Sharp decay estimates for massless Dirac fields on a Schwarzschild background

Abstract

We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin s=±12 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−sτ−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.