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66 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
The two constituents of the double–Kerr solution will be in equilibrium
if the metric functions and satisfy the following conditions on
the symmetry axis
where‘+’, ‘0’ and ‘-’ denote the and
parts of the symmetry axis, respectively.
The balance equations have been obtained in the explicit form and
solved in the paper [12]. It turns out that for any choice of the parameters
there always exists a set of the constants determining an
equilibrium configuration of the two constituents:
where the complex parameter is subjected to the constraint
The above very concise balance formulas for have been used in
[12] for the analysis of of various particular equilibrium states involving
either black holes or superextreme objects.
4. THE KOMAR MASSES AND ANGULAR
MOMENTA
Although formulas (4) give a mathematical solution of the double–
Kerr equilibrium problem, they still need to be complemented by the
expressions of the individual Komar masses of the constituents in order
to be able to judge whether a particular equilibrium configuration is
physical or not, i.e. whether both masses of the constituents are positive.
In [12] the individual masses were calculated anew for each particular
equilibrium state. Recently, however, we have been able to obtain the
general elegant expressions [13] for the Komar masses and angular momenta
of each balancing constituent corresponding to the equilibrium
formulas (4). The details of the derivation of these expressions can be
found in Ref. [13], here we only give their explicit form: