Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Energy Balance of a Boson Gas at Zero Temperature in Curved Spacetime

Jorge Meza-Domínguez, Tonatiuh Matos, Pierre-Henri Chavanis

Abstract

We develop a comprehensive thermodynamic description for a zero-temperature boson gas in curved spacetime, integrating energy conservation with information-theoretic principles. Using the hydrodynamic Madelung representation within the ADM formalism, we establish two fundamental relationships: an energy balance equation representing the first law of thermodynamics from a spacetime perspective, and an information-theoretic constraint connecting Fisher entropy to the dynamical evolution of the boson density. This dual formulation clearly separates energy transport from information conservation while revealing how quantum information is preserved in curved backgrounds. The introduction of a stochastic velocity provides a bridge between quantum potential effects and underlying spacetime fluctuations, suggesting a gravitational basis for quantum stochastic behavior. We demonstrate the consistency of our framework through detailed analyses of quantum systems in both Minkowski and Schwarzschild spacetimes. This work provides a unified foundation for studying relativistic bosonic systems, with direct relevance to boson stars and scalar field dark matter models.

Energy Balance of a Boson Gas at Zero Temperature in Curved Spacetime

Abstract

We develop a comprehensive thermodynamic description for a zero-temperature boson gas in curved spacetime, integrating energy conservation with information-theoretic principles. Using the hydrodynamic Madelung representation within the ADM formalism, we establish two fundamental relationships: an energy balance equation representing the first law of thermodynamics from a spacetime perspective, and an information-theoretic constraint connecting Fisher entropy to the dynamical evolution of the boson density. This dual formulation clearly separates energy transport from information conservation while revealing how quantum information is preserved in curved backgrounds. The introduction of a stochastic velocity provides a bridge between quantum potential effects and underlying spacetime fluctuations, suggesting a gravitational basis for quantum stochastic behavior. We demonstrate the consistency of our framework through detailed analyses of quantum systems in both Minkowski and Schwarzschild spacetimes. This work provides a unified foundation for studying relativistic bosonic systems, with direct relevance to boson stars and scalar field dark matter models.
Paper Structure (33 sections, 126 equations, 6 figures)

This paper contains 33 sections, 126 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Klein-Gordon-Maxwell Equations
  3. Hydrodynamic form equations
  4. Derivation of the Balance equation
  5. Energy Balance Equation: Thermodynamical First Law
  6. Integration over Spatial Hypersurface $\Sigma_{t}$
  7. First Law of Thermodynamics Form and Thermodynamical Spacetime Coupling $\mathcal{T}$
  8. Physical Interpretation:
  9. Stochastic velocity: $u_{\mu}$
  10. Fisher Entropy
  11. Fisher Entropy in Curved Spacetime
  12. Hydrodynamic Representation
  13. Continuity equation and the integrated current
  14. Fisher entropy density
  15. Information-theoretic constraint
  16. ...and 18 more sections

Figures (6)

  • Figure 1: (Color online) One-dimensional harmonic oscillator. Left: Boson density $n(x)$ for quantum states $\nu = 0, 1, 2, 3, 4$. Right: Corresponding Fisher entropy density $\mathcal{I}_F(x)$. Position $x$ is in nanometers (nm) using the convention from Appendix \ref{['Appendix:B']}.
  • Figure 2: (Color online) Hydrogen atom with fixed $\ell=0$, $m=0$ (s-states). Left: Density $n(r)$ for principal quantum numbers $\nu = 1, 2, 3, 4, 5$. Right: Fisher entropy $\mathcal{I}_F(r)$. Higher $\nu$ states extend farther from the nucleus and show more radial structure.
  • Figure 3: (Color online) Hydrogen atom with fixed $\nu=3$, $m=0$. Left: Radial density $n(r)$ for angular momentum quantum numbers $\ell = 0, 1, 2$. Right: Fisher entropy $\mathcal{I}_F(r)$. Radial coordinate is in units of Bohr radius $a_0$.
  • Figure 4: (Color online) Hydrogen atom with fixed $\nu=4$, $\ell=3$. Left: Density $n(r)$ for magnetic quantum numbers $m = -3, -2, \dots, 3$. Right: Fisher entropy $\mathcal{I}_F(r)$. The angular dependence modulates the radial profiles through spherical harmonics $Y_{\ell m}(\theta,\phi)$ evaluated at $\theta=\pi/4$.
  • Figure 5: (Color online) Klein-Gordon field in Schwarzschild geometry with fixed $\ell=0$, $m=0$ (spherically symmetric modes). Left: Density $n(r)$ for radial quantum numbers $\nu = 1, 2, 3, 4, 5$. Right: Fisher entropy $\mathcal{I}_F(r)$. Higher $\nu$ states decay faster and exhibit more spatial oscillations.
  • ...and 1 more figures