A Derivation of the Vacuum Einstein Equations: Understanding ln det g = 1

[PGaccount]

I read somewhere that the vacuum Einstein equations can be written as

ln det g = 1

Does anyone know the derivation of this?

 

[Nugatory]

I read somewhere...

Where?

 

[PGaccount]

Green Schwarz Witten, Volume II. page 440

 

[martinbn]

I read somewhere that the vacuum Einstein equations can be written as

ln det g = 0

Does anyone know the derivation of this?

That is not the same as Ricci=0.

 

[haushofer]

How can one equation contain all the information of 1/2*D*(D+1) equations?

 

[PGaccount]

That is what GSW says. Can someone look at it?

 

[martinbn]

There is context that you haven't provided. They don't say that this is the vacuum Einsein equations. They say that the Ricci = 0 for a Kahler manifold is equivalent to ln(det g) = 1.

 

[Martin Scholtz]

I read somewhere that the vacuum Einstein equations can be written as

ln det g = 1

Does anyone know the derivation of this?

This cannot be true since it is not a differential equation for the metric. #Ric=0# are vacuum Einstein's equations and thus 2nd derivatives must appear. I don't have the book you cite, can you give us some context?

 

[Martin Scholtz]

There is context that you haven't provided. They don't say that this is the vacuum Einsein equations. They say that the Ricci = 0 for a Kahler manifold is equivalent to ln(det g) = 1.

Thanks, I didn't know this.