Sharp decay estimates for massless Dirac fields on a Schwarzschild background

TL;DR

This paper establishes sharp decay rates for massless Dirac fields on a Schwarzschild background, confirming the Price's law conjecture through explicit energy bounds and pointwise decay estimates.

Contribution

It provides the first rigorous proof of the precise decay rates for massless Dirac fields outside a Schwarzschild black hole, including both upper and lower bounds.

Findings

Uniform energy bounds for Dirac components

Integrated local energy decay estimates

Explicit pointwise decay rates matching Price's law

Abstract

We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin $s=\pm \frac{1}{2}$ components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay $fv^{-3/2-s}\tau^{-5/2+s}$ as both an upper and a lower bound outside the black hole, with function $f$ finite and explicitly expressed in terms of the initial data and the coordinates. This establishes the validity of the conjectured Price's law for massless Dirac fields outside a Schwarzschild black hole.