Revisiting the fastest way to circle a black hole

Abstract

The shortest time for a null particle traveling between two arbitrary points outside a static spherically symmetric black hole is revisited. We introduce a functional for the time taken by a null particle in traveling on the path between the two points. By variating the time functional, we analyze the possible path with the shortest travel time for the null particle. It is analytically proven that the Euler-Lagrange equation corresponding to the time-functional for finding the path with the shortest traveling time is equivalent to the geodesics equation. This is in agreement with Hod's conjecture on the fastest way to circle a black hole. We apply the formalism to the dirty black hole in Einstein-square-root nonlinear electrodynamics-dilaton theory. We calculate explicitly the time measured by an asymptotic observer which is needed for a null particle to circle the dirty black hole. Accordingly, a null particle circling the dirty black hole on an almost circular path of radius infinity achieves the shortest time.