Behaviour of Weyl–Levi Civita Coordinates for a Class of Solutions approximating the Schwarzschild Metric

Abstract

THE search for energy sources for the ultra-high-energy phenomena in astrophysics discovered recently (quasars and pulsars, for example) has led to renewed interest in the study of general-relativistic gravitational collapse. (For a survey of relativistic astrophysics, see ref. 1.) Important unsolved problems include whether realistic models of collapsing bodies must pass through an event horizon, and, if so, whether they must continue collapsing until the inevitable development of a singularity. It has been conjectured² that, once the simplifying assumption of spherical symmetry is dropped, it may be possible to avoid the formation of an event horizon (the generalization of the Schwarzschild radius) during collapse. In the hope that in the late stages of collapse the exterior field of the collapsing object might be well approximated by a static field, Israel² has attempted to prove that “… there exists a general class of Weyl [static, axially-symmetric exterior gravitational] fields which deviate arbitrarily little from Schwarzschild's solution for r ≥ (1 + √(2)) m, which are analytic except for a particle-like singularity (well within the gravitational radius r = 2m) and which are free of event horizons”.