A Neumann Boundary Term for Gravity

TL;DR

This paper introduces a new boundary term for gravity under Neumann conditions, providing a well-defined variational principle and connecting to Chern-Simons theory and the Einstein-Hilbert action.

Contribution

It proposes a novel boundary term for gravity with Neumann boundary conditions, linking it to Chern-Simons theory and clarifying its role in the Einstein-Hilbert action.

Findings

In 3D, reduces to a half GHY term.

In 4D, boundary term vanishes, giving a natural Neumann interpretation.

Connects to Chern-Simons action without extra boundary terms.

Abstract

The Gibbons-Hawking-York (GHY) boundary term makes the Dirichlet problem for gravity well defined, but no such general term seems to be known for Neumann boundary conditions. In this paper, we view Neumann {\em not} as fixing the normal derivative of the metric ("velocity") at the boundary, but as fixing the functional derivative of the action with respect to the boundary metric ("momentum"). This leads directly to a new boundary term for gravity: the trace of the extrinsic curvature with a specific dimension-dependent coefficient. In three dimensions this boundary term reduces to a "one-half" GHY term noted in the literature previously, and we observe that our action translates precisely to the Chern-Simons action with no extra boundary terms. In four dimensions the boundary term vanishes, giving a natural Neumann interpretation to the standard Einstein-Hilbert action without boundary terms. We argue that in light of AdS/CFT, ours is a natural approach for defining a "microcanonical" path integral for gravity in the spirit of the (pre-AdS/CFT) work of Brown and York.