Observer Time as a Coordinate in Relativistic Spherical Hydrodynamics

It is shown that space-time events at which 2Um/c3R> 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 is an invariantly defined "circumference" coordinate and rn(r,t) is a function obtained from a certain curvature invariant.J The singularities which follow the formation of such a "trapped surface" or "Schwarzschild surface," although unobservable in principle, halt numerical computations of spherically symmetric hydrodynamics problems using a diagonal metric before the more slowly moving regions outside the Schwarzschild surface have completed their observable motions. By using a retarded time coordinate ` we reformulate the relativistic equations in such a way that time dilation effects prevent the formation of Schwarzschild surfaces at finite values of s# while allowing all observable aspects of the dynamics to proceed. The treatment of energy transfer by radially outward-moving radiation is also greatly simplified by the use of this retarded time coordinate.