Non-conservative gravitational fields around spinning spherical objects (and the absence of gravitomagnetic fields) as revealed by the Kerr metric

Abstract

The Kerr metric in the equatorial plane of a spinning, spherical mass is scrutinized. It turns out that the gravitational field lines of a spinning, massive sphere are not strictly straight, but are, in the equatorial plane, curved because of a tangential component. This was mentioned by N.A. Sharp in 1979, although without any proof or reference. An equation for determining the tangential component of the gravitational field is provided. Thereby it is shown that a gravitational Lorentz force and hence a gravitomagnetic field do not exist, although this had been postulated by Heaviside and Thirring. Because of the tangential component, the gravitational field around a spinning spherical body is not conservative. Hence, there is an analogy to the electric field, which, too, can either be conservative (as is the case for the electrostatic field), or non-conservative (which is the case whenever the magnetic flux en-circled by a path in space is subject to change). An orbiter held on a circular trajectory in the equatorial plane and circling with the spin of the central mass thus experiences a steady onward force like a charged particle in a ring − shaped particle accelerator does. The gain in kinetic energy of an orbiter is at the expense of the rotational kinetic energy of the central, spinning mass. This is because the gravitational field lines of any orbiter, too, are curved, and thus exert a torque on the central, spinning mass.