Observer Time as a Coordinate in Relativistic Spherical Hydrodynamics

Abstract

It is shown that space-time events at which 2Gm/c 2 R > 1 holds in a spherically symmetric, inward-moving fluid satisfy Penrose's criterion for a " trapped surface " so that light rays leading from these events can never escape to infinity but instead terminate in a singularity. [Here 2tcR is an invariantly defined " circumference " coordinate and m(r,t) is a function obtained from a certain curvature invariant.] The singularities which follow the formation of such a " trapped surface " or " Schwarzschild surface, " al-though unobservable in principle, halt numerical computations of spherically symmetric hydrodynamics problems using a diagonal metric before the more slowly moving regions outside the Schwarzschild sur-face have completed their observable motions. By using a retarded time coordinate u we reformulate the relativistic equations in such a way that time dilation effects prevent the formation of Schwarzschild surfaces at finite values of u while allowing all observable aspects of the dynamics to proceed. The treat-ment of energy transfer by radially outward-moving radiation is also greatly simplified by the use of this retarded time coordinate. Hydrodynamics with spherical symmetry has been studied within the framework of general relativity by many authors. The early investigations of Datt (1938) and of Oppenheimer and Snyder (1939) ignored pressure gradient forces. More recently the problem has been reconsidered, retaining these mechanical forces (Podurets 1964; Misner and Sharp 1964; Bardeen 1965; May and White 1965). The relativistic equations are necessary to develop the supernova models proposed by Colgate and White (1965) and have been applied in a number of simple examples by May and White (1966); these relativistic equations may also be relevant to the mechanisms for quasi-stellar radio sources suggested by Hoyle and Fowler (1963; see also Fowler 1965). Under some conditions gravitational collapse will proceed not toward a final equilib-rium but to a state of continuing collapse (Oppenheimer and Snyder 1939) where some of the matter is enveloped by a " Schwarzschild surface^ from the interior of which no light ray or other signal can escape. In this latter situation two classes of questions may be distinguished: first, those which center upon the description and ultimate fate of the matter inclosed by the Schwarzschild surface (Wheeler 1965 ; see also Harrison, Thorne, Wakano, and Wheeler 1965, chaps, viii and xi), and, second, those directed toward the external manifestations of this catastrophic central collapse, that is, the observable phenomena outside the Schwarzschild surface. For this second category of questions one finds that the hydrodynamic equations as previously formulated are unsuitable, as a consequence of the formation of Schwarzschild surfaces as discussed in I. In II the equations of relativistic spherical hydrodynamics are reformulated in a different co-ordinate system more suitable to the study of observable phenomena. Section III dis-cusses some aspects of this new system of equations. I. schwarzschild surfaces