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 $K = -λ^2$ between the Gaussian curvature $K$ and the Lyapunov exponent $λ$, 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 $\mathcal{G} \propto -λ^2$ 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 $λ$ and yields a critical exponent $δ=1/2$. Our results demonstrate that spacetime geometry encodes thermodynamic information, opening a new pathway for studying black hole phase transitions from a geometric perspective.

Figures (1)

• Figure 1: Phase transition signatures in dynamics and geometry for timelike orbits, with $\tilde{g}_c=0.09002$, $a=0.6$, $L=20\ell$. Left column ($\tilde{g}=0.061538$, $\tilde{g}<\tilde{g}_c$): (a) free energy $\tilde{F}_{HL}$ versus $\tilde{T}_{HL}$, (c) Lyapunov exponent $\lambda_{HL}$ versus $\tilde{T}_{HL}$, (e) geometric quantity $|\mathcal{G}_{HL}|$ versus $\tilde{T}_{HL}$ (log scale). Right column ($\tilde{g}=0.092308$, $\tilde{g}>\tilde{g}_c$): (b) $\tilde{F}_{HL}$ versus $\tilde{T}_{HL}$, (d) $\lambda_{HL}$ versus $\tilde{T}_{HL}$, (f) $|\mathcal{G}_{HL}|$ versus $\tilde{T}_{HL}$ (log scale). The synchronized multivalued behavior of $\lambda_{HL}$ and $|\mathcal{G}_{HL}|$ in the spinodal region $\tilde{T}_{HL} \in (\tilde{T}_{1},\, \tilde{T}_{2})$ corresponds to the swallowtail structure in the free energy, with the phase transition occurring at $\tilde{T}_p$. All quantities become monotonic in the absence of a phase transition.