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

Solving equilibrium problem for the double–Kerr spacetime 65
treating equilibrium of any combination of the subextreme (black hole)
and superextreme constituents. The complex Ernst potential [10] and
corresponding metric functions and entering the Papapetrou axisymmetric
line element
( are the Weyl–Papapetrou cylindrical coordinates and time)
for the extended double–Kerr solution have the form [11, 12]
where a bar over a symbol means complex conjugation, and the subindices
vary from 1 to 4.
The arbitrary parameters entering the formulas (2) are which
assume arbitrary complex values, and which can assume
arbitrary real values or occur in complex conjugate pairs. From (2)
follows that instead of the parameters and one can use a set of
the constant objects that facilitates the solution of
the balance equations. Without any lack of generality we can assume
then in the case of the real–valued
the parts of the symmetry axis and represent
the Killing horizons of two black holes, while a pair of complex conjugate
say and represents a superextreme Kerr constituent.