Minkowski Metric and lorentz transformation

1. The problem statement, all variables and given/known data

So I have just been introduced to indices, four vectors and tensors in SR and I’m having trouble knowing exactly what I am being asked in some questions.

So the first question asks to write explicitly how a covariant two tensor transforms under a lorentz boost.

The next questions asks to show that the covariant form of the minkowski metric tensor is invariant under this lorentz boost.

2. Relevant equations

Now I know that it transforms like [itex]A’_{\mu \nu}=(L^{-1})_{\mu}\ ^{\kappa}(L^{-1})_{\nu}\ ^{\lambda}A_{\kappa \lambda}[/itex]
where L is the matrix with [itex]L_{00/11}=\gamma[/itex] and [itex]L_{12/21}=-\gamma \beta[/itex]

Is this written explicitly or do I need to write it out in matrix form?

3. The attempt at a solution

What I have so far for part 2 is [itex]\eta’_{\mu \nu}=(L^{-1})_{\mu}\ ^{\kappa}(L^{-1})_{\nu}\ ^{\lambda}\eta_{\kappa \lambda}[/itex]

[itex]\eta’_{\mu \nu}=(L^{-1})_{\mu}\ ^{\kappa}(L^{-1})_{\nu \kappa}[/itex]

I’m not really sure how to proceed from there, I’m pretty sure there is some relationship between raising/lowering indices and the inverse/transpose of lorentz transformation I’m not aware of.

Also I’ve seen the minkowski metric written both -1,1,1,1 and 1,-1,-1,-1 on various pdfs/notes online why is this?

And finally is there a link to some proof that the L I described acts on contravariant components and not covariant, i.e somethat shows the basis vectors of minkowski space transform by L^-1?


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