Geometric unification of timelike orbital chaos and phase transitions in black holes

Abstract

The deep connection between black hole thermodynamics and spacetime geometry remains a central focus of general relativity. While recent studies have revealed a precise correspondence for null orbits, given by $K = -\lambda^2$ between the Gaussian curvature $K$ and the Lyapunov exponent $\lambda$, its validity for timelike orbits had remained unknown. Our work introduces the massive particle surface (MPS) framework and constructs a new geometric quantity $\mathcal{G}$. We demonstrate that $\mathcal{G} \propto -\lambda^2$ on unstable timelike orbits, thus establishing the geometry-dynamics correspondence for massive particles. Crucially, near the first-order phase transition of a black hole, $\mathcal{G}$ displays synchronized multivalued behavior with the Lyapunov exponent $\lambda$ and yields a critical exponent $\delta=1.0244$. Our results demonstrate that spacetime geometry encodes thermodynamic information, opening a new pathway for studying black hole phase transitions from a geometric perspective.