Price’s Law for Spin Fields on a Schwarzschild Background

In this work, we derive the globally precise late-time asymptotics for the spin-\({\mathfrak {s}}\) fields on a Schwarzschild background, including the scalar field \(({\mathfrak {s}}=0)\), the Maxwell field \(({\mathfrak {s}}=\pm 1)\) and the linearized gravity \(({\mathfrak {s}}=\pm 2)\). The conjectured Price’s law in the physics literature which predicts the sharp rates of decay of the spin \(s=\pm {\mathfrak {s}}\) components towards the future null infinity as well as in a compact region is shown. Further, we confirm the heuristic claim by Barack and Ori that the spin \(+1, +2\) components have an extra power of decay at the event horizon than the conjectured Price’s law. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation that is satisfied by all these components.