Axially symmetric two-body problem in general relativity. IV. Boundary conditions, time scales, and the quadrupole formula

Abstract

The analysis of the field equations and the conservation laws is extended into the skin boundary region which maintains the initially static configuration of the two fluid spheres. Although this leads to a correction of the gravitational-radiation energy-loss rate from a dependence of $E\ensuremath{\sim}{\ensuremath{\alpha}}^{\ensuremath{-}4}{{\ensuremath{\rho}}_{0}}^{\ensuremath{-}4}$ to ${\ensuremath{\alpha}}^{\ensuremath{-}6}{{\ensuremath{\rho}}_{0}}^{\ensuremath{-}2}$, the importance of the nonlinear structure-dependent terms remains and the essential conclusion, that the quadrupole formula does not apply to this problem, is unaltered. The hydrodynamic, stress-breaking, and free-fall time scales are considered. It is shown that insofar as the quadrupole-formula comparison for free-fall is concerned, only the contribution from bulk motion of the fluid spheres need be considered since tidal quadrupole deformation contributes negligibly to the quadrupole formula. With reference to our problem, it is shown that a recent derivation of the quadrupole formula for free-fall by Walker and Will is incorrect and it is suggested that certain other derivations may be applicable only to the radiation damping of a single body.