Weighing the Vacuum Energy

Abstract

We discuss the weight of vacuum energy in various contexts. First, we compute the vacuum energy for flat spacetimes of the form ${\mathbb{T}}^3\times \mathbb{R}$, where ${\mathbb{T}}^3$ stands for a general 3-torus. We discover a quite simple relationship between energy at radius $R$ and energy at radius $\frac{l_s^2}{R}$. Then we consider quantum gravity effects in the vacuum energy of a scalar field in ${\mathbb{M}}_3\times S^1$ where ${\mathbb{M}}_3$ is a general curved spacetime, and the circle $S^1$ refers to a spacelike coordinate. We compute it for General Relativity and generic transverse {\em TDiff} theories. In the particular case of Unimodular Gravity vacuum energy does not gravitate.

32 pages. Minor corrections