The complete exterior spacetime of spherical Brans-Dicke stars
arXiv:2404.13887 · doi:10.1016/j.physletb.2024.138803
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 γ_{exact}=((ω+1+(ω+2)Θ)/(ω+2+(ω+1)Θ)), where Θ is the ratio of the star's "total pressure" integral over its energy integral. This non-perturbative result reproduces the usual Post-Newtonian ((ω+1)/(ω+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 (γ_{GR}=1) as the star's equation of state approaches that of ultra-relativistic matter (in which case Θ 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. ω>-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.
1 figure. Correcting a typo in Eq (19) (an unjustified 4*pi factor has been removed)