Luís Filipe O. Costa

Centro de Física do Porto

Aspects of the motion of gyroscopes around Schwarzschild and Kerr Black Holes - exact gravito-electromagnetic analogies

We write a covariant form for the electromagnetic force exerted on a moving magnetic dipole, and compare it with Papapetrou's equation for the gravitational force exerted on a spinning test particle. We show that if Pirani supplementary spin condition holds, there is an exact, covariant, and fully general analogy relating these two forces: both are determined by a contraction of the spin 4-vector with a magnetic-type tidal tensor. Moreover, these tidal tensors obey strikingly analogous equations which are shown to be a covariant form of Maxwell's equations and (some of) Einstein's field equations. We exemplify by considering gyroscopes in Schwarzschild and Kerr spacetimes, and comparing with the analogous situation of magnetic dipoles moving in the electromagnetic field of non-spinning and spinning charges. It is shown that, in the special case that the gyroscope/dipole are at rest and far from the source, the two forces are similar (which is in accordance with the results known from linearized theory, e.g. [arXiv:gr-qc/0207065]); but that for generic dynamics key differences (overlooked in the literature) arise. These differences, which are transparent in the symmetries of the tidal tensors, prove to be particularly illuminating to the understanding of many aspects of spin curvature coupling. In particular we show that in the Kerr spacetime there are velocities for which there is no gravitational force on the gyroscope, which in the electromagnetic case is forbidden by Maxwell's equations; that the time projection of the force on a dipole is the power transferred to it by Faraday's induction, whereas the fact that the force on a gyroscope is spatial signals the absence of an analogous gravitational effect; that whereas the total work done on a magnetic dipole by a stationary magnetic field is zero, a stationary gravitomagnetic field, by contrast, does work on mass currents, which is shown to quantitatively explain the Hawking-Wald Spin Interaction Energy [PRL. 26 1344, 1971; PRD 6 406, 1972].