Massless Majorana spinors in the Kerr spacetime

TL;DR

This paper investigates the existence and properties of massless Majorana spinors in Kerr and Schwarzschild spacetimes, establishing conditions for their existence, explicit solutions, and long-term behavior.

Contribution

It proves the nonexistence of certain massive and non-extreme Kerr massless Majorana spinors, and characterizes the conditions under which massless solutions exist and are unique.

Findings

Massive Majorana spinors do not exist if t- or φ-dependent in Kerr.

Explicit solutions for massless Majorana spinors when one parameter is zero.

In Schwarzschild spacetime, spinors disperse over time with probability tending to zero.

Abstract

In this paper, we show that massive Majorana spinors \eqref{1.2} do not exist if they are $t$-dependent or $\phi$-dependent in Kerr, or Kerr-(A)dS spacetimes. For massless Majorana spinors in the non-extreme Kerr spacetime, the Dirac equation can be separated into radial and angular equations, parameterized by two complex constants $\epsilon_1$, $\epsilon_2$. If at least one of $\epsilon_1$, $\epsilon_2$ is zero, massless Majorana spinors can be solved explicitly. If $\epsilon_1$, $\epsilon_2$ are nonzero, we prove the nonexistence of massless time-periodic Majorana spinors in the non-extreme Kerr spacetime which are $L^p$ outside the event horizon for $ 0<p\le\frac{6}{|\epsilon_1|+|\epsilon_2| +2}$. We then provide the Hamiltonian formulation for massless Majorana spinors and prove that the self-adjointness of the Hamiltonian leads to the angular momentum $a=0$ and spacetime reduces to the Schwarzschild spacetime, moreover, the massless Majorana spinor must be $\phi$-independent. Finally, we show that, in the Schwarzschild spacetime, for initial data with $L^2$ decay at infinity, the probability of the massless Majorana spinors to be in any compact region of space tends to zero as time tends to infinity.