All Questions
8
questions
0
votes
2
answers
151
views
Dimension of a vector space of all tensors of rank $(k,l)$ in 4D
Dual vector space is the set of all linear functionals defined on a given vector space. The vector space and dual vector space is isomorphic and hence have the same dimension. A rank $(k,l)$ tensor is ...
- 310
3
votes
2
answers
397
views
Why is it difficult to define conserved quantities in general relativity as in special relativity?
How exactly are conserved quantities, i.e. mass, energy and momenta of a system computed in special relativity and why doesn't it work in general relativity? I know that the curvature of the spacetime ...
- 113
4
votes
1
answer
454
views
Notation of Mixed Tensors: Risk of Confusing Index Positions?
The convention for notating indices of a tensor is to write a contravariant index superscript and a covariant index subscript. If one has a pure contravariant or a pure covariant tensor of $2$nd order,...
- 43
3
votes
4
answers
2k
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4-Vector Definition
In most places I've looked, I see that 4-vectors are defined as 4-element vectors that transform like the 4-position under lorentz transformation. This is typically accompanied by generally,
$$\...
- 255
0
votes
0
answers
103
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Contracted product of matrices with Lorentz indices
Let $\mathbf{A}_\mu$ and $\mathbf{B}_\mu$ be two matrix-valued spacetime vectors, i.e. $(A_\mu)_{ab}$ and $(B_\mu)_{ab}$ and let $\mathbf{C}$ be a matrix in the same space (external to spacetime), i.e....
user20250
-1
votes
2
answers
1k
views
Off-diagonal terms in metric for 4D space-time [closed]
Consider a delta between two events in 4D space-time written as a 4-vector, $x^\mu=(dt, dR)$. The time $dt$ is a scalar difference in time. The 3-vector $dR$ points some direction in space. One ...
- 81
4
votes
1
answer
310
views
Proper time along path in Minkowski Space
Consider the path $x^\mu(u)$ in Minkowski space; such that:
$$t = \frac{a}{c} \sinh(u) , \quad x = a \cosh(u) ,\quad y = 0 ,\quad z = 0 $$
where $a$ is a positive constant and $u$ is a parameter
...
- 615
5
votes
4
answers
3k
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How to prove the raising/lowering indices operation?
I've read this related question, though it didn't satisfy me; I hope this complements it. I know that if I contract a covariant tensor ${A_{\alpha\beta}}$ with a vector ${B^\beta}$, I get some other ...
user24999