All Questions
22
questions
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
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
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
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
171
views
Why not define tensors under Galilean or Poincare transformations?
I have seen vectors (and tensors, in general) defined under rotations,
$$V^i=R^i_{~j}V^j$$
and under Lorentz transformations,
$$V^{\prime\mu}=\Lambda^\mu_{~~\nu}V^\nu$$
where $R,\Lambda$ are the ...
- 11.8k
0
votes
1
answer
80
views
About the Lorentz transformation in Spacetime and Geometry
In Spacetime and Geometry by Sean Carroll, page 18, he said
"We will therefore introduce a somewhat subtle notation, by using the same symbol for both matrices, just with primed and unprimed ...
- 77
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
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
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
0
votes
1
answer
1k
views
Proof that Kronecker's Delta is invariant under Lorentz transformation
There's an exercise in my book that says
"Prove that the Kronecker $\delta$ is invariant under Lorentz transformations".
The solution says that from the property
$$\Lambda^Tg\Lambda=g$$ of ...
- 693
3
votes
2
answers
316
views
Lorentz boost matrix
In my textbook, there is a proof that the dot product of 2 four-vectors is invariant under a Lorentz transformation. While I understood most of the derivation (I am a beginner and we haven't done any ...
- 1,398
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
4
votes
3
answers
5k
views
Proving that the Minkowski metric tensor is invariant under Lorentz transformations
I'm studying special relativity.
A general Lorentz transformation is defined by $\Lambda^T\eta\Lambda=\eta$.
Now,
\begin{align}
\eta'^{\mu\nu} &= \Lambda^\mu_{\;\;\alpha}\Lambda^\nu_{\;\;\beta}\...
- 1,951
1
vote
2
answers
1k
views
Lorentz invariance of the Lorentz force law
I'm self-studying Friedman and Susskind's book Special Relativity and Classical Field Theory. The following question popped up while reading section 6.3.4 Lorentz Invariant Equations.
In this Lecture, ...
- 1,951