Self-interaction of a point charge in a Kerr spacetime

Abstract

As is known, the expression of the electromagnetic field from a charge at rest on the symmetry axis of a Kerr spacetime, first given as a multi-polar expansion, has been later calculated in closed form. This allowed the derivation, in closed form, of the electromagnetic 4-potential and, consequently, the electrostatic self-interaction force (“self-force”) on the charge. We point out that, in order to derive the electromagnetic 4-potential and, consequently, the electrostatic self-force, a particular reference frame is required, in which the Kerr metric “locally” (i.e., in a small neighbourhood of the charge world-line) reduces to that of a static homogeneous gravitational field (SHGF). The starting point of this paper is the realization that the choice of such a reference frame in the literature is incorrect, because the phenomenon of dragging of inertial frames is neglected. This demands a new derivation both of the electromagnetic 4-potential and of the electrostatic self-force. To this end, we construct the correct reference frame, “locally“ transforming the Kerr metric into that of an SHGF. In such a frame, we derive the correct expressions both of the electromagnetic 4-potential and of the electrostatic self-force. In particular: i) the time component of the 4-potential differs from that known in the literature by an additional term, induced by the dragging of inertial frames, vanishing only on the symmetry axis; ii) the self-force turns out to be the sum of two terms, of which only the first (involving both the mass and the angular momentum per unit mass of the Kerr spacetime) is known in the literature. The second term (involving only the angular momentum per unit mass), which arises from a careful calculation and not from the additional term in the 4-potential, is a significant improvement provided by this paper.