All Questions
Tagged with special-relativity tensor-calculus
389
questions
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72
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Transforming the field strength tensor - Why do we need to use $\Lambda^T$, instead of a row times row multiplication?
The transformation law is
\begin{align}
F'^{\mu\nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta} = {\Lambda^{\mu}}_{\alpha} F^{\alpha \beta} {\Lambda^{\nu}}_{\beta}
\end{align}
...
- 6,763
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votes
1
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65
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Fraction with components of Lorentz transformation
I want to show how partial derivative transforms under a Lorentz transformation.
Since the partial derivative has a fixed definition with respect to the $x$-coordinate it stays unchanged: $\partial_\...
- 425
2
votes
1
answer
171
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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
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2
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53
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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
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1
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80
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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
0
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0
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50
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How is tensor analysis useful to Relativity? [duplicate]
How does the knowledge of tensor analysis and Differential Geometry help us understand the equations of General and Special Relativity?
3
votes
1
answer
109
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Proof that a Lorentz-invariant scalar function can only depend on scalar products
If we have a Lorentz-invariant scalar function $f$ of a single four-vector $x^{\mu}$ we can show that $f$ can only depend on $x^2$ (see Argument of a scalar function to be invariant under Lorentz ...
- 1,129
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
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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
1
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0
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66
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Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory
1) Introduction
From standard tensor calculus, here restricted to Minkowski spacetime, we learned that:
A scalar field is a object that transforms as:
$$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$
A ...
- 3,085
2
votes
2
answers
265
views
Determinant of the inverse of a Lorentz transformation
In many text book (Ashok Das Quantum Field Theory) $$(\Lambda^T)_\nu{}^\mu=\Lambda^\mu{}_\nu$$ that gives $\Lambda^T$ = $\Lambda^{-1}$, where $\Lambda$ is Lorentz Transformation matrix. However, this ...
- 45
1
vote
1
answer
219
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How to contract spinor indices?
In normal vector representation, vectors can be contracted as follows:
$$v^\mu v_\mu$$
with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of ...
- 1,324
6
votes
1
answer
527
views
Tensor manipulations in Landau & Lifschitz "Classical theory of fields"
Landau & Lifshitz "Classical theory of fields" section 6 p. 19 define:
$$
df^{ik} = dx^i dx'^k - dx^k dx'^i
$$
and
$$
df^{*ik}=\frac{1}{2} \; \epsilon^{iklm}df_{lm} \tag{6.11}
$$
and ...
- 63
1
vote
1
answer
151
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Is four-vector product always Lorentz invariant?
Let's say we have two four-vectors $a^{\mu}$ and $b^{\nu}$.Is it always true that any combination of those 4-vectors (1-rank tensors) multiplied together will yield an invariant quantity (0-rank ...
- 41
-1
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2
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61
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Confusion about raising and lowering indices
Is it possible to take the following expression:
$$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