The complete exterior spacetime of spherical Brans-Dicke stars

TL;DR

This paper derives an explicit, exact exterior spacetime solution for spherical Brans-Dicke stars, linking interior structure to observable gravitational effects and providing a non-perturbative formula for the Eddington parameter.

Contribution

It presents a novel, generic expression for the Brans-Dicke exterior spacetime in terms of energy and pressure profiles, including an exact formula for the Eddington parameter that accounts for interior stellar properties.

Findings

Derived a closed-form exterior spacetime solution for Brans-Dicke stars.

Provided an exact formula for the Eddington parameter involving interior pressure and energy.

Showed the solution reduces to known limits in Newtonian and General Relativity regimes.

Abstract

We derive the complete expression for the Brans Class I exterior spacetime explicitly in terms of the energy and pressures profiles of a stationary spherisymmetric gravity source. This novel and generic expression is achieved in a parsimonious manner, requiring only a subset of the Brans-Dicke field equation and the scalar equation. For distant orbiting test particles, this expression promptly provides a simple, closed and exact formula of the [textgreek]<LaTeX>\textgreek{g}</LaTeX> Eddington parameter, which reads {\gamma}_{exact}=(({\omega}+1+({\omega}+2){\Theta})/({\omega}+2+({\omega}+1){\Theta})), where {\Theta} is the ratio of the star's "total pressure" integral over its energy integral. This non-perturbative result reproduces the usual Post-Newtonian (({\omega}+1)/({\omega}+2)) expression in the case of a "Newtonian star", in which the pressure is negligible with respect to the energy density. Furthermore, it converges to the General Relativity value ({\gamma}_{GR}=1) as the star's equation of state approaches that of ultra-relativistic matter (in which case {\Theta} approaches 1), a behavior consistent with broader studies on scalar-tensor gravity. Our derivation underscores the essence of these results involving (1) the key relevant portion of the Brans-Dicke field equations, (2) the uniqueness of the Brans Class I vacuum solution for the non-phantom action, viz. {\omega}>-3/2, and (3) the involvement of only two free parameters in this solution, hence requiring two quantities (energy and pressure integrals) of the mass source to fully characterize the solution. From a practical standpoint, it elucidates how a given stellar interior structure model determines the star's exterior gravitational field and impacts the motions of light objects (such as planets and accretion disks) orbiting it.