The discussion centers on the implications of ignoring the boundary term in the action when varying the Einstein-Hilbert action, specifically the term $$\int d^{4}x \sqrt{-g}g^{ab} \delta R_{ab}$$. It is established that this term can be omitted only when the manifold is both compact and without boundary, as per the Gibbons-Hawking-York boundary term. The conversation highlights the confusion surrounding the conditions under which boundary terms are necessary, particularly in relation to Minkowski and Schwarzschild spacetimes, which are examples of non-compact manifolds without boundaries at finite distances.
PREREQUISITESThe discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students of General Relativity seeking to deepen their understanding of manifold properties and their implications in gravitational theories.
davidge said:What I mean is, let the natural coordinates --just label of the points of ##\mathbb{R}^2 \times \mathbb{S}^2##-- be ##(t,r, \theta, \phi)##. Now, let's consider a metric space, namely ##\mathbb{R}^4## with its usual metric.