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

TL;DR

This work introduces a generating-function based redefinition of four Newman-Penrose spin coefficients, $\rho,\mu,\tau,\pi$, enabling a coordinate-independent, universal perturbation framework for massless fields with spins $s\le 2$ in Petrov type-D spacetimes. By deriving a spin-coefficient connection $L_{\mu}$ and formulating the unified equation $\{(\nabla^{\mu}+pL^{\mu})(\nabla_{\mu}+pL_{\mu})-4p^{2}\psi_{2}+\tfrac{1}{6}R\}\Phi_p = 2\kappa_s T_p$ for $\chi_p^{(s)}=H^{p-s}\Phi_p$, the paper unifies the perturbation dynamics of scalar, neutrino, electromagnetic, Rarita-Schwinger, and gravitational fields within type-D backgrounds. It verifies the definitions across general spherically symmetric, Vaidya-type, Plebański-Demiański, complete black-hole-like, and Kerr-Newman-de Sitter spacetimes, providing explicit forms for the generating function $H$, the transformed wave functions, and the terms entering the unified equation. This unification broadens applicability beyond the Teukolsky framework, enabling EMRI modeling and cross-spin analyses in a wide class of type-D spacetimes. The results offer a solid foundation for exploring shared structural properties of massless fields in strong gravity and motivate extensions to non-type-D geometries as a working hypothesis.

Abstract

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 establishing a new definition of spin coefficients. Specifically, we introduce a new definition for the spin coefficients $ρ$, $μ$, $τ$, and $π$, which are defined as the directional derivatives of the logarithm of a generating function along the null tetrad ($l^μ$, $n^μ$, $m^μ$, $\bar{m}^μ$), respectively. This is the first discovery that these spin coefficients are interconnected through a generating function. Using the newly defined spin coefficients, we find that the field equations for massless particles with spins 0, 1/2, 1, 3/2, and 2 in arbitrary type-D black hole spacetimes can be described by a single unified equation. This finding is particularly surprising, 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.