Causality bounds chaos in geodesic motion

Abstract

Predictability is grounded by causality while may be practically restricted by the occurrence of chaos. To reveal the relation between these two popular notions, we study chaos in geodesic motion in generic curved spacetimes with external potentials, where causality is controlled by a scalar potential. We develop a reparametrization-independent method to analytically estimate the Lyapunov exponent $\ensuremath{\lambda}$ of particle motion. We show that causality gives the universal upper bound $\ensuremath{\lambda}\ensuremath{\propto}E\text{ }(E\ensuremath{\rightarrow}\ensuremath{\infty})$, which coincides with the chaos energy bound proposed by Murata, Tanahashi, Watanabe, and one of the authors (K. H.). We also find that the chaos bound discovered by Maldacena, Shenker, and Stanford can be violated in particular potentials, even with causality. Our estimates, although waiting for numerical confirmation, reveal the hidden nature of physical theories: causality bounds chaos.