On Continued Gravitational Contraction

TLDR

When a massive star exhausts its thermonuclear fuel, it collapses, and unless its mass is reduced by fission, rotation, or radiation, the contraction continues indefinitely. This paper investigates the solutions of the gravitational field equations that describe this continued contraction. The authors first present qualitative arguments that the star’s radius asymptotically approaches its gravitational radius, light becomes progressively redshifted, and escape angles narrow, then derive an analytic pressure‑free solution confirming these behaviors. They find that the collapse time for a comoving observer is finite—about a day for typical stellar masses—while an external observer sees the star shrink asymptotically toward its gravitational radius.

Abstract

When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely. In the present paper we study the solutions of the gravitational field equations which describe this process. In I, general and qualitative arguments are given on the behavior of the metrical tensor as the contraction progresses: the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles. In II, an analytic solution of the field equations confirming these general arguments is obtained for the case that the pressure within the star can be neglected. The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius.