Proving that Levi-Civita tensor density is invariant

[baba26]

TL;DR Summary

It's a problem from the textbook Supergravity ( Freedman, Proeyen ). We are asked to prove that under any infinitesimal change in frame-fields, there is no change in the Levi-Civita tensor density i.e. the variation equals zero.

This is a problem from the textbook Supergravity ( by Daniel Z. Freedman and Antoine Van Proeyen ). I am trying to learn general relativity from this book. I am attempting to do the later part of the Exercise 7.14 ( on page 148 ). Basically it asks us to explicitly show that the Levi-Civita tensor density doesn't change under any variation of frame fields. I am supposed to use the formula: variation of determiant of matrix M = determinant * trace ( M_inverse * variation in M ). But I can not even think of how to begin with the problem. Any hint will be appreciated.

 

[ergospherical]

For the frame field you have $\delta e = e e^{\mu}_a \delta e^a_{\mu}$ where $e = \mathrm{det}(e^{\mu}_a)$, which you can use when you take the variation of $\epsilon^{a_1 \dots} = e \epsilon^{b_1 \dots} ({e^{a_1}}_{b_1})(\dots)$

 

[baba26]

Thanks for the reply. I just added some more details to the question. Since I was having trouble with use Latex here, I posted a question on Physics Stack Exchange : https://physics.stackexchange.com/q...-that-levi-civita-tensor-density-is-invariant
Hopefully that will be helpful.

 

[Meir Achuz]

That $\epsilon^{\mu\nu\rho\sigma}$ is a actually a tensor is confirmed by the identity for the determinant of a $4\times 4$ matrix that

\begin{equation}

\epsilon'^{\mu'\nu'\rho'\sigma'}{\rm Det[L]}=L^{\mu'}_{\mu} L^{\nu'}_{\nu} L^{\rho'}_{\rho} L^{\sigma'}_{\sigma}\epsilon^{\mu\nu\rho\sigma}.

\end{equation}

This shows that $\epsilon^{\mu\nu\rho\sigma}$ is an idempotent pseudotensor of rank four.