All Questions
35
questions
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:
$$...
- 111
0
votes
0
answers
38
views
Extrinsic curvature of constant time hypersurfaces in Minkowski
Along the geodesic of a stationary observer in Minkowski spacetime we have the following tangent vector
$$t^\mu = (1,0,0,0)$$
We have that hypersurfaces of constant time along this are just 3D ...
- 49
1
vote
1
answer
87
views
Electromagnetic tensor and its components
I'm dealing with the covariant formulation of electromagnetism and I've come across the Electromagnetic tensor after learning a bit about the covariant notation.
In particular I've problems ...
1
vote
1
answer
188
views
Null surfaces in Lorentzian manifold
Null Hypersurface of Lorentzian Manifold: A hypersurface that admits a null-like normal vector field($N^a$) to it. i.e. $g_{ab}N^a N^b=0$ (metric signature$(-1,1,1,1,...)$)
In Minkowski spacetime the ...
- 27
2
votes
3
answers
627
views
Four-velocities, geodesics and antisymmetry in Christoffel symbols
It might be just a basic confusion, but couldn't find an answer. Given the geodesic equation:
$$\frac{d^{2}x^{\lambda}}{d\tau^{2}}+\Gamma_{\mu\nu}^{\lambda} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\...
- 679
2
votes
2
answers
132
views
Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?
On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
- 737
0
votes
0
answers
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?
5
votes
2
answers
451
views
What is the idea behind 2-spinor calculus?
In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
- 7,980
0
votes
1
answer
154
views
Four-divergence of the wedge product
I have maybe simple question, but I cannot find it anywhere in the internet.
When I calculate four-divergence on tensor, being wedge product, do I calculate it the way A or B?
$$
\partial_{\alpha} \, ...
- 9
1
vote
1
answer
304
views
How to determine sub and superscript for divergence theorem in Minkowski Space?
Suppose we are given a type $(0,2)$ tensor $T_{\mu\nu}$ in a Minkowski space with $(-,+,+,+)$ signature. Consider a closed 3-dimensional hypersurface $\partial \Omega$ which encloses a volume $\Omega$ ...
- 838
1
vote
1
answer
142
views
If $g_{ij}$ is a tensor of type $(0,2)$, what is kind of tensor is $\partial_{i}g_{jk}$?
Suppose $g_{ij}$ is a tensor of type $(0,2)$, then what type of object is $\partial_{i}g_{jk}$? Is it even a tensor, and if so, of what type? Is the $\partial_{i}$ still a differential with respect to ...
- 123
3
votes
0
answers
90
views
Tetrad basis: a doubt on "Comoving" and "Static" tetrads
In the awesome paper $[1]$, Müller then gives us a plethora of spacetimes and their basic geometrical objects like the form of the line element, Christoffel symbols, Krestchmann scalars and so on. ...
- 3,085
4
votes
2
answers
161
views
If $ \partial_a F_{bc} + \partial_b F_{ca} + \partial_c F_{ab} = 0 $ then $ F_{ab} $ is the curl of a 4-vector
Any skew-symmetric tensor $ F_{\alpha\beta} $ which is the curl of a 4-vector $A_\mu$,
that is each tensor having the form
$ F_{\alpha\beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha $,
...
- 303
0
votes
1
answer
124
views
Covariant derivate of constant vector
We know that $$\frac{dv}{d t}=\frac{d\left(v^{i} e_i\right)}{d t}=\partial_{j} v^{i} v^{j} e_{i}+v^{i} v^{j} \partial_{j} e_{i}$$
As $\partial_{j} e_{i}$ is another vector we can expand it in the same ...
- 135
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,...
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