Violation of γ in Brans–Dicke Gravity

Abstract

The Brans Class I solution in Brans–Dicke gravity is a staple in the study of gravitational theories beyond General Relativity. Discovered in 1961, it describes the exterior vacuum of a spherical Brans–Dicke star and is characterized by two adjustable parameters. Surprisingly, the relationship between these parameters and the properties of the star has not been rigorously established. In this article, we bridge this gap by deriving the complete exterior solution of Brans Class I, expressed in terms of the total energy and total pressure of the spherisymmetric gravity source. The solution allows for the exact derivation of all post-Newtonian parameters in Brans–Dicke gravity for far field regions of a spherical source. Particularly for the γ parameter, instead of the conventional result γ_PPN =(ω+1)/(ω+2), we obtain the analytic expression γ_exact =(ω+1+(ω+2) Θ)/(ω+2+(ω+1) Θ), where Θ is the ratio of the total pressure P^*_‖+ 2P^*_⊥ and total energy E^* contained within the mass source. Our non-perturbative γ formula is valid for all field strengths and types of matter comprising the mass source. Consequently, observational constraints on γ thus set joint bounds on ω and Θ, with the latter representing a global characteristic of the mass source. More broadly, our formula highlights the importance of pressure (when Θ ̸ = 0) in spherical Brans–Dicke stars, and potentially in stars within other modified theories of gravitation.