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It is well known that Teukolsky equation of gravitational perturbationsprovides a powerful tool for the investigation of the ringdown phase of abinary black hole merger, but it is applicable only for final configurations inthe Kerr cases. Here we introduce a new concept: the spin-coefficientconnection. Employing it and a new transformation function, we find that thegravitational perturbations can be written as a single equation. Eachcoefficients of this equation has explicit physical significance. Moreimportantly, the equation is universally applied to virtually any black holesince it does not depend on specific metrics and coordinate system. We alsofind that any massless field of nonzero spin $s<2$ obeys the gravitationalwave-like equation, which shows that the fields can be used as the analoguemodels of gravitational wave.

A Fundamental Equation for Gravitational Wave and Its Analogues in Type D Spacetimes