Necessary and sufficient condition for hydrostatic equilibrium in general relativity

We present explicit examples to show that the ‘compatibility criterion’ [recently obtained by us towards 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), whereM is the total mass andR is the radius of the configuration] of any static configuration cannot exceed the compactness ratio, uh, of the homogeneous density sphere (that is, 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 coreenvelope 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 centre. 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, umax ≤ 0.3406, and (Γ1)0 ≤ 2.5946, which impose strong constraints on these parameters of such models.