Quasi-local energy and compactification

TL;DR

This paper investigates how quasi-local energy, based on Brown and York's definition, can distinguish between uncompactified Minkowski space and compactified Kaluza-Klein spacetime by analyzing the extrinsic curvature integral.

Contribution

It demonstrates that the quasi-local energy measure varies with the size of the compact dimension, effectively differentiating between uncompactified and compactified spacetimes.

Findings

Quasi-local energy interpolates between zero and the uncompactified value as compact dimension size increases.

The measure can discriminate between Minkowski and Kaluza-Klein spacetimes.

The integral of the trace of extrinsic curvature serves as an effective diagnostic tool.

Abstract

Based on the quasi-local energy definition of Brown and York, we compute the integral of the trace of the extrinsic curvature over a codimension-2 hypersurface. In particular, we study the difference between the uncompactified Minkowski spacetime and the toroidal Kaluza-Klein compactification. For the latter, we find that this quantity interpolates between zero and the value for the uncompactified spacetime, as the size of the compact dimension increases. Thus, the quasi-local energy is able to discriminate between the two spacetimes.