Violation of γ in Brans-Dicke gravity
The Brans Class I solution in Brans-Dicke gravity, discovered in 1961, is a pivotal element in the examination of gravitational theories extending beyond General Relativity. This solution characterizes the external vacuum of a spherical Brans-Dicke star and is distinguished by two adjustable parameters. Notably, a rigorous establishment of the relationship between these parameters and the star's properties is currently lacking. In this work, we endeavor to address this gap by deriving the comprehensive exterior solution of Brans Class I, articulated in relation to the total energy and total pressure of the spherically symmetric gravity source. This derived solution facilitates the precise computation of all post-Newtonian parameters in Brans-Dicke gravity for the distant field regions of a spherical source. Specifically, we obtain an alternative analytical expression for the γ parameter, denoted as γ exact = ω+1+(ω+2) Θ ω+2+(ω+1) Θ, where Θ represents the ratio of the total pressure P∗∥ + 2P∗⊥ to the total energy E∗ contained within the mass source. Our non-perturbative γ formula is universally applicable across varying field strengths and types of matter comprising the mass source. As a result, observational constraints on γ concurrently establish limits on ω and Θ. Significantly, the latter serves as a global characteristic of the mass source. On a broader scope, our formula underscores the significance of pressure (when Θ ̸= 0) in spherical Brans-Dicke stars and potentially in stars within other modified theories of gravitation.
Abstract
Introduction
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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 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 analytical 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.