Deflection of Massive Spin-$\frac{1}{2}$ Particles around Kerr Black Hole

TL;DR

The paper addresses how intrinsic quantum spin of massive spin-$\frac{1}{2}$ particles, such as neutrinos, alters their trajectories when deflecting around a Kerr black hole. By deriving a MPD-like framework where the intrinsic spin is encoded in an effective spin tensor with $J_m=\frac{\hbar}{2}$ and $S^{\mu\nu}S_{\mu\nu}=\frac{\hbar^2}{2}$, the authors compute quantum spin corrections to the critical radius $r_{\mathrm{crit}}$ of polarized beams in Kerr spacetime, focusing on the equatorial plane. They find that relative corrections can exceed $10^{-12}$ for $m=1\ \mathrm{eV}/c^2$ around a solar-mass black hole, with the sign depending on whether the spin is aligned or anti-aligned with the orbital angular momentum, and they identify a lower bound $m_{\mathrm{crit}}$ below which the corrected radius would lie inside the horizon, implying a potential neutrino-mass bound if a full critical ring were observed. While primarily theoretical and observationally challenging for astrophysical black holes, the work suggests a novel gravitational lensing approach to constrain ultra-light spin-$\frac{1}{2}$ particles and motivates further study of related effects such as gravitational spin Hall corrections.

Abstract

The exact measurement of neutrino mass remains a longstanding issue. So far, there has been much success in providing an upper bound for the neutrino rest mass, both theoretically and experimentally. In this work, by exploring the critical radius of a beam of polarized quantum spin-$\frac{1}{2}$ particle deflecting around a classical Kerr black hole, we attempt to provide an additional testing ground for neutrino mass, as well as the mass of other proposed ultra-light particles yet to be determined. Notably, the quantum Dirac equation is used to derive a MPD-like equation satisfied by the polarized beam of massive spin-$\frac{1}{2}$ particles and identify the effective spin in the spin tensor with the particle's intrinsic quantum spin, confirming the previous theoretical result that the MPD equation can be in fact applied to particles' intrinsic spin. The result of this work shows that corrections of relative magnitude $>10^{-12}$ can be achieved for spin-$\frac{1}{2}$ particles with rest mass equal to $1 eV/c^2$ deflecting around a solar mass Kerr black hole. Although highly theoretical, a new method of extracting the lower bound for the neutrino mass individually is also proposed due to the behavior of the quantum spin correction.

Figures (2)

• Figure 1: Relation between particle speed $v$ and the spin correction to the critical radius.
• Figure 2: The main results of our work: (a) The relative correction of the polarized particle spin to the critical radius of the particle trajectory with respect to the size of the black hole horizon when fixing particle mass. (b) Detailed comparison showing when the radius of the black becomes small, the quantum spin correction to the particle trajectory become large, and the direction of the correction takes opposite sign when the particle spin shares the same direction with its orbital angular momentum or opposite the direction of its orbital angular momentum. (c) The relative correction of the polarized particle spin to the critical radius of the particle trajectory with respect to particle mass when fixing the size of the black hole horizon. (d) Detailed comparison showing when the radius of the black becomes small, the quantum spin correction to the particle trajectory become large, and the direction of the correction takes opposite sign when the particle spin shares the same direction with its orbital angular momentum or opposite the direction of its orbital angular momentum. (e) Comparison between the quantum spin corrections of particles passing through black holes with different rotation parameters. (f) Comparison between the relative difference induced by the quantum spin corrections of particles passing through black holes with different rotation parameters.