Price’s Law for Spin Fields on a Schwarzschild Background

Abstract

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.