Nonexistence of time-periodic solutions of the Dirac equation in nonextreme Kerr-Newman-AdS spacetime

Xiao Zhang (Chinese Academy of Sciences)

Abstract

In non-extreme Kerr-Newman-AdS spacetime, we prove that there is no nontrivial Dirac particle which is $L^p$ for $0 \lt p\leq\frac{4}{3}$ with arbitrary eigenvalue $\lambda$, and for $\frac{4}{3}\lt p\leq\frac{4}{3-2 q}$, $0\lt q \lt \frac{3}{2}$ with eigenvalue $|\lambda| \gt |Q|+q \kappa $, outside and away from the event horizon. By taking $q=\frac{1}{2}$, we show that there is no normalizable massive Dirac particle with mass greater than $|Q|+\frac{\kappa}{2} $ outside and away from the event horizon in non-extreme Kerr-Newman-AdS spacetime, and they must either disappear into the black hole or escape to infinity, and this recovers the same result of Belgiorno and Cacciatori in the case of $Q=0$ obtained by using spectral methods. Furthermore, we prove that any Dirac particle with eigenvalue $|\lambda| \lt \frac{\kappa}{2} $ must be $L^2$ outside and away from the event horizon. This is joint work with Yaohua Wang.