How Far Away are Gravitational Lens Caustics? Wrong Question

Abstract

It has been a persistent question at least for a decade where the gravitational lens caustics are in the radial direction: whether in front of the lensing mass, behind the lensing mass, or on the plane normal to the line of sight that passes through the lensing mass, the radiation source, or the observer. It is a wrong question. And, the truth angers certain referees who somehow possess the ability to write lengthy rubbish referee reports and delay certain papers indefinitely. General relativity is a metric theory, particularly of Riemannian geometry, which is characterized by the existence of an inner product -- or, the invariance of the proper time. According to Einstein field equations, a compact mass defines a spherical geometry around it and focuses photons from a distant source to an observer with the source and observer as the two focal points. When the mass is spherically symmetric, the two dimensional lens equation that relates the angular positions of a source and its images defines a point caustic at the angular position of the lensing mass. The third (radial) position of the point caustic is not defined. For an arbitrary mass, the caustic extends into a web of piecewise smooth curves punctuated by cusps and again its notion exists only within the context of the lens equation. We point out a few errors in a couple of papers, published in the Astrophysical Journal, which may be influential.