Scalar-tensor corrections and observational signatures of hairy black holes in Horndeski gravity

Abstract

We investigate specific physical properties of a previously derived hairy black hole solution in a particular Horndeski gravity model, focusing on observational signatures that might distinguish it from standard General Relativity (GR) solutions. Working with the metric function $f\left(r\right)=1-\frac{2m}{r}+\frac{h}{r}ln\left(\frac{r}{2m}\right)$ derived by Perez Bergliaffa et al., where $h$ represents the scalar hair parameter, we analyze its horizon structure and thermodynamic behavior. We demonstrate how the parameter $h$ modifies the Hawking temperature according to $T_H=\frac{2m+h}{16\pi m^2}$, with negative values suppressing temperature below the Schwarzschild baseline while positive values enhance it, potentially leading to altered evaporation processes compared to standard black holes. Using the Hamilton–Jacobi formalism modified by generalized uncertainty principle (GUP) considerations, we explore wave propagation and particle motion in this spacetime, deriving particle-dependent temperature corrections that introduce species-specific thermodynamic behavior. We derive analytical expressions for gravitational deflection angles in three distinct contexts: light rays in vacuum, electromagnetic waves in plasma, and massive particles, applying both the Gauss–Bonnet theorem and the Jacobi metric approach. For each case, we present explicit formulas showing the characteristic logarithmic terms introduced by the scalar hair, with plasma effects amplifying these signatures through frequency-dependent modifications. Through numerical analysis illustrated in our figures, we demonstrate how the scalar hair parameter influences the magnitude of these effects, revealing that negative $h$ values produce dramatically different phenomenology compared to positive values. Our entropy analysis reveals logarithmic corrections to the Bekenstein–Hawking area law consistent with quantum gravity predictions, supporting remnant formation scenarios that could resolve the information paradox.