- #1
Loro
- 80
- 1
Homework Statement
I'm confused about writing down the equation: [itex] \Lambda \eta \Lambda^{-1} = \eta [/itex] in the Einstein convention.
Homework Equations
The answer is: [itex] \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho\sigma}[/itex]
However it's strange because there seems to be no distinction between [itex]\Lambda[/itex] and [itex]\Lambda^{-1}[/itex] if we write it this way.
However we know that:
[itex](\Lambda^{-1})^{\mu}{}_{\nu} = \Lambda_{\nu}{}^{\mu} [/itex]
The Attempt at a Solution
If the equation was instead [itex] \Lambda B \Lambda^{-1} = B [/itex]
Where [itex] B [/itex] is a tensor given in the form [itex] B^{\mu}{}_{\nu}[/itex] then it's clear to me how to write it:
[itex] \Lambda^{\rho}{}_{\mu} B^{\mu}{}_{\nu} \Lambda_{\sigma}{}^{\nu} = B^{\rho}{}_{\sigma}[/itex]
But [itex] \eta [/itex] is given in the form [itex] \eta^{\mu\nu} [/itex] and I don't understand how I can contract it with both [itex] \Lambda^{\mu}{}_{\nu} [/itex] and [itex] \Lambda_{\nu}{}^{\mu} [/itex] in order to arrive eventually at the result quoted in (2).