The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity.

In this paper the theory of the infinitesimal, baryon-number conserving, adiabatic, radial oscillations of a gas sphere is developed in the framework of general relativity. A variational base for determining the characteristic frequencies of oscillation is established. It provides a convenient method for obtaining sufficient conditions for the occurrence of dynamical instability. The principal result of the analysis is the demonstration that the Newtonian lower limit T4, for the ratio of the specific heats , for insuring dynamical stability is increased by effects arising from general relativity; indeed, is increased to an extent that, so long as is finite, dynamical instability will intervene before a mass contracts to the limiting radius (>2.25 GM/c2) compatible with hydrostatic equilibrium. Moreover, if should exceed only by a small amount, then dynamical instability will occur if the mass should contract to the radius Rc = K 4 C2 -3 where K is a constant depending, principally, on the density distribution in the configuration. The value of the constant K is explicitly evaluated for the homogeneous sphere of constant energy density and the polytropes of indices n = 1, 2, and 3.