Massless Majorana spinors in the Kerr spacetime
Tianyuan Cai, Xiao Zhang
TL;DR
The work analyzes Majorana spinors in Kerr-type spacetimes, showing nonexistence of massive spinors with time or azimuthal dependence and deriving a separable framework for massless spinors with parameters epsilon1, epsilon2. It establishes precise L^p obstructions for nonzero epsilon1, epsilon2 and demonstrates that self-adjointness of the Dirac Hamiltonian forces the Kerr geometry to reduce to Schwarzschild, with the spinor becoming phi-independent. In Schwarzschild, massless Majorana spinors with decaying initial data disperse over time, as the probability in any fixed exterior region tends to zero. These results provide rigorous constraints on the dynamics of Majorana fields in rotating black-hole backgrounds and their asymptotic behavior.
Abstract
In this paper, we show that massive Majorana spinors \eqref{1.2} do not exist if they are $t$-dependent or $φ$-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 $ε_1$, $ε_2$. If at least one of $ε_1$, $ε_2$ is zero, massless Majorana spinors can be solved explicitly. If $ε_1$, $ε_2$ 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}{|ε_1|+|ε_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 $φ$-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.