All Questions
68
questions
0
votes
1
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71
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On covariant form of Lorentz equation
The non-relativistic version of Lorentz equation has the form
$$m\frac{d\vec{v}}{dt}=q(\vec{E}+\vec{v}\times\vec{B}) $$
Where $\vec{v}, \vec{E}, \vec{B}$ refers to the velocity of charged particle, ...
- 1,752
2
votes
0
answers
43
views
Fundamental invariants of a Lorentz tensor
As answered in this question, an antisymmetric tensor on 4D Minkowski space has two Lorentz-invariant degrees of freedom. These are the two scalar combinations of the electromagnetic tensor (as proven ...
- 21
2
votes
0
answers
75
views
How do Maxwell's equations follow from the action of Lorentz generators on field strength?
Following Warren Siegel's book on Field theory (pg. 223), one might derive the action of Lorentz generators $S_{ab}$ on an antisymmetric 2-tensor field strength $F_{cd}$ which arises for example in ...
- 785
2
votes
1
answer
191
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
0
votes
1
answer
209
views
Formulation of the Bianchi identity in EM
I'm trying to understand, as a self learner, the covariant formulation of Electromagnetism. In particular I've been stuck for a while on the Bianchi identity. As I've come to understand, when we ...
- 540
2
votes
0
answers
124
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Weinberg's proof that $F^{\alpha \beta}$ is a tensor
I am interested in Weinberg's approach to proving that $F^{\alpha \beta}$ is a tensor in his book Gravitation and Cosmology. He begins by rewritting the Maxwell equations as:
$$\frac{\partial}{\...
- 157
1
vote
1
answer
87
views
Electromagnetic tensor and its components
I'm dealing with the covariant formulation of electromagnetism and I've come across the Electromagnetic tensor after learning a bit about the covariant notation.
In particular I've problems ...
5
votes
3
answers
2k
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Why do we need to make a tensor for the electromagnetic field?
I was wondering why we need the electromagnetic field tensor $F_{\mu\nu}$ to be a tensor and why can't we work with the electric and magnetic fields while dealing with the electromagnetic field ...
- 53
13
votes
1
answer
318
views
Is there a Lorentz invariant electromagnetic quadrupole moment tensor?
I'm familiar with the electric and magnetic quadrupole moment tensors. However, I'm bothered that these objects are tensors only in the sense of spatial rotations. After all, Maxwell's equations and ...
- 1,900
0
votes
1
answer
72
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Transforming the field strength tensor - Why do we need to use $\Lambda^T$, instead of a row times row multiplication?
The transformation law is
\begin{align}
F'^{\mu\nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta} = {\Lambda^{\mu}}_{\alpha} F^{\alpha \beta} {\Lambda^{\nu}}_{\beta}
\end{align}
...
- 6,763
1
vote
1
answer
173
views
Is there only one convention to define the electromagnetic field tensor?
I know that the electromagnetic field tensor depends on which metric is used. For example wikipedia uses the $(+---)$ sign convention, but in the Griffiths we have the $(-+++)$ sign convention.
That's ...
- 1,100
2
votes
1
answer
1k
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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
-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
1
vote
0
answers
239
views
Lorentz Transform of the Electromagnetic Field
I'm trying to prove that the Transformation of the electric and magnetic field under a boost of arbitrary direction is
$$
E_i^{\prime\parallel}=E_i^\parallel\qquad B_i^{\prime\parallel}=B_i^\parallel
$...
0
votes
1
answer
813
views
Lorentz Transforming the electric field and the change of its directions
This is a two part question about the Lorentz transformation of the electromagnetic field, the electric field to specific. The Lorentz transformation will be a simple boost in the x direction.
first ...
- 97