A Neumann Boundary Term for Gravity

TL;DR

Problem: a generally covariant Neumann boundary term for gravity was lacking. Approach: redefine Neumann conditions by fixing the functional derivative of the action with respect to the boundary metric (boundary momentum), yielding a simple boundary term $S_N$ with coefficient $(4-D)/2\kappa$. Key results: in $D=3$ the term is half of the GHY term and aligns with the Chern-Simons description, while in $D=4$ the term vanishes, giving a natural Neumann interpretation of the Einstein–Hilbert action without boundary terms. Implications: frames a covariant microcanonical path integral in AdS/CFT, connects to Brown–York boundary stress tensor and holographic renormalization, with finite on-shell actions in AdS$_3$ and a counter-term structure identified for AdS$_4$.

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.