All Questions
Tagged with special-relativity tensor-calculus
389
questions
2
votes
3
answers
627
views
Four-velocities, geodesics and antisymmetry in Christoffel symbols
It might be just a basic confusion, but couldn't find an answer. Given the geodesic equation:
$$\frac{d^{2}x^{\lambda}}{d\tau^{2}}+\Gamma_{\mu\nu}^{\lambda} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\...
- 679
2
votes
0
answers
104
views
Wald General Relativity Exercice 4.5 - Derivation of Tensor Calculus Identity Relevant to "Effective Gravitational Stress Tensor"
This is a lot of text so I apologise, its hard to pose this question concisely while still being clear.
In the text, Wald derives to second order deviation from flatness an expression for the "...
- 307
5
votes
3
answers
2k
views
Why do we need to make a tensor for the electromagnetic field?
I was wondering why we need the electromagnetic field tensor $F_{\mu\nu}$ to be a tensor and why can't we work with the electric and magnetic fields while dealing with the electromagnetic field ...
- 53
0
votes
1
answer
63
views
How to convert the following Matrix equation to tensor notation?
Consider the following equation :
$$\Lambda^{-1}\Lambda^T \Lambda=A$$
Here $\Lambda$ are my lorentz transformations such that $\Lambda^T \eta \Lambda=\eta$. $A$ is some matrix.
I know that in terms of ...
- 1,791
1
vote
1
answer
101
views
Question on the spinor Indices, in non-relativistic quantum mechanics
I've caught by a loop of:
Standard texts of Non-Relativistic Quantum Mechanics $\to$ Representation theory of Lie groups and Lie algebras of $SO(3)$ and $SU(2)$ $\to$ Discussions of infinitesimal ...
- 315
0
votes
2
answers
151
views
Dimension of a vector space of all tensors of rank $(k,l)$ in 4D
Dual vector space is the set of all linear functionals defined on a given vector space. The vector space and dual vector space is isomorphic and hence have the same dimension. A rank $(k,l)$ tensor is ...
- 310
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
0
votes
0
answers
561
views
Bilinear covariants of Dirac field
In the book "Advanced quantum mechanics" by Sakurai there is a section (3.5) about bilinear covariants, however i can't really find a definition of these objects, neither in the book nor ...
- 59
2
votes
1
answer
110
views
Four-vector and Notation significance [closed]
As the title suggest, this has to do, on the most part, with four vector notation. I have a series of questions, the majority, related to this topic:
1- If we assume a lorentz boost in the x direction ...
- 1,398
3
votes
2
answers
193
views
Problem with proving the invariance of dot product of two four vectors
I am having a spot of trouble with index manipulation (its not that I am very unfamiliar with this, but I keep losing touch). This is from an electrodynamics course - we're just getting started with 4 ...
- 322
0
votes
1
answer
59
views
Not so trivial indeces in isometries of special relativity
I am trying to understand isometries and how to work with tensors.
I know that in special relativity metric transforms as follows
$$
g_{\alpha^{\prime} \beta^{\prime}}=g_{\alpha \beta} \Lambda_{\...
2
votes
2
answers
132
views
Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?
On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
- 737
0
votes
2
answers
258
views
Product of Lorentz Transformation with metric tensor and inverse metric tensor with different indexes
I am trying to understand the following product:
$$\eta_{\mu\lambda}\eta^{\nu\rho}\Lambda^{\lambda}_\rho.$$
I understand that the first metric lowers the $\lambda$ and changes it for a $\mu$, while ...
- 538
0
votes
2
answers
254
views
Why is Lorentz Transformation defined with one super and one sub index?
I came across the Lorentz transformation in tensor form, usually written as
$$\Lambda ^\mu _{\nu}$$
I understand that the first index usually corresponds to rows and the second to columns, and while I ...
- 538
2
votes
1
answer
116
views
Contravariant Components (Susskind's book)
In his book about SR & classical field theory, Susskind generalizes from the differential of $X'$ (function differential) to any 4-vector. I got stuck there trying to figure out why it is ...
- 21