Impact of Star Pressure on $γ$ in Modified Gravity beyond Post-Newtonian Approach

Abstract

We provide a concrete example exhibiting marked deviation from the PPN approximation in a modified theory of gravity. Specifically, we derive the exact formula for the Robertson parameter $γ$ in Brans-Dicke gravity for compact mass sources, explicitly incorporating the pressure content of these sources. We achieve this by exploiting the $\textit integrability$ of the 00-component of the Brans-Dicke field equation. In place of the conventional PPN result $γ_{PPN}=\frac{ω+1}{ω+2}$, we obtain the analytical expression $γ_{\,exact}=\frac{ω+1+(ω+2)\varTheta}{ω+2+(ω+1)\varTheta}$ where $\varTheta$ is the ratio of the total pressure $P_\parallel^*+2P_\perp^*$ and total energy $E^*$ contained within the mass source. Our $\textit non\text{-}perturbative$ formula is valid for all field strengths and types of matter comprising the mass source. We draw four key conclusions: (1) The usual $γ_{PPN}$ formula is violated in the presence of pressure, viz. when $\varTheta\neq0$, revealing a limitation of the PPN approximation in Brans-Dicke gravity. (2) The PPN result mainly stems from the assumption of pressureless matter. Even in the weak-field star case, non-zero pressure leads to a violation of the PPN $γ$ formula. Conversely, the PPN result is a good approximation for low-pressure matter, i.e. when $\varTheta\approx0$, for all field strengths. (3) Observational constraints on $γ$ set $\textit joint$ bounds on $ω$ and $\varTheta$, with the latter representing a global characteristic of a mass source. If the equation of state of matter in the mass source approaches the ultra-relativistic form, entailing $\varTheta\simeq1$, $γ_{\,exact}$ converges to 1 $\textit irrespective$ of $ω$. (4) In a broader context, our findings indicate the latent significance of considering the interior structure of stars in observational astronomy.