Einstein Hilbert Action

The EH action is an alegbraic expression which when extremized gives the Einstein field equations. This is according to the variational principle. Do a Google, or go straight to the Wiki article which is adequate.

 

The Einstein-Hilbert action is a generalization of the classical least-action principle, which uses variational calculus.

Historically, Hilbert figured it out before Einstein. Never tell a mathematician what you are working on. :-)

 

You need to be careful here. The Einstein-Hilbert action gives you the correct field equations for general relativity if and only if you restrict attention to spacetimes without boundary and which are compact (a dubious practice in itself). If you need to work with spacetimes with boundary or which are non-compact then the Einstein-Hilbert action needs to be significantly modified.

 

Do you mean that Hilbert found the field equations via a variational principle before Eintein, or that Hilbert found the field equations before Eintein, because there is some controversy associated with the latter claim.

See http://www.sciencemag.org/cgi/content/abstract/278/5341/1270" [Broken]. Not everyone accepts the validity of this paper, though.

 

The former. As I remember the story, Einstein gave a lecture on his progress with GR and got into a discussion with Hilbert. Hilbert then went away and derived the equations (I believe for empty space only) and wrote Einstein about it. Einstein then redoubled his efforts and solved the whole thing. Their papers came out at about the same time. It was a bit of a race.

Thanks for the reference. I had not realized that Hilbert may not have appreciated the importance of covariance.

Hilbert once said something like "Physics is too hard to be left to physicists." Major ego.

 

Did you check out the wikipedia article?

 

You might want to look at chapter 4 of https://www.amazon.com/gp/reader/0521830915/ref=sib_dp_pt/103-2145582-2430211#reader-link" by Eric Poisson.

It isn't completely rigorous in the sense of coalquay404's post, but it does deal with boundary terms.

 

You might also check out A. S. Eddington's "The Mathematical Theory of Relativity" (Cambridge, 1st edition 1923). Eddington was one of the pioneers in GR and gives a complete (early) treatment. Section 60 on "Action" starts from the 4-D action based on classical action and shows how it converts to an integral over the scalar G. He then does the variation, but gets bogged down in a seemingly non-sensical tirade over the validity of the principle of stationary action. Fun reading.

You should also look at how the great man himself did it, without appealing to least action. See Einstein's 1916 GR paper in "The Principle of Relativity" (Dover, 1952). The simplicity of his derivation of Guv=0, when he finally gets to it, is unnerving. In a following paper in this volume he treats the Hilbert derivation (and gives some credit to Lorentz).

 

There's a thorough discussion on the lagrangian formulation of GR in one of the appendices to R.Wald's "General Relativity".

 

It's also an incorrect discussion.

 

Why's that ? More precisely where are the flaws ? Anyway, i'll reread it and try to see if I find something fishy on my own...

 

Wald makes a hash of several things, the treament of electromagnetism being (surprisingly) one of them. However, far more serious than that is the fact that he pays no proper attention to the sort of manifold that he's attempting to formulate an action principle for. Granted, he does treat the possibility that the boundary to the spacetime is non-empty, but he doesn't address at all the simple fact that the Einstein-Hilbert action is actually undefined in the case of a non-compact manifold (the E-H action essentially becomes infinite even for flat space). There are various correction terms one must add to the action in the presence of both non-empty boundaries and, crucially, a regularisation term to correct for the divergence of the E-H action whenever [itex]M[/itex] is non-compact.

I think someone pointed out earlier that Poisson's book contains a nice discussion of these points. It doesn't, however, deal with more sophisticated techniques such as actions for asymptotically flat spaces which have a number of ends diffeomorphic to [itex]\mathbb{R}^3-\lbrace0\rbrace[/itex]. Then again, these are reasonably advanced topics so maybe I'm being a bit picky. It does pay to be aware of them though.