Chaos as a Possible Probe for Scalar Hair in Horndeski Gravity

TL;DR

The study probes whether chaotic dynamics of a spinning test particle in Horndeski hairy BH spacetimes can reveal scalar hair beyond general relativity. By solving the MPD equations under the Tulczyjew-Dixon SSC and analyzing Lyapunov exponents, Poincaré sections, and effective potentials, it demonstrates that increasing hair strength $h$ systematically suppresses chaos by flattening the effective potential and removing saddle points. It extends the analysis to gravitational waves via the quadrupole formula, finding that hair-induced damping of chaos restores phase coherence between GW polarizations $h_+$ and $h_\times$, linking orbital dynamics to observable signals. The results suggest that combining chaotic observables with GW measurements provides a powerful, complementary route to testing strong-field gravity and constraining scalar hair with current and future detectors.

Abstract

The detection of black hole scalar hair, a possible deviation from general relativity's "no-hair" theorem, requires sensitive probes beyond conventional methods. This study proposes chaotic dynamics as a novel indicator for scalar hair in Horndeski gravity. We investigate the motion of a spinning test particle in a static, spherically symmetric hairy black hole spacetime. Our results show that increasing scalar hair systematically suppresses orbital chaos, as evidenced by regularized precession, reduced Lyapunov exponents, and contracted Poincare sections. Furthermore, scalar hair enhances the correlation between the two gravitational wave polarization modes, restoring phase coherence. These findings demonstrate that chaotic observables and gravitational wave signatures can jointly serve as sensitive probes for black hole hair, offering a complementary approach to testing gravity in strong-field regimes.

Figures (11)

• Figure 1: Orbital evolution of a spinless particle for hairy parameter values $h = 0$, $0.5$, and $1$ (left to right).
• Figure 2: Orbital evolution of a spinning particle ($S=0.5$) for $h = 0, 0.5, 1$ (top to bottom). Left column: $x$–$y$ projections. Right column: full three-dimensional trajectories.
• Figure 3: Orbital evolution of a spinning particle ($S=1$) for $h = 0, 0.5, 1$ (top to bottom). Left column: $x$–$y$ projections. Right column: full three-dimensional trajectories.
• Figure 4: Shown are $\tilde{r}$ (left) and the polar angle $\theta$ (right) as functions of the sampling index $\tilde{t}$, for $S=1$.
• Figure 5: Effective potentials in cylindrical coordinates, varying over the hairy parameter $h$ (columns: $0,0.5,1$) and the particle spin $S$ (rows: $0,0.5,1$).