Hawking Radiation via Complex Geodesics

We describe in detail the quantum tunneling of massive particles from Kerr black hole by using complex trajectories, which are solutions to the Hamilton’s equations of motion with imaginary proper time. The trajectories are smooth and cover the inner and outer horizon regions. Following the worldline approach, we compute the energy flux at the event horizon as a summation over these complex trajectories. Density of states is given with the aid of Carter’s constant and it is shown to be linear in momenta in the leading order, as long as the phase portrait of the system stays uniform. Under this assumption, we obtain the thermal spectrum $\sim (T_H^{+}{)}^4$.