Surface Vacuum Energy in Cutoff Models: Pressure Anomaly and Distributional Gravitational Limit

TL;DR

This paper investigates the vacuum energy and pressure anomalies near ideal reflecting boundaries, proposing a cutoff-dependent model that yields consistent gravitational equations through distributional analysis and renormalization.

Contribution

It introduces a distributional approach to vacuum energy with cutoff regularization, resolving pressure-energy inconsistencies and providing a mathematically consistent framework for boundary effects.

Findings

Cutoff-dependent stress tensor models boundary effects.

Pressure-energy inconsistency is resolved with spatial point separation cutoff.

Distributional limits lead to renormalized boundary properties.

Abstract

Vacuum-energy calculations with ideal reflecting boundaries are plagued by boundary divergences, which presumably correspond to real (but finite) physical effects occurring near the boundary. Our working hypothesis is that the stress tensor for idealized boundary conditions with some finite cutoff should be a reasonable ad hoc model for the true situation. The theory will have a sensible renormalized limit when the cutoff is taken away; this requires making sense of the Einstein equation with a distributional source. Calculations with the standard ultraviolet cutoff reveal an inconsistency between energy and pressure similar to the one that arises in noncovariant regularizations of cosmological vacuum energy. The problem disappears, however, if the cutoff is a spatial point separation in a "neutral" direction parallel to the boundary. Here we demonstrate these claims in detail, first for a single flat reflecting wall intersected by a test boundary, then more rigorously for a region of finite cross section surrounded by four reflecting walls. We also show how the moment-expansion theorem can be applied to the distributional limits of the source and the solution of the Einstein equation, resulting in a mathematically consistent differential equation where cutoff-dependent coefficients have been identified as renormalizations of properties of the boundary. A number of issues surrounding the interpretation of these results are aired.