Massless Majorana spinors in the Kerr spacetime

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.

💡 Research Summary

The paper investigates the existence of Majorana spinors—self‑conjugate solutions of the Dirac equation—in rotating black‑hole spacetimes. Starting from the standard Dirac equation (D\Psi+i\lambda\Psi=0) on a four‑dimensional Lorentzian manifold, the authors distinguish massive ((\lambda\neq0)) and massless ((\lambda=0)) cases. They first point out that Chandrasekhar’s classic separation ansatz for Kerr does not respect the Majorana reality condition, and therefore introduce a new time‑periodic Majorana spinor ansatz (equation 1.4) that incorporates a diagonal phase matrix (E).

Massive Majorana spinors. By inserting the ansatz into the Dirac equation on Kerr, Kerr–(A)dS, and Kerr–Newman backgrounds, the authors derive four coupled first‑order equations. Analyzing these equations they prove (Propositions 2.1–2.3) that any differentiable massive Majorana spinor that depends on either the time coordinate (t) or the azimuthal angle (\phi) must vanish identically. Consequently, massive Majorana fields cannot exist in these rotating spacetimes unless they are completely static and axis‑independent, a situation that is ruled out by the underlying geometry.

Massless Majorana spinors in non‑extreme Kerr. For (\lambda=0) the Dirac equation separates into radial and angular parts after the usual Boyer‑Lindquist decomposition. The separation introduces two complex constants (\epsilon_{1}) and (\epsilon_{2}) through the operators \