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The most promising way to compute the gravitational waves emitted by binaryblack holes (BBHs) in their last dozen orbits, where post-Newtonian techniquesfail, is a quasistationary approximation introduced by Detweiler and beingpursued by Price and others. In this approximation the outgoing gravitationalwaves at infinity and downgoing gravitational waves at the holes' horizons arereplaced by standing waves so as to guarantee that the spacetime has a helicalKilling vector field. Because the horizon generators will not, in general, betidally locked to the holes' orbital motion, the standing waves will destroythe horizons, converting the black holes into naked singularities that resembleblack holes down to near the horizon radius. This paper uses a sphericallysymmetric, scalar-field model problem to explore in detail the following BBHissues: (i) The destruction of a horizon by the standing waves. (ii) Theaccuracy with which the resulting naked singularity resembles a black hole.(iii) The conversion of the standing-wave spacetime (with a destroyed horizon)into a spacetime with downgoing waves by the addition of a ``radiation-reactionfield''. (iv) The accuracy with which the resulting downgoing waves agree withthe downgoing waves of a true black-hole spacetime (with horizon). The modelproblem used to study these issues consists of a Schwarzschild black holeendowed with spherical standing waves of a scalar field. It is found that thespacetime metric of the singular, standing-wave spacetime, and itsradiation-reaction-field-constructed downgoing waves are quite close to thosefor a Schwarzschild black hole with downgoing waves -- sufficiently close tomake the BBH quasistationary approximation look promising fornon-tidally-locked black holes.Comment: 12 pages, 6 figure

Geometry of a naked singularity created by standing waves near a Schwarzschild horizon, and its application to the binary black hole problem