Dirac dynamical resonance states around Schwarzschild black holes

Recently, a novel kind of scalar wigs around Schwarzschild black holes\char22{}scalar dynamical resonance states\char22{}was introduced in [Phys. Rev. D 84, 083008 (2011)] and [Phys. Rev. Lett. 109, 081102 (2012)]. In this paper, we investigate the existence and evolution of Dirac dynamical resonance states. First, we look for stationary resonance states of a Dirac field around a Schwarzchild black hole by using the Schro\ifmmode \ddot{}\else \"{}\fi{}dinger-like equations reduced from the Dirac equation in Schwarzschild spacetime. Then Dirac pseudostationary configurations are constructed from the stationary resonance states. We use these configurations as initial data and investigate their numerical evolutions and energy decay. These dynamical solutions are the so-called ``Dirac dynamical resonance states.'' It is found that the energy of the Dirac dynamical resonance states shows an exponential decay. The decay rate of energy is affected by the resonant frequency, the mass of the Dirac field, the total angular momentum, and the spin-orbit interaction. In particular, for an ultralight Dirac field, the corresponding particles can stay around a Schwarzschild black hole for a very long time, even for cosmological time scales.