Boundary Terms of the Einstein–Hilbert Action

Abstract

The Einstein–Hilbert action for general relativity is not well posed in terms of the metric $g_{ab}$ as a dynamical variable. There have been many proposals to obtain an well posed action principle for general relativity, e.g., addition of the Gibbons–Hawking–York boundary term to the Einstein–Hilbert action. These boundary terms are dependent on what one fixes on the boundary and in particular on spacetime dimensions as well. Following recent works of Padmanabhan we will introduce two new variables to describe general relativity and the action principle with these new dynamical variables will turn out to be well posed. Then we will connect these dynamical variables and boundary term obtained thereof to existing literature and shall comment on a few properties of Einstein–Hilbert action which might have been unnoticed earlier in the literature. Before concluding with future prospects and discussions, we will perform a general analysis of the boundary term of Einstein–Hilbert action for null surfaces as well.

Dedication

One of the great quality of Prof. Padmanabhan is his ability to ask the correct question. This work stems from such questions asked by him during our discussions: “What is the connection between various action principles for general relativity? Why N and $N^{\alpha }$ are non-dynamical? What one should fix on a null surface?”. I have tried to answer them in this work and I respectfully dedicate it to Prof. Padmanabhan on the occasion of his 60th birthday.