On the Smarr formula for rotating dyonic black holes

Abstract

We revisit the derivation by Tomimatsu of the generalized Komar integrals giving the mass and angular momentum of rotating Einstein-Maxwell black holes. We show that, contrary to Tomimatsu's claim, the usual Smarr formula relating the horizon mass and angular momentum still holds in the presence of both electric and magnetic charges. The simplest case is that of dyonic Kerr-Newman black holes, for which we recover the modified Smarr formula relating the asymptotic mass and angular momentum, the difference between asymptotic and horizon masses being equal to the sum of the two Dirac string masses. Our results apply in particular to the case of dyonic dihole solutions which have been investigated recently.Comment: Journal version. Two references adde

On the Smarr formulas for electrovac spacetimes with line singularities

“…However, care should be taken in using (3.8) in the presence of Dirac or Misner strings extending to infinity, which is necessarily the case if the total magnetic or NUT charge is non-zero. As previously noted in [27], the electromagnetic contributions to (3.8)…”

“…Although the Tomimatsu approach for the strings breaks down for n = 0 because Ω ± diverges, we can nevertheless recover the results of [27] for the dyonic Kerr-Newman black hole by taking with due care the limit n → 0. In this limit the string electric charges Q ± = nu ± go to zero, so that Q H = Q, but the string potentials Φ ± = −p/2n diverge, their product going to the finite limit −(p/2)u ± .…”

“…A convenient setting for calculating Komar integrals in spaces with weak line singularities on the axis is a scheme developed by Tomimatsu [28,29] and corrected in [27]. Here we reformulate the approach of [27] in terms of the rod structure.…”

“…defining the electric potential in the corotating frame (for more details, see [27]), which for n = H differs from Tomimatsu's [29] Eq. (52) by the presence of the second term.…”

“…where, as advocated in [27], we have set the gauge so that the two Dirac strings are symmetric. The calculation of the electric charge, mass and angular momentum of the horizon rod, and of the conjugate variables, gives:…”