2
votes
1answer
95 views

Question on index notation and metric tensor

I found this expression in my SR notes: $$ (\Lambda^{-1})^{\lambda}_{\ \ \ \sigma} = g^{\lambda\mu}~\Lambda^{\rho}_{\ \ \ \mu} ~g_{\rho\sigma} = \Lambda_\sigma^{\ \ \ \lambda}$$ I know where it ...
13
votes
7answers
696 views

Is it foolish to distinguish between covariant and contravariant vectors?

A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a ...
1
vote
1answer
55 views

Covariant derivative as a tensor

$$\nabla_{j} v^{i}~=~g^{ik}\nabla_{j}v_{k}.$$ Does this equality involve an intermediate step, where I take the metric inside the derivative, and then use the fact that covariant derivative of the ...
3
votes
2answers
138 views

Partial Differentiation of a Tensor

I have doubts in the statement that the partial or ordinary differentiation of tensor is not a tensor. The argument for this is that the partial differentiation of the tensor involves evaluating the ...
6
votes
2answers
241 views

Is there any physics behind covariance and contravariance of indices of tensors?

Is there any physics behind covariance and contravariance (up and down) of indices of tensors?
4
votes
5answers
361 views

Why define four-vectors to be quantities that transform only like the position vector transforms?

A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation ...
0
votes
3answers
358 views

Relativistic basic question - four vector, Lorentz matrix

I have heard relativistics only very compressed during my student time. Now I looked up the definitions again and a question comes into my mind: A contravariant vector is transformed like this: ...
1
vote
1answer
236 views

Covariant derivative with upper index

I just need clarification, that is, to see that I'm doing the right thing. When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have ...
4
votes
1answer
854 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 ...