Gravitational Spin Interaction

The gravitational spin interaction is investigated by studying the deviation from geodesic motion of spinning test bodies. The force on a spinning test body at rest in the exterior field of an arbitrary stationary, rotating source is evaluated and found to be given by

F⃗=-∇⃗(-S⃗·J⃗+3(r̂·J⃗)(r̂·S⃗)/r³)+O(1/r⁵) where S⃗ is the spin of the test body, J⃗ is the angular momentum of the source of the gravitational field, and geometrized units G=c=1 are used. Thus, the gravitational spin-spin force has the same form as the force between two dipoles in electromagnetism except that its sign is opposite, i.e., "north pole" attracts "north pole" in gravitational spin-spin interaction. The gravitational spin-spin torque, previously investigated by Schiff, Mashhoon, and Wilkins, also has opposite sign to the corresponding electromagnetic dipole-dipole torque. The sign of the spin-spin force agrees with that predicted by Hawking on the basis of the fact that less energy can be extracted from colliding black holes if their spins are parallel rather than antiparallel. Furthermore, it is shown that the spin-interaction energy quantitatively accounts for the angular momentum dependence in Hawking's formula for the upper limit for energy released from colliding black holes. The gravitational spin-orbit force is also investigated, and it is found to differ in form from the corresponding electromagnetic spin-orbit force.