Im taking GR for the first time and it is definitely throwing me for a loop. The question I am working on is this:
Prove that if a contravariant tensor $A^{uv}$ is symmetric, then it remains symmetric under Lorentz transformations. Is this also true for a mixed tensor $A_{u}{}^{v}$?
I'm working on the first part. I have a tensor: $$A^{uv} = A^{vu}$$
And I want to show that it is symmetric under a Lorentz transformation. For reasons which I do not know, I need to multiply by 2 lorentz transforms and show that the outcome is still symmetric. I am very bad with tensor notation and as such I am sure what I am going to write next will make experts in the field cringe. Is this the correct way to multiply $A^{uv}$ by 2 lorentz transforms?
$$A'^{uv} = {{a_\alpha}^\beta}{{a_\rho}^\sigma}A^{uv}$$
I know this is wrong, however I'm not sure why. Here is my thought process: The two a's correspond to the 2 different lorentz transforms, and they are "operating" on the original tensor A to give the new, transformed tensor. The reason why I know this is wrong is because there is no "cancellation" of any of the sub/superscripts, however, I'm not sure how to fix it.
I'll try to be brief, but while I am at it, what does it mean for a tensor (in this case a matrix) to be contravariant? I was struggling with the difference between contravariant and covariant vectors, and someone much more knowledgeable explained it to me in this way: "A contravariant vector is a covariant vector which has been operated on by the metric you're working in. Normally we don't think about it because the euclidean metric is the identity matrix, however in SR the metric commonly used is the minkowski metric". What would be the analogy with a matrix?