Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse

The Einstein equations for a spherically symmetrical distribution of matter are studied. The matter is described by the stress-energy tensor of an ideal fluid (heat flow and radiation are therefore excluded). In this case, the Einstein equations give a generalization of the Oppenheimer-Volkoff equations of hydrostatic equilibrium so as to include an acceleration term and a contribution to the effective mass of a shell of matter arising from its kinetic energy. A second equation also appears in this time-dependent case; it gives the rate of change of an appropriate "total energy" $m(r, t)$ of each fluid sphere in terms of the work done on this sphere by the fluid surrounding it. These equations would be an appropriate starting point for a study of relativistic gravitational collapse in which an adiabatic equation of state more realistic than the $p=0$ form of Oppenheimer and Snyder could be used.