Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

64 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
attraction and spin–spin repulsion forces. Although it was shown later
that this claim was only partially correct in a sense that the black hole
equilibrium configurations necessarily implied the negative mass of one
of the constituents, the Kramer–Neugebauer paper gave a tremendous
impetus to the study of different two–body equilibrium problems within
the framework of general relativity.
A detailed analysis of the physical properties of the double–Kerr solution
had been probably started by the paper of Oohara and Sato [2] in
which the existence of two separated stationary limit surfaces associated
with each of the two black hole constituents was demonstrated, and it
was shown that the Tomimatsu–Sato solution [3] was a special case
of the double–Kerr spacetime. The balance conditions were first derived
and analyzed by Kihara and Tomimatsu [4, 5] and Tomimatsu produced
formulas [6] for the calculation of the individual masses and angular momenta
of the black holes; he also claimed the possibility of equilibrium
between two extreme black holes, but later on it was demonstrated by
Hoenselaers [7] that Tomimatsu’s equilibrium case involved a ring singularity
arising from the negative mass of one of the constituents. The main
result of the paper [7] was, however, the derivation of the expressions
for the Komar masses of the black hole constituents in equilibrium and
the numerical check that these masses could not simultaneously assume
positive values; this demonstrated (but a rigorous proof still was needed)
that two normal Kerr black holes with positive individual masses could
not be in gravitational equilibrium.
The non–existence of the black hole equilibrium configurations motivated
a search for the equilibrium states of the superextreme Kerr constituents.
Dietz and Hoenselaers used a complex continuation of the parameters
in order to pass from the subextreme case to the superextreme
one, and they found equilibrium positions of two equal superextreme
spinning masses [8]. Dietz and Hoenselaers passed over the possibility
to consider equilibrium configurations between a black hole and a
superextreme object, and in the paper [9] the existence of such configurations
was claimed for the first time, however without any analysis of
the positivity of the Komar masses of the constituents.
3. A NEW APPROACH TO THE
DOUBLE–KERR
EQUILIBRIUM PROBLEM
A new approach to solving the equilibrium problem for the double–
Kerr solution consists in using the extended version of this solution
and searching for the analytic balance formulas equally applicable for