All Questions
52
questions
-1
votes
1
answer
74
views
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\...
- 99
2
votes
0
answers
81
views
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:
$$...
- 111
0
votes
0
answers
432
views
How to know if a matrix is a (0,2) tensor, a (2,0) tensor or a (1,1) tensor?
For example:
$$
X = \begin{bmatrix}
1 & -1 & 0 & 0 \\
-1 & 0 & 5 & 3 \\
-2 & 1 & 0 & 0 \\
0 & 1 & 0 & 2
...
- 157
1
vote
1
answer
516
views
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:
$$
\eta_{\mu\nu} = \mathrm{diag}\{-1, 1, 1, 1\}
$$
...
- 157
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:
$$X^{\mu \nu} =
\pmatrix{
a & b & c & d \\
e & f & g & h \\
i & j & k & l \\
m &...
- 337
2
votes
0
answers
74
views
Find the energy-impulse tensor of a fluid of charged dust from the action principle
I have the total action given by:
$$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
vote
1
answer
188
views
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 ...
- 27
2
votes
3
answers
627
views
Four-velocities, geodesics and antisymmetry in Christoffel symbols
It might be just a basic confusion, but couldn't find an answer. Given the geodesic equation:
$$\frac{d^{2}x^{\lambda}}{d\tau^{2}}+\Gamma_{\mu\nu}^{\lambda} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\...
- 679
2
votes
0
answers
104
views
Wald General Relativity Exercice 4.5 - Derivation of Tensor Calculus Identity Relevant to "Effective Gravitational Stress Tensor"
This is a lot of text so I apologise, its hard to pose this question concisely while still being clear.
In the text, Wald derives to second order deviation from flatness an expression for the "...
- 307
0
votes
2
answers
53
views
Are these two "notations" the same? [closed]
Say we have a tensor $T^{\sigma\tau}$ and I want to now how it transforms, the transformation coefficients in terms of Lorentz transformation matrices would be: $$T^{\mu'\nu '} = L^{\mu '}{}_{\sigma}L^...
- 11
0
votes
0
answers
50
views
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?
1
vote
1
answer
86
views
General relativity algebraic manipulation help
I'm having difficulty understanding a lot of the fundamentals behind the algebra of general relativity. I have a specific question I'm trying to understand but any pointers about how any of it works ...
- 47
0
votes
0
answers
382
views
Deriving relativistic equations of motion using scalar field stress-energy tensor
Question: Stress energy tensor of a minimally coupled scalar field is $T_{\mu\nu} = \partial_\mu\phi\partial_\nu\phi - \left[\frac{1}{2}(\nabla\phi)^2+V(\phi)\right]g_{\mu\nu}$.
Derive the scalar ...
- 13
1
vote
1
answer
304
views
How to determine sub and superscript for divergence theorem in Minkowski Space?
Suppose we are given a type $(0,2)$ tensor $T_{\mu\nu}$ in a Minkowski space with $(-,+,+,+)$ signature. Consider a closed 3-dimensional hypersurface $\partial \Omega$ which encloses a volume $\Omega$ ...
- 838
0
votes
2
answers
371
views
Riemannian and Weyl tensors as spinor representation
There is the way of converting vector indices to spinor indices, for example, Maxwell stress tensor $F_{[\mu\nu]}$ can be decomposed to $(1,0) \oplus (0,1)$ irreducible representations of $\mathfrak{...
- 5,707