Unified equation for massless spin fields and new definitions of key spin coefficients

Whether studying gravitational waves from extreme mass ratio inspirals or exploring the analogy between massless spin-particle waves, black hole perturbation theory proves indispensable. At the heart of developing a universal perturbation framework for such problems lies the challenge of formulating a coordinate-independent, unified wave equation that is universally applicable to any black hole spacetime. This paper resolves this central issue in type-D spacetimes by introducing a generating function $H$ and establishing new definitions for the key spin coefficients. Specifically, the spin coefficients $\rho$, $\mu$, $\tau$, and $\pi$ are redefined, respectively, as the directional derivatives of the logarithm of the generating function along the null tetrad ($l^{\mu}$, $n^{\mu}$, $m^{\mu}$, $\bar{m}^{\mu}$), and the field quantities are rescaled using $H$. It is thereby found that the field equations governing massless particles of spins $0$, $1/2$, $1$, $3/2$, and $2$ in arbitrary type-D black hole spacetimes can all be described by a single, unified equation. This finding is particularly remarkable, as unifying these field equations is already a significant challenge in flat spacetime, let alone in the intricate spacetime around black holes. Consequently, this work will inevitably prompt a re-examination of the shared characteristics among various types of particles in black hole spacetimes. Meanwhile, we verify the correctness of the new definition for the spin coefficients, and provide the explicit form of the unified equation for nearly all known type-D black hole backgrounds. This lays a solid foundation not only for studying gravitational waves from extreme mass ratio inspirals but also for exploring the analogy between massless spin-particle waves in any type-D black hole background.