Lyapunov exponents of scalar and spinor fields in the Schwarzschild spacetime

Abstract A semiclassical particle moving near the horizon of a Schwarzschild black hole is chaotic, and its Lyapunov exponent saturates the chaos bound proposed by Maldacena, Shenker, and Stanford, with the temperature being the Hawking temperature. Motivated by this, we consider the Lyapunov exponents of scalar and spinor fields in Schwarzschild spacetime by calculating their out-of-time-ordered commutators along the radial direction. Numerically, we find that the Lyapunov exponent of the scalar field is smaller than that of the spinor field. They are mainly contributed by the bound states near the horizon and lie below the chaos bound.