NECESSARY AND SUFFICIENT CONDITION FOR HYDROSTATIC EQUILIBRIUM IN GENERAL RELATIVITY

Abstract

We present explicit examples to show that the "compatibility criterion" (recently obtained by us toward providing equilibrium configurations compatible with the structure of general relativity) which states that for a given value of σ[≡ (P0/E0) ≡ the ratio of central pressure to central energy-density], the compactness ratio u[≡ (M/R), where M is the total mass and R is the radius of the configuration] of any static configuration cannot exceed the compactness ratio, uh, of the homogeneous density sphere (i.e., u ≤ uh) is capable of providing a necessary and sufficient condition for any regular configuration to be compatible with the state of hydrostatic equilibrium. This conclusion is drawn on the basis of the finding that the M–R relation gives the necessary and sufficient condition for dynamical stability of equilibrium configurations only when the compatibility criterion for these configurations is appropriately satisfied. In this regard, we construct an appropriate sequence composed of core-envelope models on the basis of compatibility criterion such that each member of this sequence satisfies the extreme case of causality condition v = c = 1 at the center. The maximum stable value of u ≃ 0.3389 (which occurs for the model corresponding to the maximum value of mass in the mass–radius relation) and the corresponding central value of the local adiabatic index, (Γ1)0 ≃ 2.5911, of this model are found fully consistent with those of the corresponding absolute values, u max ≤ 0.3406 and (Γ1)0 ≤ 2.5946, which impose strong constraints on these parameters of such models. In addition to this example, we also study dynamical stability of pure adiabatic polytropic configurations on the basis of variational method for the choice of the "trial function," ξ = reν/4, as well as the mass–central density relation, since the compatibility criterion is appropriately satisfied for these models. The results of this example provide additional proof in favor of the statement regarding compatibility criterion mentioned above. Together with other results, this study also confirms the previous claim that just the choice of the "trial function," ξ = reν/4, is capable of providing the necessary and sufficient condition for dynamical stability of a mass on the basis of variational method. Obviously, the upper bound on the compactness ratio of neutron stars, u ≅ 0.3389, which belongs to two-density model studied here, turns out to be much stronger than the corresponding "absolute" upper bound mentioned in the literature.