All Questions
Tagged with special-relativity tensor-calculus
57
questions
14
votes
3
answers
4k
views
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
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
7
votes
6
answers
830
views
Deriving $\Lambda^i_{\,j}$ components of the Lorentz transformation matrix
I am trying to follow Weinberg's derivation (in the book Gravitation and Cosmology) of the Lorentz transformation or boost along arbitrary direction. I am having trouble deriving the $\Lambda^i_{\,j}$ ...
- 682
6
votes
1
answer
1k
views
Working with indices of tensors in special relativity
I'm trying to understand tensor notation and working with indices in special relativity. I use a book for this purpose in which $\eta_{\mu\nu}=\eta^{\mu\nu}$ is used for the metric tensor and a vector ...
- 723
5
votes
1
answer
1k
views
Why is not ${(\Lambda^T)^\mu}_\nu = {\Lambda_\nu}^\mu$?
I am following lecture notes on SR. The author writes that the following is equivalent:
$$\Lambda^T\eta\Lambda = \eta \iff \eta_{\mu \nu} {\Lambda^\mu}_\rho{\Lambda^\nu}_\sigma = \eta_{\rho \sigma}. \...
- 1,356
5
votes
2
answers
875
views
"Vectors", i.e. (1,0)-tensors, their definition and motivation for relativity
I'm reading Einstein Gravity in a Nutshell (by Zee) and here he defines a vector as an object which is invariant under coordinate representation; concretely, if in one coordinate representation, $V$, $...
- 249
3
votes
1
answer
1k
views
Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?
The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is:
$$(F^\mu{}_\nu)=\left(\begin{array}{cccc}0&E_x&E_y&E_z\\E_x&0&B_z&-B_y\\E_y&-B_z&0&B_x\\E_z&...
- 88.5k
8
votes
2
answers
3k
views
Understanding the difference between co- and contra-variant vectors
I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear.
One ...
- 1,224
14
votes
1
answer
14k
views
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
8
votes
2
answers
6k
views
The definition of transpose of Lorentz transformation (as a mixed tensor)
In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1)
$${{A^T}_i}^j~=~{A^j}_i.$$
This is extremely confusing for ...
- 962
7
votes
1
answer
699
views
On the Lorentz Group representation [closed]
I am going through the notes on QFT by Srednicki.
When describing fermions, from the very beginning he introduces the Lorentz Group and its algebra, and proves that it is equivalent to two copies of $...
5
votes
1
answer
322
views
Argument of a scalar function to be invariant under Lorentz transformations
I'm trying to prove that a Lorentz scalar object $\rho(k)$ which is a function of a cuadri-vector $k^{\mu}$ can only have a $k^2$ dependency in the argument.
I can imagine that this object has to ...
- 123
2
votes
1
answer
319
views
Proof that 4-potential exists from Gauss-Faraday field equation
This is a problem concerning covariant formulation of electromagnetism.
Given
$$\partial_{[\alpha} F_{\beta\gamma]}~=~ 0 $$
how does one prove that $F$ can be obtained from a 4-potential $A$ such ...
- 626
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
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