All Questions
Tagged with special-relativity tensor-calculus
389
questions
-4
votes
1
answer
66
views
Cannot understand this identity between kronecker and metric tensor [closed]
I'm working on Lorentz generators and I am really not able to understand this relation:
$$\omega_{\rho \sigma} \eta^{\rho\mu} \delta^{\alpha}_{\nu} = \frac{1}{2}\omega_{\rho \sigma} \left(\eta^{\rho\...
- 361
0
votes
0
answers
29
views
Unclear passage in Lorentz generators derivation
It's not clear to me a passage, in the extraction of the generators of Lorentz's group acting on the Minkowksi's space points: we have
\begin{equation*}
\begin{split}
x^{' \alpha} & = \Lambda^{\...
- 361
1
vote
1
answer
91
views
Questions about Lorentz Matrices and Lorentz Metric
(I use the abstract index notation convention in this post)
In $\mathbb{R}^4$, denote the Lorentz Metric as $g_{\mu\nu}=$diag$(-1,1,1,1)$, then we can define the Lorentz Matrices to be all $4\times 4$ ...
- 77
1
vote
1
answer
94
views
Why does the full contraction of a tensor and its (Hodge) dual is a pseudoscalar?
I'm trying to prove that the contraction between a tensor and its dual is a pseudoscalar, while the contraction of the dual with itself is just a scalar. I'm using index notation and every time I try ...
0
votes
1
answer
83
views
How can I calculate the square of Pauli-Lubanski vector in a rest frame?
recently I've been trying to demonstrate that, $$\textbf{W}^2 = -m^2\textbf{S}^2$$ in a rest frame, with $W_{\mu}$ defined as $$W_{\mu} = \dfrac{1}{2}\varepsilon_{\mu\alpha\beta\gamma}M^{\alpha\beta}p^...
- 1
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
0
votes
1
answer
69
views
Doubt about product of four-vectors and Minkowski metric [closed]
Given the Minkowski metric $\eta_{\mu\nu}$
And $\eta^{\mu\nu}\eta_{\mu\nu}$=4
I can write $\eta^{\mu\nu}\eta_{\mu\nu}k^{\mu}k^{\nu}$=$4k^{\mu}k^{\nu}$
But $\eta^{\mu\nu}\eta_{\mu\nu}k^{\mu}k^{\nu}$=$\...
- 13
-1
votes
2
answers
95
views
Confusion with Lorentz indices notation
Apologies in advance if this question has been asked before (if it has, I can't find it).
I am really confused with the right-left index notation of the Lorentz matrix.
In the very first exercise of ...
- 280
0
votes
0
answers
432
views
How to know if a matrix is a (0,2) tensor, a (2,0) tensor or a (1,1) tensor?
For example:
$$
X = \begin{bmatrix}
1 & -1 & 0 & 0 \\
-1 & 0 & 5 & 3 \\
-2 & 1 & 0 & 0 \\
0 & 1 & 0 & 2
...
- 157
1
vote
1
answer
516
views
Contraction of Levi-Civita tensor in Minkowski space-time (or any space-time)
I know that $\varepsilon_{i_1 i_2\cdots i_n}\varepsilon^{i_1 i_2\cdots i_n} = n!$ in Euclid space.
But in Minkowski space-time, the metric tensor is:
$$
\eta_{\mu\nu} = \mathrm{diag}\{-1, 1, 1, 1\}
$$
...
- 157
2
votes
0
answers
124
views
Weinberg's proof that $F^{\alpha \beta}$ is a tensor
I am interested in Weinberg's approach to proving that $F^{\alpha \beta}$ is a tensor in his book Gravitation and Cosmology. He begins by rewritting the Maxwell equations as:
$$\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 ...
1
vote
3
answers
636
views
Difference between upper and lower indices in Einstein notation
Consider a $(2,0)$ tensor $X^{\mu \nu}$ that can be represented in matrix form by:
$$X^{\mu \nu} =
\pmatrix{
a & b & c & d \\
e & f & g & h \\
i & j & k & l \\
m &...
- 337
2
votes
0
answers
74
views
Find the energy-impulse tensor of a fluid of charged dust from the action principle
I have the total action given by:
$$S_{tot} = -\frac{1}{16\pi c}\int d\Omega\ F^{\mu \nu}F_{\mu \nu} + \sum_{i=1}^{N}\bigg(-\frac{q}{c}\int dx_i^\mu A_\mu - mc \int ds_i\bigg) \\= \int d\Omega\ \frac{...
1
vote
1
answer
188
views
Null surfaces in Lorentzian manifold
Null Hypersurface of Lorentzian Manifold: A hypersurface that admits a null-like normal vector field($N^a$) to it. i.e. $g_{ab}N^a N^b=0$ (metric signature$(-1,1,1,1,...)$)
In Minkowski spacetime the ...
- 27