# Non-relativistic Limit of Thermodynamics of Bose Field in a Static Space-time and Bose-Einstein Condensation

## TL;DR

This paper derives the nonrelativistic thermodynamic behavior of a scalar Bose field in curved static spacetime, revealing gravitational and boundary effects on Bose-Einstein condensation.

## Contribution

It introduces a method to analyze the nonrelativistic limit of thermodynamics in curved spacetime using Mellin transform and heat kernel techniques, focusing on Bose-Einstein condensation.

## Key findings

- Gravitational effects influence Bose-Einstein condensation in finite volumes.
- Boundary effects impact the depletion coefficient of the scalar field.
- Asymptotic expansions of thermodynamic quantities are obtained for curved backgrounds.

## Abstract

We consider the grand canonical thermodynamics of a noninteracting scalar field in a static spacetime. We take the nonrelativistic limit of thermodynamic quantities in a way that leaves the curved structure of the background geometry intact. Using Mellin transform and heat kernel techniques we obtain asymptotic expansions of thermodynamic quantities appropriate for the analysis of Bose-Einstein condensation. We apply our results to investigate gravitational effects on the Bose-Einstein condensation for a scalar field in a finite volume. We also analyze the boundary effects on the depletion coefficient of the scalar field.