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.