Sharp decay estimates for massless Dirac fields on a Schwarzschild background

Siyuan Ma, Lin Zhang

Published 2020-08-26Version 1

We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove a uniform bound estimate for a positive definite energy and an integrated local energy decay estimate for the spin $s=\pm \frac{1}{2}$ components of the Dirac field. Based on these estimates and depending on the asymptotics of the initial data, we further show these components have pointwise decay $c_1v^{-3/2-s}\tau^{-3/2+s}$ or $c_2v^{-3/2-s}\tau^{-5/2+s}$ as both an upper and a lower bound, with constants $c_1$ and $c_2$ explicitly expressed in terms of the initial data. This establishes the validity of the conjectured Price's law for massless Dirac fields outside a Schwarzschild black hole.