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

  • Tonatiuh Matos
  • Ana Avilez
  • Tula Bernal
  • Pierre-Henri ChavanisEmail author
Research Article
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Abstract

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.

Keywords

Scalar field theory Klein–Gordon equation Gross–Pitaevskii equation Bose–Einstein condensates Bohm’s interpretation Boson stars Dark matter General relativity Quantum mechanics 

Notes

Acknowledgements

We thank L.E. Padilla for useful comments and corrections in the elaboration of this manuscript, and the anonymous reviewers who helped us with valuable comments and suggestions to improve this article. This work was partially supported by Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico, under Grants: CB-2014-01 No. 240512, Projects No. 269652 and 283151, and Fronteras Project 281. Also by Xiuhcoatl and Abacus Clusters at Centro de Investigación y de Estudios Avanzados, Instituto Politécnico Nacional. A.A. acknowledges financial support from CONACyT Mexico within “Retención y Repatriación” Program and from Vicerrectoría de Investigación y Estudios de Posgrado, Benemérita Universidad Autónoma de Puebla.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Tonatiuh Matos
    • 1
  • Ana Avilez
    • 2
  • Tula Bernal
    • 3
  • Pierre-Henri Chavanis
    • 4
    Email author
  1. 1.Departamento de FísicaCentro de Investigación y de Estudios Avanzados del IPNMexico CityMéxico
  2. 2.Facultad de Ciencias Físico-MatemáticasCiudad Universitaria, Benemérita Universidad Autónoma de PueblaPueblaMéxico
  3. 3.Universidad Autónoma ChapingoTexcocoMéxico
  4. 4.Laboratoire de Physique Théorique, CNRSUniversité Paul SabatierToulouseFrance

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