Resolving "dirty" effects around black holes by decoupling the Teukolsky equation

Detecting the environment around the supermassive black holes and tests of general relativity are important applications of extreme-mass-ratio inspirals (EMRIs). There is still a challenge to efficiently describe various "dirty" impacts on the inspirals like dark matter, gas, dipole radiation, electromagnetic interaction, and so on. In this Letter, we find the inherent linearity of the asymptotic solution of the inhomogeneous Teukolsky equation. Based on this property, we completely decouple the factors of the perturber and the background spacetime in the energy fluxes and waveforms. With the new decoupling form, the waveforms of EMRIs with non-geodesic motion in Kerr spacetime can be calculated conveniently. This will help to resolve the environment (including gas, field, dark matter, electromagnetic interaction, etc.) around the supermassive black holes and test general relativity.