All Questions
22
questions
1
vote
3
answers
187
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Confused about tensor notations of how vector and covectors act on each other
I'm learning/playing around with tensors and somehow got this contradiction,
suppose $\{v_i\}$ and $\{w_i\}$ are basis for a vector space $V$ and $\{ v^i \}$ and $\{w^i\}$ are basis for the dual space ...
- 33
1
vote
1
answer
173
views
Is there only one convention to define the electromagnetic field tensor?
I know that the electromagnetic field tensor depends on which metric is used. For example wikipedia uses the $(+---)$ sign convention, but in the Griffiths we have the $(-+++)$ sign convention.
That's ...
- 1,100
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
vote
2
answers
322
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
0
votes
1
answer
181
views
What is the convention for tensor indices for matrices?
The Lorenz-Transformation of the EM-Tensor F is given by the equation
$$ F'^{\mu \nu} = \Lambda^{\mu}_{\ \ \rho} \Lambda^\nu_{\ \ \sigma} F^{\rho \sigma}$$
Then it says that this is equivalent to the ...
- 986
0
votes
0
answers
53
views
Which basis 2-form elements represent positive traversals in Minkowski 4-space?
I'm certain some of this relies on arbitrary choice, for even in Euclidean 3-space, there is no a priori preferred choice of left versus right hand coordinates. In fact according to Einstein:
There ...
- 2,015
0
votes
0
answers
184
views
Contravariant rank-2 tensor transformation in index notation
I'm slightly confused about the placement of upper and lower indices for the transformation of a rank-2 contravariant tensor.
A contravariant rank-2 tensor transforms as $$M' = \Lambda M \Lambda^{T}$$....
- 361
4
votes
2
answers
441
views
Tensor and Matrices
Suppose that we are dealing with the following matrix:
$$A=\begin{bmatrix}a_{00} & a_{01} \\ a_{10} & a_{11}\end{bmatrix}$$
but I don't want to use matrix notation, insted I want to use tensor ...
- 4,577
1
vote
2
answers
346
views
The definition of quantities in special relativity as upper-index or lower-index
My question is for Minkowski metric $\eta_{\alpha\beta}=\mathrm{diag}(1,-1,-1,-1)$. While defining quantities like the four potential, four momentum or even space-time interval for that matter, why do ...
- 1,212
1
vote
0
answers
267
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Difference between position of indexes in tensor notation (SR) [duplicate]
I am learning SR, and don't understand the difference between the following notations of a Lorentz transformation $\Lambda$
$$\Lambda_{\mu\nu} , \Lambda_{\mu}\ ^\nu , \Lambda^{\mu\nu}$$
I know that ...
- 538
4
votes
1
answer
515
views
Superscript vs. subscript indices in Euler-Lagrange equation in relativistic field theories
In the literature of field theories in flat spacetime, both forms of Euler Lagrange equation are used.
e.g Consider a real scalar field $\phi$
$$\partial_\mu\frac{\partial L}{\partial \phi,_\mu}=\...
- 399
2
votes
2
answers
277
views
Transferring from vectors and tensors in $(-+++)$ signature to $(+---)$ signature?
Minkowski space with the signature $(+---)$ can be described by $\Bbb{C}^{1,3}$ whilst with the signature $(-+++)$ by $\Bbb{C}^{3,1}$ (I am using $\Bbb{C}$ instead of $\Bbb{R}$ to allow for Wick ...
2
votes
1
answer
94
views
Notation used in Tensor analysis
I have come across the following notations for matrices in relativity: $X_{\nu}{}^{\mu}$ and $X^{\mu}{}_{\nu}$. How are these things related?
- 592
0
votes
1
answer
503
views
Is the correct way to Lorentz transform a tensor $T$ by writing $\Lambda T \Lambda^T$ or $\Lambda^T T \Lambda$?
Is the correct way to Lorentz transform a tensor T by writing $\Lambda T \Lambda^T$ or $\Lambda^T T \Lambda$? Is it ${\Lambda_\rho}^\mu{\Lambda_\sigma}^\nu T_{\mu \nu}$ or ${\Lambda^\mu}_\rho{\Lambda^\...
- 1,356
5
votes
1
answer
1k
views
Why is not ${(\Lambda^T)^\mu}_\nu = {\Lambda_\nu}^\mu$?
I am following lecture notes on SR. The author writes that the following is equivalent:
$$\Lambda^T\eta\Lambda = \eta \iff \eta_{\mu \nu} {\Lambda^\mu}_\rho{\Lambda^\nu}_\sigma = \eta_{\rho \sigma}. \...
- 1,356