Laws of motion and precession for black holes and other bodies

Abstract

Laws of motion and precession are derived for a Kerr black hole or any other body which is far from all other sources of gravity (``isolated body'') and has multipole moments that change slowly with time. Previous work by D'Eath and others has shown that to high accuracy the body moves along a geodesic of the surrounding spacetime geometry, and Fermi-Walker transports its angular-momentum vector. This paper derives the largest corrections to the geodesic law of motion and Fermi-Walker law of transport. These corrections are due to coupling of the body's angular momentum and quadrupole moment to the Riemann curvature of the surrounding spacetime. The resulting laws of motion and precession are identical to those that have been derived previously, by many researchers, for test bodies with negligible self-gravity. However, the derivation given here is valid for any isolated body, regardless of the strength of its self-gravity. These laws of motion and precession can be converted into equations of motion and precession by combining them with an approximate solution to the Einstein field equations for the surrounding spacetime. As an example, the conversion is carried out for two gravitationally bound systems of bodies with sizes much less than their separations. The resulting equations of motion and precession are derived accurately through ${\mathrm{post}}^{1.5}$-Newtonian order. For the special case of two Kerr black holes orbiting each other, these equations of motion and precession (which include couplings of the holes' spins and quadrupole moments to spacetime curvature) reduce to equations previously derived by D'Eath. The precession due to coupling of a black hole's quadrupole moment to surrounding curvature may be large enough, if the hole lives at the center of a very dense star cluster, for observational detection by its effects on extragalactic radio jets. Unless the hole rotates very slowly, this quadrupole-induced precession is far larger than the spin-down of the hole by tidal distortion (``horizon viscosity''). When the hole is in orbit around a massive companion, the quadrupole-induced precession is far smaller than geodetic precession.