Where the Gauss--Bonnet coupling enters the Bondi--Sachs hierarchy

Abstract

We integrate the Bondi--Sachs hierarchy of scalar--Gauss--Bonnet gravity with an arbitrary coupling function $f(Φ)$ and a radiating massless scalar, and determine the first order in $1/r$ at which the coupling $λ$ becomes visible in each field equation. We obtain the asymptotic falloff of the effective Gauss--Bonnet stress tensor together with its leading coefficients in covariant form on the sphere: every component decays between one and three powers of $1/r$ faster than the slowest falloff compatible with a smooth null infinity. As a consequence, (i) the Bondi mass-loss formula and the evolution equation of the angular-momentum aspect are those of the minimally coupled Einstein--scalar system, a statement that is exact in $λ$ under the assumption of a smooth expansion; (ii) the coupling generates no logarithmic terms through the orders considered, even though the Horndeski representation of the theory involves logarithmic functionals; and (iii) the coupling first enters a dynamical equation in the evolution of the $O(r^{-3})$ coefficient of the angular metric, through the cross term $-2λf'(\vp_0)\,\vp_1\,\partial_u N_{AB}$ between the asymptotic scalar charge aspect and the time derivative of the news. In Newman--Penrose form, the evolution equation of the leading coefficient of $Ψ_0$ acquires the source $-12λf'(\vp_0)\,\vp_1\barΨ_4^0$, while those of $Ψ_1$ and $Ψ_2$ are unchanged. The Gauss--Bonnet invariant decays as $r^{-6}$ and affects the scalar hierarchy only from the evolution of the $r^{-5}$ coefficient onward. For vacuum black-hole spacetimes, where the scalar is sourced by the coupling itself, all these effects are of order $λ^2$; if $f'(\vp_0)=0$ they are pushed one further order in $1/r$.