Bobbing and kicks in electromagnetism and gravity

We study systems analogous to binary black holes with spin in order to gain some insight into the origin and nature of “bobbing” motion and “kicks” that occur in this system. Our basic tool is a general formalism for describing the motion of extended test bodies in an external electromagnetic field in curved spacetime and possibly subject to other forces. We first show that bobbing of exactly the type as observed in numerical simulations of the binary black hole system occurs in a simple system consisting of two spinning balls connected by an elastic band in flat spacetime. This bobbing may be understood as arising from the difference between a spinning body’s “lab frame centroid” and its true center of mass, and is purely “kinematical” in the sense that it will appear regardless of the forces holding two spinning bodies in orbit. Next, we develop precise rules for relating the motion of charged bodies in a stationary external electromagnetic field in flat spacetime with the motion of bodies in a weakly curved stationary spacetime. We then consider the system consisting of two orbiting charges with magnetic dipole moment and spin at a level of approximation corresponding to 1.5 post-Newtonian order. Here we find that considerable amounts of momentum are exchanged between the bodies and the electromagnetic field; however, the bodies store this momentum entirely as “hidden” mechanical momentum, so that the interchange does not give rise to any net bobbing. The net bobbing that does occur is due solely to the kinematical spin effect, and we therefore argue that the net bobbing of the electromagnetic binary is not associated with possible kicks. We believe that this conclusion holds in the gravitational case as well.

Figure 1

An illustration of kinematical bobbing. The lab frame centroid of a body (solid line) is confined to the $x^1\mathrm{\text{−}}{x}^2$ plane. However, since the body has a (constant) spin vector lying in the plane, its center of mass (solid dot) is displaced in the $x^3$ direction by a velocity-dependent factor [Eq. (7)], giving rise to bobbing. The body is shown at two phases in its orbit. In the foreground, the bottom portion of the body contains extra lab frame kinetic energy (illustrated with shading), skewing the lab frame centroid downward so that the center of mass must have bobbed up. One-half of an orbit later (shown in the background), the situation is reversed and the center of mass has bobbed down. If two bodies with anti-aligned spins were considered, a net bobbing would be observed.