Erratum: Remarks on the mass-angular momentum relations for two extreme Kerr sources in equilibrium [Phys. Rev. D82, 124042 (2010)]

Abstract

The general analysis of the relations between masses and angular momenta in the configurations composed of two balancing extremal Kerr particles is made on the basis of two exact solutions arising as extreme limits of the well-known double-Kerr spacetime. We show that the inequality M 2 ≥ |J| characteristic of an isolated Kerr black hole is verified by all the extremal components of the Tomimatsu and Dietz-Hoenselaers solutions. At the same time, the inequality can be violated by the total masses and total angular momenta of these binary systems, and we identify all the cases when such violation occurs.

Stationary configurations of two extreme black holes obtainable from the Kinnersley-Chitre solution

“…This is apparently in contrast to the single Kerr extreme geometry characterized by the equality |J| = M 2 , but, at the same time, in full agreement with the recent numerical study of the interacting black holes carried out in [7] and [21]. It is worth noting that in the equilibrium configurations of two extreme Kerr sources recently considered in the paper [5], the individual Komar quantities verify the inequality |J i | ≤ M 2 i characterizing a single Kerr black hole [22], but this of course should be attributed to the presence of a negative mass in all such configurations from [5].…”

“…i characterizing a single Kerr black hole [22], but this of course should be attributed to the presence of a negative mass in all such configurations from [5].…”

“…From our analysis it follows in particular that the equality m 2 − a 2 − e 2 = 0 relating the mass, angular momentum per unit mass and electric charge of a single Kerr-Newman extreme black hole is no longer verified by the analogous extreme black-hole constituents in binary configurations.The well-known Kinnersley-Chitre (KC) solution [1] represents the extremal limit of the double-Kerr spacetime [2], and as such permits one to describe stationary axisymmetric configurations of two extreme Kerr sources. In the case of the balancing constituents [3][4][5] it can be shown that at least one of these constituents is endowed with a negative mass, which spoils the interpretation of those binary configurations as describing two black holes. This naturally motivates the search for (and subsequent analysis of) the systems composed of two extreme black holes separated by a massless strut (conical singularity [6]) which may arise for instance in the context of some more general nonstationary axisymmetric scenarios as momentary stationary data [7] for interacting black holes.…”

On a Simple Representation of the Kinnersley-Chitre Metric

“…We end up this section by noting that the balance condition γ(x = 1) = 0 which, if satisfied, leads to the disappearance of a strut in the configurations from the subfamilies A, B and C considered above, reduces to taking the limit p 2 = q 2 = 1/2; however, in no one case this limit leads to the equilibrium states with positive masses of both constituents. For instance, the equilibrium configuration due to Dietz and Hoenselaers [21] which recently has been analyzed in the paper [29] follows from (23) as the limiting case…”

“…Note that E and Ē in the numerator of the function ω from Eq. (14) of Ref [29]. must have the subscript '-'.…”