Why is the boundary term in the Einstein-Hilbert action, the Gibbons-Hawking-York term, generally "missing" in General Relativity courses, IMPORTANT from the variational viewpoint, geometrical setting and the needs of Black Hole Thermodynamics? Shouldn't it be also included in modern courses of General Relativity despite its global effect on the equations of motion is irrelevant (at least in the classical theory of relativistic gravity)?
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2$\begingroup$ Because it's distracting when you're first learning the topic, and easy enough to add back into discussions when it's relevant. Also, note that MTW, which is still the source book for many intro graduate GR courses, predates the publication of this term in the action. I'm sure others would disagree. $\endgroup$– Zo the RelativistCommented Sep 16, 2014 at 18:40
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1$\begingroup$ Eric Poisson's lecture notes on Advanced General Relativity (pdf), which formed the basis of his book Relativist's Toolkit, discusses this (in chapter 4)... $\endgroup$– Alex NelsonCommented Sep 16, 2014 at 18:56
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Two points. First, the variation of the GHY term is discussed in detail in this post:
Explicit Variation of Gibbons-Hawking-York Boundary Term
Second, the GHY term on its own is enough to render the boundary value problem well-defined, but it is not sufficient for a physically interesting variational principle. This requires additional surface terms in the action, discussed by Regge and Teitelboim in the Hamiltonian formulation
http://adsabs.harvard.edu/abs/1974AnPhy..88..286R
and by Mann and Marolf in the Lagrangian case
