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

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.