Converting between Covariant and Contravariant matrices

[LoopQG]

Homework Statement

Given a matrix {latex] A_11 =A_22 = 0 A_12 =A_21 = x/y +y/x [ /latex] Find the contravariant components in polar coordinates.

Answer:

[itex] A_11 = 2 A_22 = -2/r^2 A_12 = 2cot(2 /theta)/r [ /latex]

Homework Equations

I used the polar coordinates metric to raise indecies but i do not get the correct answer I get:

[itex] A^11 = A^22 = 0 A^12 = csc( /theta)sec( /theta) / r^4 A^21 = csc( /theta)sec( /theta) [ /latex]Any clue on what I am doing wrong?

The Attempt at a Solution

 

[grey_earl]

You should use [ t e x ] and [ / t e x ] to write in tex.

What you forget (I think) is that you should give the matrix also in the basis of polar coordinates (i.e. d/dr and d/dθ) instead of d/dx and d/dy as you did.

 

[AlexChandler]

Leonard Susskind does an example of this in lecture 3 of his general relativity lectures.



He does an example with polar coordinates at 1:37

It may be helpful