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

The most promising way to compute the gravitational waves emitted by binary black holes (BBHs) in their last dozen orbits, where post-Newtonian techniques fail, is a quasistationary approximation introduced by Detweiler and being pursued by Price and others. In this approximation the outgoing gravitational waves at infinity and downgoing gravitational waves at the holes’ horizons are replaced by standing waves so as to guarantee that the spacetime has a helical Killing vector field. Because the horizon generators will not, in general, be tidally locked to the holes’ orbital motion, the standing waves will destroy the horizons, converting the black holes into naked singularities that resemble black holes down to near the horizon radius. This paper uses a spherically symmetric, scalar-field model problem to explore in detail the following BBH issues: (i) The destruction of a horizon by the standing waves. (ii) The accuracy 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-reaction field.” (iv) The accuracy with which the resulting downgoing waves agree with the downgoing waves of a true black-hole spacetime (with horizon). The model problem used to study these issues consists of a Schwarzschild black hole endowed with spherical standing waves of a scalar field, whose wave frequency and near-horizon energy density are chosen to match those of the standing gravitational waves of the BBH quasistationary approximation. It is found that the spacetime metric of the singular, standing-wave spacetime, and its radiation-reaction-field-constructed downgoing waves are quite close to those for a Schwarzschild black hole with downgoing waves—sufficiently close to make the BBH quasistationary approximation look promising for non-tidally-locked black holes.

Figure 1

The standing-wave scalar field in a Schwarzschild background (solid curve) and the effective potential (dashed curve) for angular frequency $\omega =0.049$.

Figure 2

Metric perturbations for a standing-wave scalar field in a Schwarzschild background with angular frequency $\omega =0.19$ and amplitude ${\Psi }_0=0.015$ far from the black hole, corresponding to a binary separation $a\approx 6M$ [Eq. (12a)].

Figure 3

Embedding diagram for the spacetime with time-averaged standing-wave scalar field of angular frequency $\omega =0.19$ and amplitude ${\Psi }_0=0.015$ at large radii [corresponding to the binary-black-hole separation $a\approx 6M$; Eq. (12a)]. The solid line represents embedding in Euclidean space; the dashed line, embedding in Minkowski space. Regions I, II and III are labeled on plot.

Figure 4

(a) Redshift $z=\delta \lambda /\lambda$ of light emitted from radius $r$ and received by an observer at $r=10$. (b) Redshift for an observer at $r=0.0001$. A distant observer would see light emitted from $r=0.0001$ redshifted by $\ln (z+1)\approx {10}^5$. These curves are drawn for the spacetime with time-averaged standing-wave scalar field that has angular frequency $\omega =0.19$ and amplitude ${\Psi }_0=0.015$ at large radii [corresponding to the binary-black-hole separation $a\approx 6M$; Eq. (12a)].

Figure 5

(a) Fractional differences of the metric components $g_{tt}=-g_{r^{*}{r}^{*}}$ (solid curve) and $g_{\theta \theta }$ (dashed curve) between Schwarzschild spacetime $D$ and standing-wave scalar-field spacetime $S$ with scalar-wave amplitude and frequency chosen to model BBH separation $a\approx 6M$ [Eq. (12a)]. (b) Same quantities plotted for scalar-field parameters chosen to model BBH separation $a\approx 15M$ [Eq. (12b)].

Figure 6

(a) The fractional difference in the amplitudes of the reconstructed scalar field and downgoing scalar field $\delta d/d\equiv (d^{\mathrm{SW}+\mathrm{RR}}-d^{\mathrm{down}})/d^{\mathrm{down}}$ (solid curve) and the phase difference between the two fields $\delta {\varphi }_d={\varphi }_d^{\mathrm{SW}+\mathrm{RR}}-{\varphi }_d^{\mathrm{down}}$ (dashed curve), plotted vs $r^{*}$. Scalar-wave amplitude and frequency chosen to model BBH separation $a\approx 6M$. (b) Same quantities plotted for scalar-field parameters chosen to model BBH separation $a\approx 15M$.