All Questions
61
questions
0
votes
1
answer
69
views
Doubt about product of four-vectors and Minkowski metric [closed]
Given the Minkowski metric $\eta_{\mu\nu}$
And $\eta^{\mu\nu}\eta_{\mu\nu}$=4
I can write $\eta^{\mu\nu}\eta_{\mu\nu}k^{\mu}k^{\nu}$=$4k^{\mu}k^{\nu}$
But $\eta^{\mu\nu}\eta_{\mu\nu}k^{\mu}k^{\nu}$=$\...
- 13
1
vote
3
answers
636
views
Difference between upper and lower indices in Einstein notation
Consider a $(2,0)$ tensor $X^{\mu \nu}$ that can be represented in matrix form by:
$$X^{\mu \nu} =
\pmatrix{
a & b & c & d \\
e & f & g & h \\
i & j & k & l \\
m &...
- 337
2
votes
1
answer
110
views
Four-vector and Notation significance [closed]
As the title suggest, this has to do, on the most part, with four vector notation. I have a series of questions, the majority, related to this topic:
1- If we assume a lorentz boost in the x direction ...
- 1,398
2
votes
2
answers
132
views
Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?
On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
- 737
0
votes
2
answers
254
views
Why is Lorentz Transformation defined with one super and one sub index?
I came across the Lorentz transformation in tensor form, usually written as
$$\Lambda ^\mu _{\nu}$$
I understand that the first index usually corresponds to rows and the second to columns, and while I ...
- 538
0
votes
2
answers
53
views
Are these two "notations" the same? [closed]
Say we have a tensor $T^{\sigma\tau}$ and I want to now how it transforms, the transformation coefficients in terms of Lorentz transformation matrices would be: $$T^{\mu'\nu '} = L^{\mu '}{}_{\sigma}L^...
- 11
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
2
votes
2
answers
265
views
Determinant of the inverse of a Lorentz transformation
In many text book (Ashok Das Quantum Field Theory) $$(\Lambda^T)_\nu{}^\mu=\Lambda^\mu{}_\nu$$ that gives $\Lambda^T$ = $\Lambda^{-1}$, where $\Lambda$ is Lorentz Transformation matrix. However, this ...
- 45
-1
votes
2
answers
61
views
Confusion about raising and lowering indices
Is it possible to take the following expression:
$$U^\mu U^v\partial_\mu\partial_v$$
Where $U$ is the four-velocity, and simplify it the following way?:
$$U^\mu U^v \eta_{\mu v}\partial^v\partial_v =c^...
- 633
0
votes
1
answer
56
views
Space-time metric in tensor form
In space time metric in tensor form:
The distance is given by $$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$$
Using Einstein ...
1
vote
1
answer
166
views
Index convention according to Schwartz
On page 15 of his QFT book, Schwartz writes that all the following contractions are equivalent as long as the flat metric is used: $$v^\mu w_\mu=v_\mu w^\mu=v_\mu w_\mu=v^\mu w^\mu.$$ Isn't this false?...
- 89
1
vote
2
answers
323
views
Transpose of a 2x2 Tensor
This question arises after reading through several Stack Exchange posts and after a long chat with another user in a previous question I asked about this topic. The following "contradiction" ...
- 2,599
1
vote
3
answers
1k
views
What does it mean to contract the indices of a Lorentz matrix?
The metric tensor in SR obeys the transformation law (I am using Schutz's bar notation for different frame indices):
$$\eta_{\bar{\alpha}\bar{\beta}}=\Lambda^\mu_{~\bar{\alpha}}~\Lambda^\nu_{~\bar{\...
- 1,063
6
votes
4
answers
650
views
Ordering of Indices in $\Lambda^\mu_{\space\space\nu}$ [duplicate]
I am having some questions on the ordering of indices that are both upstairs and downstairs. Let's take an example: $\Lambda^\mu_{\space\space\nu}$ is a Lorentz transfom if the following equation is ...
- 1,335
1
vote
1
answer
433
views
Tensors index notation Symmetrisation/antisymmetrisation
I am having trouble figuring out what is the development of the following equation due to its notation
Its a definition for the <> notation, and all that was previously stated was that $u^{(ab)...
- 154