Abstract
Abstract
We study an axisymmetric metric satisfying the Petrov type D property with some additional
ansatze, but without assuming the vacuum condition. We find that our metric in turn becomes
conformal to the Kerr metric deformed by one function of the radial coordinate. We then study the
gravitational-wave equations on this background metric in the case that the conformal factor is
unity. We find that under an appropriate gauge condition, the homogeneous wave equations admit the
separation of the variables, which is also helpful for solving the nonhomogeneous equations. The
resultant ordinary differential equation for the radial coordinate gives a natural extension of
the Teukolsky equation.