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 Proceeding, we bridge this gap by deriving $\textit{the}$ 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 $\textit{exact}$ derivation of $\textit{all}$ post-Newtonian parameters in Brans-Dicke gravity for far field regions of a spherical source. Particularly for the $\gamma$ parameter, instead of the conventional result $\gamma_{\,\text{PPN}}=\frac{\omega+1}{\omega+2}$, we obtain the analytical expression $\gamma_{\,\text{exact}}=\frac{\omega+1+(\omega+2)\,\Theta}{\omega+2+(\omega+1)\,\Theta}$ where $\Theta$ is the ratio of the total pressure $P_{\parallel}^{}+2P_{\perp}^{}$ and total energy $E^{*}$ contained within the mass source. Our $\textit{non-perturbative}$ $\gamma$ formula is valid for all field strengths and types of matter comprising the mass source. Consequently, observational constraints on $\gamma$ thus set $\textit{joint}$ bounds on $\omega$ and $\varTheta$, with the latter representing a global characteristic of the mass source. More broadly, our formula highlights the importance of pressure (when $\varTheta\neq0$) in spherical Brans-Dicke stars, and potentially in stars within other modified theories of gravitation.