Tagged Questions
1
vote
2answers
56 views
Kronecker delta confusion
I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same):
...
3
votes
1answer
100 views
When a variation of a tensor is not a tensor?
In a comment about variation of metric tensor it was shown that
$$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$
which is contrary to the usual rule of lowering indeces of a ...
2
votes
1answer
125 views
Stress energy tensor of a perfect fluid and four-velocity
In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units).
We consider a metric $g_{\mu\nu}$ with a signature $(-, +, +, +)$ :
...
2
votes
1answer
70 views
Sign crazyness on the stress energy tensor?
I would like to know on what depends the sign of the stress energy tensor in the following formula :
$T_{\mu\nu}=\pm(\rho c^2+P)u_{\mu}u_{\nu} \pm P g_{\mu\nu}$
In my case the metric is equal to ...
1
vote
0answers
62 views
Lecture Notes confusion: Constructing the Einstein Equation
This question is on the construction of the Einstein Field Equation.
In my notes, it is said that
The most general form of the Ricci tensor $R_{ab}$ is $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$
...
0
votes
1answer
50 views
Contraction of the metric tensor
This is perhaps a simple tensor calculus problem -- but I just can't see why...
I have notes (in GR) that contains a proof of the statement
In space of constant sectional curvature, $K$ is ...
0
votes
1answer
36 views
Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?
More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:-
$$
\Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left(
\frac{\partial ...
2
votes
1answer
137 views
Ricci identity/Riemann curvature tensor and covectors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
4
votes
2answers
117 views
Difference between slanted indices on a tensor
In my class, there is no distinction made between,
$$
C_{ab}{}^{b}
$$
and
$$
C^{b}{}_{ab}.
$$
All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. ...
0
votes
0answers
109 views
covarient derivative of electromagnetic field tensor
I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is Divergence-less directly(Without using general lie derivative method which can prove any energy momentum ...
2
votes
1answer
44 views
Non-diagonal elements when switching metric signature?
Considering a metric tensor with the signature $(-,+,+,+)$:
$g_{\mu\nu}=
\begin{pmatrix}
-c^2 & g_{01} & g_{02} & g_{03}\\
g_{10} & a^2 & g_{12} & g_{13}\\
g_{20} & g_{21} ...
1
vote
2answers
173 views
What is the Riemann curvature tensor contracted with the metric tensor?
Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example
$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
7
votes
3answers
714 views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu ...
5
votes
4answers
298 views
Are covariant vectors representable as row vectors and contravariant as column vectors
I would like to know what are the range of validity of the following statement:
Covariant vectors are representable as row vectors. Contravariant
vectors are representable as column vectors.
...
4
votes
1answer
563 views
Covariant derivative and Leibniz rule
I read the Wikipedia page about the covariant derivative, my main problem is in this part:
http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description
Some of the formulas seem to lead ...
1
vote
1answer
241 views
metric signature explanation
Can anyone explain what metric signature is?
I have a basic knowledge regarding tensors, btw.
Also, how is it related to fundamental understanding of general relativity?
Thanks.
2
votes
1answer
96 views
Question from Schutz's
In q. 22 in page 141, I am asked to show that if $U^{\alpha}\nabla_{\alpha} V^{\beta} = W^{\beta}$, then $U^{\alpha}\nabla_{\alpha}V_{\beta}=W_{\beta}$.
Here's what I have done:
$V_{\beta}=g_{\beta ...
2
votes
1answer
127 views
Are there any clear and expressive plainword sense of metric tensor components?
Can the following groups of components of metric tensor can assigned a clear sense?
https://docs.google.com/drawings/pub?id=1kVqkN1gT-a2fDy2S851l9iQKaMfaatCDo517OSZBHEo&w=467&h=228
I have ...
5
votes
2answers
87 views
Torsion and gauge invariant EM kinetic term
Everytime I hear about adding torsion to GR, something struggles me: how do you create a kinetic term for the electromagnetic field that is still gauge-invariant? One of the consequences of torsion is ...
0
votes
3answers
281 views
Need some basic help with notation and the Christoffel symbols
Apologies in advance if some of the questions below seem overly simple.
In an introductory GR book, I find the following expression for the autoparallel of the affine connection (the upper bound of ...
4
votes
1answer
311 views
How do we know the geodesic is a minimum?
The geodesic equation is derived from the Euler-Lagrange equation, which (as I understand it) is a necessary but not sufficient condition to ensure that the geodesic is a minimum.
The introductory GR ...
2
votes
0answers
128 views
Using the area element in derivation of geodesic
In the derivation of the geodesic, one starts with the integral of the line element (arclength):
$$L(C)=\int_{\tau_1}^{\tau_2}d\tau\sqrt{g_{\mu \nu}\dot{x}^{\mu} \dot{x}^{\nu}}$$
The integrand is ...
15
votes
3answers
2k views
What is the physical meaning of the connection and the curvature tensor?
Regarding general relativity:
What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)?
What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
