Gravitational soliton solutions to self-coupled Klein-Gordon and Schr\"odinger equations

TL;DR

This paper develops a relativistic model of gravitational solitons for scalar particles using the Klein-Gordon equation in curved spacetime, extending the Schrödinger-Newton framework to include general relativity effects.

Contribution

It introduces a self-consistent relativistic approach to gravitational solitons by solving the Klein-Gordon equation in a curved spacetime generated by the particle's own probability distribution.

Findings

Found static, spherically symmetric soliton solutions with various radial excitations.

Compared relativistic solutions with nonrelativistic Schrödinger-Newton results.

Modeled the gravitational potential using both direct Einstein equations and a perfect-fluid approximation.

Abstract

We use the Klein-Gordon equation in a curved spacetime to construct the relativistic analog of the Schr\"odinger-Newton problem, where a scalar particle lives in a gravitational potential well generated by its own probability distribution. A static, spherically symmetric metric is computed from the field equations of general relativity, both directly and as modeled by a perfect-fluid assumption that uses the Tolman-Oppenheimer-Volkov equation for hydrostatic equilibrium of the mass density. The latter is appropriate for a Hartree approximation to the many-body problem of a bosonic star. Simultaneous self-consistent solution of the Klein--Gordon equation in this curved spacetime then yields solitons with a range of radial excitations. We compare results with the nonrelativistic case.