All Questions
Tagged with special-relativity tensor-calculus
389
questions
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Physical intuition for the Minkowski space?
As the title suggests, I am looking for physical intuition to better understand the Minkowski metric.
My original motivation is trying to understand the necessity for distinguishing between co-variant ...
-1
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1
answer
74
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Is there an "obvious" reason for why the second derivative of an antisymmetric tensor with respect to coordinates over both of its indices equal to 0?
It was kind of difficult to word the title so I'll restate the question here. My professor took it almost as a given that
$$\frac{\partial T^{\mu\nu}}{\partial X^{\mu}\partial X^\nu} = 0$$
If $T^{\mu\...
5
votes
2
answers
251
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Do I sum over these indices?
Question 2.11 from A First Course In General Relativity, 3rd Edition by Bernard Schutz, asks the reader to verify the following equation:
$$ \Lambda^\nu_\beta(v) \Lambda^\beta_\alpha(-v) = \delta^\...
0
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1
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71
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On covariant form of Lorentz equation
The non-relativistic version of Lorentz equation has the form
$$m\frac{d\vec{v}}{dt}=q(\vec{E}+\vec{v}\times\vec{B}) $$
Where $\vec{v}, \vec{E}, \vec{B}$ refers to the velocity of charged particle, ...
1
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2
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262
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Determinant of Rank-2 Tensor using Levi-Civita notation
In my Professor's notes on Special Relativity, the determinant of a rank-two tensor $[T]$ (a $4\times 4$ matrix, basically) is given using the Levi Civita Symbol as: $$T=-\epsilon_{\mu\nu\rho\lambda}T^...
0
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0
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55
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Degrees of freedom in stress-energy tensor
The stress-energy tensor has 16 components, but this question is only about the 9 components $T^{ij}$ with $i,j=1,2,3$. According to Wikipedia, these components are defined as follows:
The components ...
2
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0
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43
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Fundamental invariants of a Lorentz tensor
As answered in this question, an antisymmetric tensor on 4D Minkowski space has two Lorentz-invariant degrees of freedom. These are the two scalar combinations of the electromagnetic tensor (as proven ...
2
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How do Maxwell's equations follow from the action of Lorentz generators on field strength?
Following Warren Siegel's book on Field theory (pg. 223), one might derive the action of Lorentz generators $S_{ab}$ on an antisymmetric 2-tensor field strength $F_{cd}$ which arises for example in ...
2
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0
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81
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Help understanding Gauss' and Stoke's Theorem in Minkowski space in index notation
My professor said that it's easy to generalize to these definitions of Stoke's and Gauss' theorem from the 3 dimensional versions but didn't say much else. He threw the following on the chalk board:
$$...
0
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1
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51
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Trouble understanding tensor notation for relativistic transformations
For $x^\mu$ with $\mu_0=t, \mu_i = x^i$ and $\eta_{\mu v}$ the metric tensor with diagonal $(-1,1,1,1)$ and zeros elsewhere, the condition for equivalence of inertial frames is stated as for some &...
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0
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38
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Extrinsic curvature of constant time hypersurfaces in Minkowski
Along the geodesic of a stationary observer in Minkowski spacetime we have the following tangent vector
$$t^\mu = (1,0,0,0)$$
We have that hypersurfaces of constant time along this are just 3D ...
2
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1
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191
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Dot product of the electric and magnetic field as the contraction of the electromagnetic tensor and its dual
I've see in some examples, e.g. here, that $$-4i\vec{E}\cdot \vec{B}=\tilde{F}_{\mu\nu}F^{\mu\nu}$$
How would you show such a relation? By inserting terms by terms inside this equation I've seen it is ...
6
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1
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300
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Inverse of anti-symmetric rank 4 tensors?
I am trying to find an inverse of a tensor of the form
$$M_{\mu\nu\rho\sigma}$$ such that $M$ is anti-symmetric in the $(\mu, \nu)$ exchange and $(\rho, \sigma)$ exchange. The inverse should be such ...
3
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0
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68
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Counting independent components of Lorentz tensor
Say I have Lorentz tensors $A^{\mu\nu}$ and say this Lorentz tensor is symmetric under $\mu \Leftrightarrow \nu$ and there are only $p^\mu$ and $q^\mu$ as the physical Lorentz vectors involved. If so, ...
3
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2
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408
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Self-studying GR. Stuck on Q3.20 in the 3rd edition of Schultz. Orthogonal coordinate transforms in Euclidian space
I am self-studying GR using "A first course in general relativity, 3rd edition". I'm doing my best to be diligent and work though the problems at the end of the chapter. But question 3.20 ...
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1
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66
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Cannot understand this identity between kronecker and metric tensor [closed]
I'm working on Lorentz generators and I am really not able to understand this relation:
$$\omega_{\rho \sigma} \eta^{\rho\mu} \delta^{\alpha}_{\nu} = \frac{1}{2}\omega_{\rho \sigma} \left(\eta^{\rho\...
0
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0
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29
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Unclear passage in Lorentz generators derivation
It's not clear to me a passage, in the extraction of the generators of Lorentz's group acting on the Minkowksi's space points: we have
\begin{equation*}
\begin{split}
x^{' \alpha} & = \Lambda^{\...
1
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1
answer
91
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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 $4\times 4$ ...
1
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1
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94
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Why does the full contraction of a tensor and its (Hodge) dual is a pseudoscalar?
I'm trying to prove that the contraction between a tensor and its dual is a pseudoscalar, while the contraction of the dual with itself is just a scalar. I'm using index notation and every time I try ...
0
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1
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83
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How can I calculate the square of Pauli-Lubanski vector in a rest frame?
recently I've been trying to demonstrate that, $$\textbf{W}^2 = -m^2\textbf{S}^2$$ in a rest frame, with $W_{\mu}$ defined as $$W_{\mu} = \dfrac{1}{2}\varepsilon_{\mu\alpha\beta\gamma}M^{\alpha\beta}p^...
0
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1
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209
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Formulation of the Bianchi identity in EM
I'm trying to understand, as a self learner, the covariant formulation of Electromagnetism. In particular I've been stuck for a while on the Bianchi identity. As I've come to understand, when we ...
0
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1
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69
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Doubt about product of four-vectors and Minkowski metric [closed]
Given the Minkowski metric $\eta_{\mu\nu}$
And $\eta^{\mu\nu}\eta_{\mu\nu}$=4
I can write $\eta^{\mu\nu}\eta_{\mu\nu}k^{\mu}k^{\nu}$=$4k^{\mu}k^{\nu}$
But $\eta^{\mu\nu}\eta_{\mu\nu}k^{\mu}k^{\nu}$=$\...
-1
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2
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95
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Confusion with Lorentz indices notation
Apologies in advance if this question has been asked before (if it has, I can't find it).
I am really confused with the right-left index notation of the Lorentz matrix.
In the very first exercise of ...
0
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0
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432
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How to know if a matrix is a (0,2) tensor, a (2,0) tensor or a (1,1) tensor?
For example:
$$
X = \begin{bmatrix}
1 & -1 & 0 & 0 \\
-1 & 0 & 5 & 3 \\
-2 & 1 & 0 & 0 \\
0 & 1 & 0 & 2
...
1
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1
answer
516
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Contraction of Levi-Civita tensor in Minkowski space-time (or any space-time)
I know that $\varepsilon_{i_1 i_2\cdots i_n}\varepsilon^{i_1 i_2\cdots i_n} = n!$ in Euclid space.
But in Minkowski space-time, the metric tensor is:
$$
\eta_{\mu\nu} = \mathrm{diag}\{-1, 1, 1, 1\}
$$
...
2
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0
answers
124
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Weinberg's proof that $F^{\alpha \beta}$ is a tensor
I am interested in Weinberg's approach to proving that $F^{\alpha \beta}$ is a tensor in his book Gravitation and Cosmology. He begins by rewritting the Maxwell equations as:
$$\frac{\partial}{\...
1
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1
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87
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Electromagnetic tensor and its components
I'm dealing with the covariant formulation of electromagnetism and I've come across the Electromagnetic tensor after learning a bit about the covariant notation.
In particular I've problems ...
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3
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636
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Difference between upper and lower indices in Einstein notation
Consider a $(2,0)$ tensor $X^{\mu \nu}$ that can be represented in matrix form by:
$$X^{\mu \nu} =
\pmatrix{
a & b & c & d \\
e & f & g & h \\
i & j & k & l \\
m &...
2
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0
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74
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Find the energy-impulse tensor of a fluid of charged dust from the action principle
I have the total action given by:
$$S_{tot} = -\frac{1}{16\pi c}\int d\Omega\ F^{\mu \nu}F_{\mu \nu} + \sum_{i=1}^{N}\bigg(-\frac{q}{c}\int dx_i^\mu A_\mu - mc \int ds_i\bigg) \\= \int d\Omega\ \frac{...
1
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1
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188
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Null surfaces in Lorentzian manifold
Null Hypersurface of Lorentzian Manifold: A hypersurface that admits a null-like normal vector field($N^a$) to it. i.e. $g_{ab}N^a N^b=0$ (metric signature$(-1,1,1,1,...)$)
In Minkowski spacetime the ...