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Energy balance of a Bose gas in a curved space-time. (English) Zbl 1437.83017

Summary: Classical solutions of the Klein-Gordon equation are used in astrophysics to model galactic halos of scalar field dark matter and compact objects such as cores of neutron stars. These bound solutions are interpreted as Bose-Einstein condensates whose particle number density is governed by the Gross-Pitaevskii (GP) equation. It is well known that the Gross-Pitaevskii-Poisson (GPP) system arises as the non-relativistic limit of the Klein-Gordon-Einstein (KGE) equations and, conversely, the KGE system may be interpreted as a generalization of the GPP equations in a curved space-time. In the present work, we consider a \(3+1\) ADM foliation of the space-time in order to construct a general-relativistic version of the GP equation. Besides, we derive a general energy balance equation for the boson gas in the hydrodynamic variables, where different energy potentials are identified as kinetic, quantum, electromagnetic and gravitational. In addition, we find a correspondence between the energy potentials in the balance equation and actual components of the scalar energy-momentum tensor. We also study the Newtonian limit of the hydrodynamic formulation and the balance equation. As an illustrative case, we study the effects in the energy potentials of a relativistic correction in the GP equation.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83B05 Observational and experimental questions in relativity and gravitational theory
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
85A15 Galactic and stellar structure
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
83C56 Dark matter and dark energy
83C47 Methods of quantum field theory in general relativity and gravitational theory
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