Entropy Bounds in $R\times S^3$ Geometries

TL;DR

This paper calculates the Casimir energy in $R\times S^3$ geometries using Green's function and statistical methods, demonstrating their equivalence and exploring entropy-energy ratios, with implications for high and low temperature regimes.

Contribution

It provides exact Casimir energy calculations in $R\times S^3$ geometries and compares different computational approaches, highlighting discrepancies with previous work.

Findings

Green's function and statistical methods are equivalent in calculating Casimir energy.

Entropy to energy ratio is unbounded even for supersymmetric cases.

Discrepancies with prior results are identified and discussed.

Abstract

Exact calculations are given for the Casimir energy for various fields in $R\times S^3$ geometry. The Green's function method naturally gives a result in a form convenient in the high-temperature limit, while the statistical mechanical approach gives a form convenient for low temperatures. The equivalence of these two representations is demonstrated. Some discrepancies with previous work are noted. In no case, even for ${\cal N}=4$ SUSY, is the ratio of entropy to energy found to be bounded. This deviation, however, occurs for low temperature, where the equilibrium approach may not be relevant. The same methods are used to calculate the energy and free energy for the TE modes in a half-Einstein universe bounded by a perfectly conducting 2-sphere.