Tidal interaction of black holes and Newtonian viscous bodies

TL;DR

This paper compares the tidal interactions of black holes and Newtonian viscous bodies, revealing a surprising quantitative similarity in their equations governing energy, angular momentum, and surface area changes.

Contribution

It demonstrates a detailed correspondence between black hole and Newtonian body tidal equations, involving the product of Love number and viscosity delay, despite black holes' unique properties.

Findings

Equations for black hole and Newtonian body tidal interactions are strikingly similar.

The product of Love number and viscosity delay, k_2 τ, is of order GM/c^3 for black holes.

Black holes exhibit a non-zero k_2 τ despite vanishing Love number k_2.

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 k_2 \tau of ``Love quantities'' that incorporate the details of the body's internal structure; k_2 is the tidal Love number, and \tau is the viscosity-produced delay between the action of the tidal forces and the body's reaction. The combination k_2 \tau is of order GM/c^3 for a black hole of mass M; it does not vanish, in spite of the fact that k_2 is known to vanish individually for a nonrotating black hole.