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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