Tidal interaction of black holes and Newtonian viscous bodies
Abstract
The tidal interaction of a (rotating or nonrotating) black hole with nearby bodies produces changes in its mass, angular momentum, and surface area. Similarly, tidal forces acting on a Newtonian, viscous body do work on the body, change its angular momentum, and part of the transferred gravitational energy is dissipated into heat. The equations that describe the rate of change of the black-hole mass, angular momentum, and surface area as a result of the tidal interaction are compared with the equations that describe how the tidal forces do work, torque, and produce heat in the Newtonian body. The equations are strikingly similar, and unexpectedly, the correspondence between the Newtonian-body and black-hole results is revealed to hold in near-quantitative detail. The correspondence involves the combination k2 τ of ``Love quantities'' that incorporate the details of the body's internal structure; k2 is the tidal Love number, and τ is the viscosity-produced delay between the action of the tidal forces and the body's reaction. The combination k2 τ is of order GM/c3 for a black hole of mass M; it does not vanish, in spite of the fact that k2 is known to vanish individually for a nonrotating black hole.