All Questions
Tagged with special-relativity tensor-calculus
389
questions
24
votes
4
answers
4k
views
Why do Maxwell's equations contain each of a scalar, vector, pseudovector and pseudoscalar equation?
Maxwell's equations, in differential form, are
$$\left\{\begin{align}
\vec\nabla\cdot\vec{E}&=~\rho/\epsilon_0,\\
\vec\nabla\times\vec B~&=~\mu_0\vec J+\epsilon_0\mu_0\frac{\partial\vec E}{\...
- 9,675
22
votes
5
answers
7k
views
Inverse and Transpose of Lorentz Transformation
I've seen this question asked a few times on Stack Exchange, but I'm still quite confused why the following "contradiction" seems to arise.
By definition:
$(\Lambda^T)^{\mu}{}_{\nu} = \...
- 736
18
votes
2
answers
741
views
In relativity, can/should every measurement be reduced to measuring a scalar?
Different authors seem to attach different levels of importance to keeping track of the exact tensor valences of various physical quantities. In the strict-Catholic-school-nun camp, we have Burke 1980,...
user4552
15
votes
3
answers
5k
views
The signature of the metric and the definition of the electromagnetic tensor
I've read the definition of the electromagnetic field tensor to be
\begin{equation}F^{\mu\nu}\equiv\begin{pmatrix}0&E_x&E_y&E_z\\-E_x&0&B_z&-B_y\\-E_y&-B_z&0&B_x\\-...
user24999
15
votes
3
answers
2k
views
Why are the metric and the Levi-Civita tensor the only invariant tensors?
The only numerical tensors that are invariant under some relevant symmetry group (the Euclidean group in Newtonian mechanics, the Poincare group in special relativity, and the diffeomorphism group in ...
- 48.4k
14
votes
2
answers
2k
views
What exactly does it mean for a scalar function to be Lorentz invariant?
If I have a function $\ f(x)$, what does it mean for it to be Lorentz invariant? I believe it is that $\ f( \Lambda^{-1}x ) = f(x)$, but I think I'm missing something here.
Furthermore, if $g(x,y)$ ...
- 3,719
14
votes
1
answer
14k
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What is the Difference between Lorentz Invariant and Lorentz Covariant? [duplicate]
Like my title, I sometimes see that my books says something is Lorentz invariant or Lorentz covariant. What's the difference between these two transformation properties? Or are they just the same ...
- 1,161
14
votes
3
answers
4k
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Staggered Indices ($\Lambda^\mu{}_\nu$ vs. $\Lambda_\mu{}^\nu$) on Lorentz Transformations
I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes.
What are the respective meanings of $\Lambda^\mu{}_\nu$ as ...
- 141
13
votes
3
answers
2k
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Why, when going from special to general relativity, do we just replace partial derivatives with covariant derivatives?
I've come across several references to the idea that to upgrade a law of physics to general relativity all you have to do is replace any partial derivatives with covariant derivatives.
I understand ...
- 1,933
13
votes
1
answer
2k
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Looking for physical intuition into the Electromagnetic Tensor:
I have done some work with the electromagnetic tensor and I'm fairly good at manipulating it and using it to transform the Maxwell Equations into tensored forms.
Admittedly though, I have no physical ...
- 431
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
11
votes
4
answers
3k
views
Is Lorentz transform a tensor?
I am confused whether Lorentz transform is a tensor or not, since it is a linear transform. If yes how can I verify that?
- 1,348
11
votes
3
answers
12k
views
Raising and Lowering Indices of Levi-Civita Symbols (+---) metric?
There are a few questions here regarding upper and lower indices of the levi-civita symbol but I haven't been able to find an answer to my exact question. I am working on a problem where I start out ...
- 14.4k
9
votes
2
answers
478
views
If $v_{a \dot{b}}$ transforms like a four-vector, what does $v_{a}^{\dot{b}}$ describe?
The $( \frac{1}{2}, 0)$ representation of the Lorentz group acts on left-chiral spinors $\chi_a$, the $( 0,\frac{1}{2} )$ representation on right-chiral spinors $\chi^{\dot a}$.
The $( \frac{1}{2}, \...
- 1,872
9
votes
1
answer
526
views
Is the distinction between covariant and contravariant objects purely for the convenience of mathematical manipulation?
Two kinds of indices, covariant and contravariant, are introduced in special relativity. This, as far as I understand, is solely for mathematical luxury, i.e. write expressions in a concise, self-...
- 26.8k