Effect of gravitational radiation reaction on circular orbits around a spinning black hole.

The effect of gravitational radiation reaction on circular orbits around a spinning (Kerr) black hole is computed to leading order in S (the magnitude of the spin angular momentum of the hole) and in the strength of gravity M/r (where M is the mass of the black hole, r is the orbital radius, and G = c = 1). The radiation reaction makes the orbit shrink but leaves it circular, and drives the orbital plane very slowly toward antialignment with the spin of the hole: tan(ι/2) = tan(ι0/2)[1+(61/72)(S/M 2)(M/r)3/2], where ι is the angle between the normal to the orbital plane and the spin direction, and ι0 is the initial value of ι, when r is very large. PACS numbers: 04.25.Nx, 04.30.Db Typeset using REVTEX 1 The earth-based LIGO/VIRGO network of gravitational wave detectors (which is now under construction) will be used to search for and study the gravitational waves from “particles”, such as neutron stars and small black holes, spiraling into massive black holes (mass M up to ∼ 300M⊙); and ESA’s planned space-based LISA [1] interferometer will do the same for inspiral into supermassive black holes (M up to ∼ 10M⊙). To search for the inspiral waves and extract the information they carry will require templates based on theoretical calculations of the emitted waveforms; and to compute the waveforms requires a detailed understanding of how radiation reaction influences the orbital evolution. For several years a stumbling block has impeded computations of the evolution, when the orbital plane of the particle is inclined to the equatorial plane of a spinning hole: No practical method has been developed to deduce how radiation reaction influences the evolution of the orbit’s “Carter constant” [2,3], which governs the orbital shape and inclination angle. This Letter describes the first progress on this problem: a “post-Newtonian” gravitational radiation reaction force is used to compute the full orbital evolution to first order in S, the magnitude of the spin angular momentum of the black hole, and leading order in the strength of gravityM/r at the orbital radius r. (Here and throughout, units withG = c = 1 are used.) The analysis is restricted to orbits that initially are “circular” (more precisely, orbits which have constant radius r—these orbits are circular in the “orbital plane” discussed below, but this plane precesses). However, the method can readily be extended to noncircular orbits [4] and (with considerably more difficulty) should be extendible to the fully relativistic regime r ∼ M . The computation of the evolution presented here proceeds as follows: First, in the absence of radiation reaction, the orbital motion and the associated constants of motion are reviewed. Then, the leading order radiation reaction accelerations that act on the orbiting particle and on the hole are derived and used to compute the radiation-reaction-induced evolution of the constants of motion. Finally, the evolution of the orbit—its shape and inclination angle—is obtained. The leading order effect of the spin on the (otherwise Newtonian) orbit was deduced long 2 ago by Lense and Thirring [5] (reviewed by Landau and Lifshitz [6])—though, of course, they regarded the central body as a star rather than a black hole. In fact, our analysis does not require the body to be a black hole, but since this is the primary case of physical interest, the discussion is phrased in terms of a black hole. Let spherical polar coordinates, r, θ, and φ, centered on the black hole, be used to describe the location of the particle (these coordinates describe the relative separation of the two bodies), with the hole’s spin along the polar axis. The Lagrangian [7] for the motion of the particle (which, for now, does not have to be circular) is given, to linear order in S but otherwise in solely Newtonian theory, by