Light focusing by gravitational fields, or the gravitational lens effect (GLE), has been under investigation in relation to General Relativity and theoretical astrophysics for over half a century. During the 196Os, the study of geometric optics in curved space-time has received a strong new impetus from the works of Ehlers, Sachs, Penrose, and others.' These recent developments have resulted in formulation of the optical scalars theory and of the focusing theorem. This theorem, which is a differential equation governing the change of the cross-sectional area of an arbitrary beam in an arbitrary gravitational field, is of singular importance.In regard to the astrophysical side of the question, one requires more information to interpret the observational data than the mere possession of the equation itsdf. One needs its solutions, obtained for some realistic situations and expressed in terms of observable quantities.The only analytic solution of the GLE obtained so far has been the approximate solution, which corresponds to what may be called a strong lens effect in a weak gravitational field. In fact, to my knowledge, all the analytic considerations of the GLE are based on the admission of two simplifying.assumptions: that the gravitational field is assumed to be weak everywhere along the ray so that the linearized Einsteinian theory can be used, and that only paraxial bundles of rays are considered, or, in other words, the light source is supposed to lie close to the straight line between the observer and deflector. Kantowski' studied light focusing in the Swiss-cheese cosmological models without these restrictions, but the differential equation was integrated numerically.In this paper, the exact solution of the geometric-optic problem of light focusing is obtained by use of a spherically symmetric gravitational field in a vacuum.In the next section, light rays in a Schwarzschild-like field are described by means of the parameter 8, which is the angle between the ray and the radial direction.In the last section, the cross-sectional area of a bundle of rays is expressed in terms of the variation, 68, of that parameter across the beam. For 68, an auxiliary differential equation is constructed and solved, thus yielding the cross-sectional area as a function of a point along the ray. Some properties of this solution are discussed.
LIGHT RAYS IN THE SCHWARZSCHILD-LIKE METRICIn this section, we shall consider null geodesics in a spherically symmetric gravitational field in a vacuum. In Schwarzschild-like coordinates, the line element is(1 ) where x = rdr, rg = 2CM/c2 is the gravitational radius of the central body with mass M, and the speed of light c = 1.We shall choose the polar axis directed from the center toward.1he observer (see FIG. 1). Then, any ray reaching the observer lies in a cp = const. plane. To avoid the 81