The brachistochrone in almost flat space

This paper is an extension, within the framework of general relativity, of the relativistic brachistochrone discussed recently by Goldstein and Bender [J. Math. Phys. 27, 507 (1986)]. Assuming that the gravitational field due to a spherically symmetric source with mass M at equilibrium is weak, it is found that the brachistochrone, for a falling particle of mass m, described by ⊖(r), with ⊖ an angle and r a distance measured from the center of symmetry, is in general a hyperelliptic integral. The latter integral can in one case be calculated exactly in terms of the normal elliptic integrals of the first and third kinds and the elementary transcendental functions. It is shown via a numerical computation using the sun's gravitational field as a reference that one can recast this exact version into a simple form, viz., √r⊖ = a, where a is a constant. © 1988 American Institute of Physics.