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## Main Question or Discussion Point

Can someone explain to me what Einstein Hilbert action is? and how it relates to the variational principle?

I appreciate any help that I can get!

I appreciate any help that I can get!

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Can someone explain to me what Einstein Hilbert action is? and how it relates to the variational principle?

I appreciate any help that I can get!

I appreciate any help that I can get!

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

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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.The EH action is an alegbraic expression which when extremized gives the Einstein field equations.

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George Jones

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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.

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

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

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

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.

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Did you check out the wikipedia article?Can someone explain to me what Einstein Hilbert action is? and how it relates to the variational principle?

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I did check out the wikipedia article. I have to write a paper on the variational principle and I was told that the Einstein Hilbert action would be a good thing to talk about. I still don't have enough stuff to talk about though.

Can you guys recommend any books or papers?

Thanks a lot once again!

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George Jones

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You might want to look at chapter 4 of https://www.amazon.com/gp/product/0521830915/?tag=pfamazon01-20 by Eric Poisson.Can you guys recommend any books or papers?

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

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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).

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There's a thorough discussion on the lagrangian formulation of GR in one of the appendices to R.Wald's "General Relativity".Can someone explain to me what Einstein Hilbert action is? and how it relates to the variational principle?

I appreciate any help that I can get!

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

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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...

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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 addressWhy'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...

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.

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