Violation of $\gamma$ in Brans-Dicke gravity

TL;DR

This paper derives an exact, non-perturbative expression for the PPN parameter gamma in Brans-Dicke gravity, revealing the influence of pressure and providing a more comprehensive understanding of gravitational effects in spherical sources.

Contribution

It provides the complete exterior solution of Brans Class I in terms of total energy and pressure, and derives an exact gamma parameter formula incorporating pressure effects, extending beyond the traditional perturbative approach.

Findings

Exact gamma formula depends on pressure-to-energy ratio Theta.

Pressure significantly affects gamma in spherical Brans-Dicke stars.

Joint constraints on omega and Theta from observational gamma bounds.

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.