All Questions
41
questions
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
3
votes
2
answers
408
views
Self-studying GR. Stuck on Q3.20 in the 3rd edition of Schultz. Orthogonal coordinate transforms in Euclidian space
I am self-studying GR using "A first course in general relativity, 3rd edition". I'm doing my best to be diligent and work though the problems at the end of the chapter. But question 3.20 ...
- 33
2
votes
0
answers
104
views
Wald General Relativity Exercice 4.5 - Derivation of Tensor Calculus Identity Relevant to "Effective Gravitational Stress Tensor"
This is a lot of text so I apologise, its hard to pose this question concisely while still being clear.
In the text, Wald derives to second order deviation from flatness an expression for the "...
- 307
1
vote
3
answers
187
views
Confused about tensor notations of how vector and covectors act on each other
I'm learning/playing around with tensors and somehow got this contradiction,
suppose $\{v_i\}$ and $\{w_i\}$ are basis for a vector space $V$ and $\{ v^i \}$ and $\{w^i\}$ are basis for the dual space ...
- 33
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
-1
votes
1
answer
45
views
Tensor algebra identity [closed]
In our course we took the following formula:
$$F^\mu{}_\lambda\partial_{\mu}F^{\lambda \nu}=\frac 1 2 F_{\mu \lambda}\partial^{\mu}F^{\lambda \nu} + \frac 1 2F_{\lambda \mu}\partial^{\lambda}F^{\mu \...
- 1,398
-1
votes
1
answer
163
views
Question in prove infinitesimal Lorentz transformation is antisymmetric
I know we need prove this property by:
\begin{align*}g_{\rho\sigma} = g_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*}
and
$$\Lambda^\mu_{\ \ \ \nu}=\delta^\mu_{\ \ \ \nu} + \...
-1
votes
1
answer
74
views
Interchange of indices [closed]
How to get the second equation, please? My result is wrong - I still have $(N-1)$. Thank you
- 103
3
votes
3
answers
2k
views
Simple derivation of the Maxwell's equations from the Electromagnetic Tensor
Lets start by considering the electromagnetic tensor $F^{\mu \nu}$:
$$F^{\mu \nu}=\begin{bmatrix}0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & ...
- 4,577
1
vote
0
answers
48
views
Lorentz symmetry and associated currents
The string Lagrangian density is invariant under Lorentz transformations. All terms that appear in this density are of the form $$\eta_{\mu\nu}\frac{\partial X^{\mu}}{\partial\xi^{\alpha}}\frac{\...
- 626
1
vote
3
answers
257
views
Question about inner products of tensors and Einstein summation convention
So I am studying Special Relativity and basic tensor calculus and got stuck at an exercise.
$$F^{\mu \nu}: = \left[ \begin {array}{cccc} 0&-{\it E_x}&-{\it E_y}&-{\it E_z}
\\ {\it E_x}&...
user63248
2
votes
1
answer
113
views
Tensors in a two-dimensional Euclidean plane
Consider a two-dimensional Euclidean plane with coordinates $(x^1,x^2)$
If we define a set of new coordinate $z^1$ and $z^2$
$$z^1=x^1+ix^2$$
$$z^2=x^1-ix^2$$
A question is if a symmetric tensor $T^...
2
votes
3
answers
299
views
Lorentz Velocity Transform With Tensor Notation [closed]
So I'm attempting to prove the Lorentz Velocity tranform:
$${v_x}' =\frac{v_x-u}{1-v_xu/c^2} $$
using tensor notation. In this case obviously $\beta = u/c$ and $\gamma=(1-\beta^2)^{-1/2}$. The ...
- 239
2
votes
1
answer
888
views
Lorentz covariance of Pauli-Lubanski pseudo-vector
The Pauli-Lubanski pseudo-vector is defined as:
$$W_{\mu}=\frac{1}{2}\epsilon_{\mu \nu \lambda \rho}J^{\nu \lambda}P^{\rho}$$
Where the rotation and translation operators transform as:
\begin{align}...
- 1,172
0
votes
1
answer
48
views
Computing the metric in the barred frame using a 2D coordinate transformation
I would like to apply the coordinate transformation $x^{\bar{1}} = 2x^1$, $x^{\bar{2}} = x^2$ in the 2D Cartesian plane. The metric in the barred frame is $g_{\overline{ij}} = \Lambda^i_{\bar{i}} \...
- 259