Spin self-force

We consider the motion of charged and spinning bodies on the symmetry axis of a non-extremal Kerr-Newman black hole. If one treats the body as a test point particle of mass, $m$, charge $q$, and spin $S$, then by dropping the body into the black hole from sufficiently near the horizon, the first order area increase, $\delta A$, of the black hole can be made arbitrarily small, i.e., the process can be done in a “reversible” manner with regard to the change of parameters of the black hole. At second order, there may be effects on the energy delivered to the black hole—quadratic in $q$ and $S$—resulting from (i) the finite size of the body and (ii) self-force corrections to the energy. Sorce and Wald have calculated these effects for a charged, non-spinning body on the symmetry axis of an uncharged Kerr black hole and showed that, when these effects are included, ${\delta }^{2}A$ also can be made arbitrarily small, i.e., this process remains reversible to second order. We consider the generalization of this process for a charged and spinning body on the symmetry axis of a Kerr-Newman black hole, where the self-force effects have not been calculated. A spinning body (with negligible extent along the spin axis) must have a mass quadrupole moment $\gtrsim S^2/m$, so at quadratic order in the spin, we must take into account quadrupole effects on the motion. After taking into account all such finite size effects, we find that the condition ${\delta }^{2}A\ge 0$ yields a nontrivial lower bound on the self-force energy, $E_{SF}$, at the horizon. In particular, for an uncharged, spinning body on the axis of a Kerr black hole of mass $M$, the net contribution of spin self-force to the energy of the body at the horizon is $E_{SF}\ge S^2/8M^3$, corresponding to an overall repulsive spin self-force. A lower bound for the self-force energy, $E_{SF}$, for a body with both charge and spin at the horizon of a Kerr-Newman black hole is given. This lower bound will be the correct formula for $E_{SF}$ if the dropping process can be done reversibly to second order.