Particle Motion and Chaos

In this note, we consider particle falling in the black hole with an additional potential. Following the proposal by Susskind (2018), we study the growth rate of the particle's Rindler momentum, which corresponds to the growth rate of the operator size in the dual chaotic system. A general analysis near the horizon shows that the growth rate of the particle's Rindler momentum of the particle falling with a regular potential is the same as that of the particle free falling, which saturates the chaos bound. However, when the potential is singular, the growth rate is suppressed such that it is below the Lyapunov exponent. It implies that the chaos suppression may be captured by an additional singular potential in the gravity side. We further explicitly study a particle falling in hyperscaling violating spacetime to confirm the general analysis results. Finally, we study the particle falling in AdS soliton geometry. It also exhibits a suppression of the growth of the Rindler momentum. It is attributed to that when the repulsive potential is introduced or the black hole horizon is absent, the particle is slowed down, and its trajectory seen by a comoving observer is timelike, which corresponds to a weak chaos system.