All Questions
Tagged with special-relativity tensor-calculus
389
questions
4
votes
2
answers
851
views
Lorentz transform of Levi-Civita Symbol
I was reading about Lorentz transformations and frequently I hear the notion of Lorentz transforming quantities like $\epsilon^{\mu \nu \rho \sigma}$. But I have never heard an explanation as to why ...
- 149
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
0
votes
2
answers
61
views
Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$
We can show that the contraction of some arbitrary $2\times2$ matrix $A_{\mu}^{\ \lambda}$ with the Levi-Civita symbol is once again antisymmetric
\begin{align*}
\varepsilon^{\mu\nu}A_\mu^{\ \lambda} ...
- 2,068
0
votes
1
answer
486
views
Show that the contraction of a covector and a vector is Lorentz invariant
I just got Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity a couple of weeks ago, and I have resolved to go through the entire book. In the first chapter, he prompts the ...
- 187
5
votes
2
answers
451
views
What is the idea behind 2-spinor calculus?
In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
- 7,980
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
1
vote
1
answer
86
views
General relativity algebraic manipulation help
I'm having difficulty understanding a lot of the fundamentals behind the algebra of general relativity. I have a specific question I'm trying to understand but any pointers about how any of it works ...
- 47
0
votes
1
answer
56
views
Orthogonality of a 2nd Rank Tensor and its Dual in Lorentz Space
I am trying to show the orthogonality of the 2nd rank anti-symmetric tensor $A^{ik}$ and its dual $A^{*ik}.$ Using a text's definition of the dual as $e^{iklm}*A_{lm}/2$, I have tried to pair ...
user325561
1
vote
0
answers
36
views
Completely antisymmetric unit tensor of fourth rank in different 4D coordinate systems [duplicate]
I am reading Landau's Classical Theory of Fields. On page 18, it is said that the completely antisymmetric unit tensor of fourth rank $\varepsilon^{iklm}$ is defined as the same in all coordinate ...
- 613
-1
votes
1
answer
45
views
Tensor algebra identity [closed]
In our course we took the following formula:
$$F^\mu{}_\lambda\partial_{\mu}F^{\lambda \nu}=\frac 1 2 F_{\mu \lambda}\partial^{\mu}F^{\lambda \nu} + \frac 1 2F_{\lambda \mu}\partial^{\lambda}F^{\mu \...
- 1,398
0
votes
1
answer
56
views
Space-time metric in tensor form
In space time metric in tensor form:
The distance is given by $$ds^2=c^2dt-dx^2-dy^2-dz^2$$
Which in tensor form is: $$ds^2=\sum_{\alpha \beta}g_{\alpha \beta}dx^\alpha dx^\beta$$
Using Einstein ...
0
votes
1
answer
80
views
About general covariance
\begin{equation} u^{\mu}=\frac{d}{d\tau}x^{\mu} \end{equation}
\begin{equation} \partial_{\lambda}(u_{\nu} u^{\nu}) = (\partial_{\lambda}u_{\nu}) u^{\nu} + u_{\nu}(\partial_{\lambda}u^{\nu}) = 0 \end{...
- 31
5
votes
3
answers
836
views
Decomposition of product of two antisymmetric Lorentz tensors
Suppose I have a tensor $A_{\mu\nu}$ in the $(3,1)\oplus (1,3)$ representation of the Lorentz group where $(a,b) =(2s_a+1,2s_b+1)$. I was wondering on how to decompose explictly in terms of tensors ...
- 2,263
2
votes
2
answers
106
views
Tensorial direct product
The direct product of two tensors is also a tensor. I would like to know if we can write a tensor as a product of only two other tensors. For example, how to find $A^{\mu}$ and $ B^{\nu}$ so that $\...
- 31
0
votes
0
answers
382
views
Deriving relativistic equations of motion using scalar field stress-energy tensor
Question: Stress energy tensor of a minimally coupled scalar field is $T_{\mu\nu} = \partial_\mu\phi\partial_\nu\phi - \left[\frac{1}{2}(\nabla\phi)^2+V(\phi)\right]g_{\mu\nu}$.
Derive the scalar ...
- 13