Impact of star pressure on $$\gamma $$ in modified gravity beyond post-Newtonian approach

Abstract

AbstractWe offer a concrete example exhibiting marked departure from the Parametrized Post-Newtonian (PPN) approximation in a modified theory of gravity. Specifically, we derive the exact formula for the Robertson parameter $$\gamma $$ γ in Brans–Dicke gravity for spherical compact stars, explicitly incorporating the pressure content of the stars. We achieve this by exploiting the integrability of the 00-component of the Brans–Dicke field equation. In place of the conventional PPN result $$\gamma _{\,\text {PPN}}=\frac{\omega +1}{\omega +2}$$ γ PPN = ω + 1 ω + 2 , we obtain the analytical expression $$\gamma _{\,\text {exact}}=\frac{\omega +1+(\omega +2)\Theta }{\omega +2+(\omega +1)\Theta }$$ γ exact = ω + 1 + ( ω + 2 ) Θ ω + 2 + ( ω + 1 ) Θ where $$\Theta $$ Θ is the ratio of the total pressure $$P_{\parallel }^{*}+2P_{\perp }^{*}$$ P ‖ ∗ + 2 P ⊥ ∗ and total energy $$E^{*}$$ E ∗ contained within the star. The dimensionless quantity $$\Theta $$ Θ participates in $$\gamma $$ γ due to the scalar degree of freedom of Brans–Dicke gravity. Our non-perturbative formula is valid for all field strengths and types of matter comprising the star. In addition, we establish two new mathematical identities linking the active gravitational mass, the ADM (Arnowitt–Deser–Misner) mass, and the Tolman mass, applicable for Brans–Dicke gravity. We draw four key conclusions:(1) The usual $$\gamma _{\,\text {PPN}}$$ γ PPN formula is violated for high-pressure mass sources, such as neutron stars, viz. when $$\Theta \ne 0$$ Θ ≠ 0 , 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 formula for $$\gamma $$ γ . Conversely, the PPN result is a good approximation for low-pressure matter, i.e. when $$\Theta \approx 0$$ Θ ≈ 0 , for all field strengths. (3) Observational constraints on $$\gamma $$ γ set joint bounds on $$\omega $$ ω and $$\Theta $$ Θ , with the latter representing a global characteristic of a mass source. If the equation of state of matter comprising the mass source approaches the ultra-relativistic form, entailing $$\Theta \simeq 1$$ Θ ≃ 1 , $$\gamma _{\,\text {exact}}$$ γ exact converges to 1 irrespective of $$\omega $$ ω . More generally, regardless of $$\omega $$ ω , ultra-relativistic matter tends to suppress the scalar degree of freedom in the exterior vacuum of Brans–Dicke stars, reducing the vacuum to the Schwarzschild solution. (4) In a broader context, by exposing a limitation of the PPN approximation in Brans–Dicke gravity, our findings indicate the significance of considering the interior structure of stars in observational astronomy when testing candidate theories of gravitation that involve additional degrees of freedom besides the metric tensor.