All Questions
46
questions
2
votes
1
answer
74
views
How to derive $\partial^{\nu}F^{\mu\alpha} + \partial^{\alpha}F^{\nu\mu} + \partial^{\mu}F^{\alpha\nu}=0$ for the Electromagnetic field tensor? [closed]
The problem says to show that
$$\partial_{[\mu}F_{\alpha\nu]}=F^{\mu\alpha, \nu} + F^{\nu\mu,\alpha} + F^{\alpha\nu,\mu}=0$$ stems from Maxwell equations.
I haven't been able to find this anywhere on ...
- 21
4
votes
2
answers
578
views
Maxwell's equations with differential form formalism
I've been reading Sean Carroll's book on GR and I stumbled upon an exercise on EM using $p$-forms. I think I've solved the problem correctly but I am having problems with my answers. I'll provide the ...
- 147
2
votes
1
answer
189
views
Dot product of the electric and magnetic field as the contraction of the electromagnetic tensor and its dual
I've see in some examples, e.g. here, that $$-4i\vec{E}\cdot \vec{B}=\tilde{F}_{\mu\nu}F^{\mu\nu}$$
How would you show such a relation? By inserting terms by terms inside this equation I've seen it is ...
- 941
1
vote
0
answers
54
views
Rewriting Maxwell Lagrangian [duplicate]
I'm having some problems with rewriting the Maxwell Lagrangian. The text states, \begin{align}\mathcal{L}&=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_\mu J^\mu \\
&= -\dfrac{1}{2}(\partial_\mu A_\nu)^...
- 11
2
votes
2
answers
2k
views
Expressing Maxwell's equations in tensor form using Electromagnetic field strength tensor [closed]
I have yet another derivation question from Carroll's General Relativity textbook. Given the electromagnetic field strength tensor is of the form: $$ F_{\mu\upsilon} =
\left(
\begin{matrix}
0 & -...
- 187
2
votes
1
answer
1k
views
Expressing Maxwell's equations in tensor notation
I've been teaching myself relativity by reading Sean Carroll's intro to General Relativity textbook, and in the first chapter he discusses special relativity and introduces the concept of tensors, ...
- 187
0
votes
2
answers
48
views
Index manipulation in Lorentz scalars
I have been trying to show that: $ \vec{B}^{2} - \vec{E}^{2} =\frac{1}{2} f^{\mu \nu }f_{\mu \nu}$
where $\vec{B}^{2}$ and $\vec{E}^{2}$ are the square of the magnitude of the magnetic field and ...
- 13
-1
votes
1
answer
45
views
Tensor algebra identity [closed]
In our course we took the following formula:
$$F^\mu{}_\lambda\partial_{\mu}F^{\lambda \nu}=\frac 1 2 F_{\mu \lambda}\partial^{\mu}F^{\lambda \nu} + \frac 1 2F_{\lambda \mu}\partial^{\lambda}F^{\mu \...
- 1,398
0
votes
0
answers
88
views
How compute the expression of electromagnetic tensor explicitly as given here?
I am trying to understand how the second line arrives at the last line of this expression.
For $F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu$
And $F^{\mu\nu} = \partial^\mu A^\nu -\partial^\nu ...
- 13
2
votes
0
answers
65
views
Starting with 1-form potential $A$, derive the relation $F_{\mu\nu}^{\nu}=A_{\nu,\mu}^{,\nu}-A_{\mu,\nu}^{,\nu}=4\pi J_\mu$ [closed]
First the problem in detail should be,
"Starting with 1-form potential $\mathbf{A}$, derive the relation $F_{\mu\nu}^{\;\;\;,\nu}=A_{\nu,\mu}^{\;\;\;\;,\nu}-A_{\mu,\nu}^{\;\;\;\;,\nu}=4\pi J_\mu$...
- 125
2
votes
2
answers
106
views
Integration of tensor to find potential
I have question given as:
$$\partial_k \varphi = -(C_k+ D_{jk}r_j)$$
where $C_k \,\&\, D_{jk}$ are constants and $D_{jk}$ is symmetric and traceless. I have to find $\varphi$.
I am getting : $\...
0
votes
1
answer
217
views
Problem 2.1(b) in Peskin and Schroeder's Introduction to QFT
In this exercise the author claims that adding $\partial_\sigma K^{\sigma \mu \nu}$ does not affect the divergence of $T^{\mu\nu}$. In other words the author claims that $\partial_\mu \partial_\sigma ...
user261609
0
votes
1
answer
70
views
The variation of the Lagrangian density for the canonical energy-momentum tensor
I expanded the Lagrangian to this form
$$ \mathcal{L} = -{1 \over 4} F^{\mu \nu} F_{\mu \nu} = ... = - {1 \over 2} (\partial^{\mu} A^{\nu} \partial_{\mu} A_{\nu} - \partial^{\mu} A^{\nu} \partial_{\nu}...
- 51
3
votes
3
answers
2k
views
Simple derivation of the Maxwell's equations from the Electromagnetic Tensor
Lets start by considering the electromagnetic tensor $F^{\mu \nu}$:
$$F^{\mu \nu}=\begin{bmatrix}0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & ...
- 4,577
2
votes
2
answers
446
views
How can we do this tensor product $F_{\mu \nu}F^{\mu \nu}$?
Iam Studying "Quantization of the electromagnetic field using Quantum Field Theory" by Lahiri and Pal.
But I don't get how they computed action in equation $8.23$.
$$A=-{1\over 4} \int d^4xF_{\mu \...
- 545