Energy balance of a Bose gas in a curved space-time

Abstract

We derive a general energy balance equation for a self-interacting boson gas at vanishing temperature in a curved spacetime. This represents a first step towards a formulation of the first law of thermodynamics for a scalar field in general relativity. By using a 3 + 1 foliation of the spacetime and performing a Madelung transformation, we rewrite the Klein-Gordon-Maxwell equations in a general curved spacetime into its hydrodynamic version where we can identify the different energy contributions of the system and separate them into kinetic, quantum, electromagnetic, and gravitational.

Hydrodynamic representation and Energy Balance for the Dirac and Weyl fermions in curved space-times

“…The main difference between the hydrodynamic representation of bosons [1] [2] and fermions concerns the form of the Bernoulli equation. For bosons, after making the Madelung transformation, we can separate the KG equation into real and imaginary parts.…”

“…The problem of the Energy Balance for boson particles in a curved spacetime is studied in [1], where the conserved 4-current associated with the KG equation describing the evolution of a complex scalar field Φ(x µ ) is defined. We can generalize this idea by defining a new 4-current J KG µ , changing the scalar field by a spinor and the complex conjugate scalar field by the conjugate transpose of the spinor.…”

“…( 35) contains the covariant derivative of γ µ which vanishes in a flat space-time. According to [1] [2] if we apply the transformation (22) to eq. ( 35), we could expect to obtain the continuity equation for the imaginary part and the Bernoulli equation for the real part.…”

“…Finally, from equation (64) we can identify the different energy contributions to the Fermi gas, and obtain an energy balance equation for fermions analogous to the one obtained for bosons in [1] [2]. In order to simplify the notations, we can re-write equation (64) in terms of the ν coefficients with the understanding that the subindex R refers to each component R = 1, 2 individually.…”

“…The Standard Model of elementary particles establishes that there exist two kinds of particles, fermions and bosons. In a previous work [1] [2], the energy balance for bosons was derived starting from the general relativistic Klein-Gordon (KG) equation. In the present work, we study a system of fermions described by the Dirac equation in arbitrary curved space-times taking into account electromagnetic effects.…”