Propagation of light in the stationary field of multipole gravitational lens

Abstract

A rigorous mathematical formalism for calculating the propagation of light rays in the stationary post-Newtonian field of an isolated celestial body (or system of bodies) considered as a gravitational lens having a complex multipole structure is developed. Symmetric trace-free tensors are used in the definition of gravitational multipoles instead of the less convenient (in general situations) scalar and vector spherical harmonics. Two types of perturbations of light rays, caused correspondingly by the mass and spin multipoles, are analyzed in full detail. A new simple method of integration for the equations of light propagation is proposed. This method enables us for the first time to obtain complete expressions both for the relativistic time delay and for the angle of the total deflection of light in any order of multipole perturbations without restriction. The results thus obtained can be applied to the interpretation of the secondary weak gravitational lens effects produced by the solar system bodies, stars, binary pulsars, and galaxies where the influence of higher-order multipoles on the propagation of null rays may be important and measurable. The methods developed in the paper can be also applied to physical optics of multipole electromagnetic lenses and for calculation of propagation of gravitational waves through the curved space–time. As a particular application of the method the generalized equation for a multipole gravitational lens is derived using Cartesian coordinates and symmetric transverse-traceless tensors.