Extended multi-soliton solutions of the Einstein field equations: II. Two comments on the existence of equilibrium states

Abstract

The results of our previous paper are applied to solving analytically the balance problem in the double-Kerr solution for all three possible types of binary systems, i.e. when a binary system is composed of two non-extreme black holes, of a non-extreme black hole and a hyperextreme object and of two hyperextreme objects. We also construct a new stationary electrovacuum metric representing binary systems of charged, magnetized, rotating, aligned masses involving one extreme object and on the basis of the numerical study of balance equations we conjecture that the equilibrium states in such systems are impossible.

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

“…It is worth noting that the aforementioned formulas (3.8) of [7] give for J 1 a positively defined quantity 4α 2 /(2 − l) 2 ; however, our choice of sign in the formula (9) for J 1 is justified not only by the expression (7) for the total angular momentum, but also by an independent numerical check of the Komar quantities that we made for the Tomimatsu solution using the metric (6) and integral formulas (23) of the paper [15]. It is easy to see that the above M i and J i verify the equilibrium formula…”

Non-Existence of Stationary Two-Black-Hole Configurations

“…The final explicit solution of the Tomimatsu-Kihara equilibrium conditions was found by Manko et al [18]. Following their idea, we start with the condition k = 0 on A ± , A 0 ,…”

“…Since a single Bäcklund transformation generates the Kerr-NUT solution that contains, by a special choice of its three parameters, the stationary black hole solution (Kerr solutions) and since Bäcklund transformations act as a "nonlinear" superposition principle, the double-Kerr-NUT solution was considered to be a good candidate for the solution of the two horizon problem and extensively discussed in the literature [8,[12][13][14][15][16][17][18][19]26,28]. However, there was no argument requiring that this particular solution be the only candidate.…”

“…Reformulations and numerical studies by Hoens-elaers [13] made plausible that the two gravitational sources (black hole candidates) of the double-Kerr-NUT solution (located at the intervals = 0, K 1 ≥ ζ ≥ K 2 and K 3 ≥ ζ ≥ K 4 ) cannot be in equilibrium if their Komar masses are positive. The first decisive step toward prove the Hoenselaers conjecture was taken by Manko et al [18], who derived an explicit and easily applicable form of the Tomimatsu-Kihara equilibrium conditions and, as an important complement, analytical formulae for the Komar masses and angular momenta of the gravitational sources. Manko and Ruiz [19] completed their non-existence proof by showing that the equilibrium conditions for the double-Kerr-NUT solution are indeed violated for positive Komar masses.…”

Realistic exact solution for the exterior field of a rotating neutron star

“…In the case of hyperextreme part, the Komar mass cannot be evaluated using the formula (23), and one more general integral expression should be used [51] …”