- 278
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
6
votes
2
answers
4k
views
Advection operator
How are exactly $u_j\partial_ju_i$ and $u_i\partial_j u_i$ related?
And what is their relation to ($\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\boldsymbol{u}\cdot(\nabla\boldsymbol{u})$ ?
I ask ...
- 1,813