The precession of particle spin in spherical symmetric spacetimes

Technical Summary: The Precession of Particle Spin in Spherically Symmetric Spacetimes

Problem Statement
The paper addresses a limitation in the standard Geometrical Optics (WKB) approximation when applied to the scattering of particles with spin in gravitational fields. While the leading-order WKB approximation successfully describes the trajectories of massless and massive particles as geodesics, it fails to account for internal degrees of freedom, specifically spin. This omission leads to a discrepancy in backward scattering scenarios: partial wave analysis predicts a divergent cross-section (a "glory spot") for massless scalar fields (spin $s=0$), whereas for massless particles with non-vanishing spin (e.g., photons, $s=1$), the backward cross-section vanishes. The authors argue that the leading-order WKB approximation is insufficient to explain this vanishing glory spot and that a sub-leading order analysis involving spin precession is necessary. Furthermore, while the behavior of massless particles is partially understood, the spin precession of massive particles, particularly in the non-relativistic limit, lacks detailed discussion in the context of strong gravitational lensing.

Methodology
The authors adopt a classical approach where the spin of a particle is characterized by a 4-vector $S^\mu$. They operate within the sub-leading order of the WKB approximation, which allows them to treat the particle trajectory as a geodesic while accounting for the parallel transport of the spin vector along that geodesic. This approach neglects spin-orbit coupling (which would require the Mathisson-Papapetrou equations and deviation from geodesics) but is sufficient to explain the vanishing glory spot for massless particles.

The core methodology involves:

1. Derivation of the Precession Equation: Starting from the parallel transport equation $DS^\mu/d\tau = 0$ in a general spherically symmetric spacetime with metric $ds^2 = f(r)dt^2 - f(r)^{-1}dr^2 - r^2d\theta^2 - r^2\sin^2\theta d\phi^2$, the authors derive a second-order differential equation for the spin components.
2. Variable Transformation: To solve the precession equation for unbounded orbits where the radial coordinate $r$ varies, they introduce a new variable $\chi$ defined by $d\chi/d\phi = E[1 + L^2/r(\phi)^2]^{-1}$. This transformation simplifies the precession equation into a harmonic oscillator form ($d^2B/d\chi^2 + B = 0$), allowing for an analytical solution without explicitly solving the orbit equation $r(\phi)$ first.
3. Definition of Precession Angle: They define an angle $\xi$ representing the rotation of the spin vector between the initial and final states of a scattering event. This angle is expressed in terms of the integrated variable $\chi_0$ (evaluated at the point of closest approach) and the orbital deflection angle $\phi_0$.
4. Specific Spacetime Applications: The general formalism is applied to Schwarzschild and Reissner-Nordström (RN) spacetimes. The authors perform explicit integrations using elliptic functions and derive strong-field expansions for the non-relativistic limit ($v \approx 0$).

Key Contributions and Results

• General Precession Equation: The paper derives a general precession equation (Eq. 2.16) applicable to both massless and massive particles in any spherically symmetric spacetime. It demonstrates that the parallel transport of spin emerges at the sub-leading order of the WKB approximation.
• Vanishing Glory Spot for Massless Particles: The authors prove that for massless particles ($v=1$) undergoing backward scattering in any spherically symmetric spacetime, the spin vector is always reversed ($\cos \xi = -1$). This reversal causes perfect cancellation of spins from symmetric orbits (one in the equatorial plane, one perpendicular), thereby explaining the absence of a glory spot for massless spinning particles, a phenomenon the leading-order WKB approximation cannot predict.
• Circular Orbits: For circular orbits, the paper shows that massless particles on the photon sphere experience no net spin precession relative to the natural basis (the spin rotates with the basis), whereas massive particles generally do experience precession unless the spacetime is asymptotically flat and the radius is large.
• Schwarzschild Spacetime (Non-Relativistic Limit): For massive particles with very small velocities ($v \approx 0$) in Schwarzschild spacetime, the authors derive an expansion showing that the spin precession angle $\xi$ depends solely on the deflection angle $\phi_0$. Specifically, $\cos \xi = -\cos(\phi_0)$. This implies that for non-relativistic particles, the glory spot does not vanish in backward scattering (unlike the massless case), as the cancellation condition is not universally met.
• Reissner-Nordström Spacetime: In RN spacetime, the spin precession for non-relativistic particles depends not only on the deflection angle but also on the black hole charge $Q$ (via the parameter $h=Q/M$). The authors provide an expansion (Eq. 5.24) showing this charge dependence, which distinguishes RN scattering from the Schwarzschild case.
• Analytical Solutions: The paper provides exact solutions for the precession angle in terms of elliptic integrals for both Schwarzschild and RN metrics and validates strong-field expansions against these exact results.

Significance and Claims
The paper claims that the parallel transport of spin along geodesics is a sufficient mechanism to explain the absence of the glory spot for massless particles in backward scattering, without needing to invoke the more complex Mathisson-Papapetrou equations (which include spin-orbit coupling and tidal forces). The authors emphasize that their results for massless particles are general and hold for any spherically symmetric spacetime, not just Schwarzschild.

For massive particles, the work fills a gap by providing a detailed analysis of spin precession in the non-relativistic limit under strong lensing conditions. The authors modestly note that their work focuses on the geodesic approximation and ignores the back-reaction of spin on the trajectory. They suggest that future work should extend these findings to the Mathisson-Papapetrou framework to verify if the vanishing glory spot persists when spin-orbit coupling is included, and to explore non-spherically symmetric spacetimes (such as Kerr) and congruences of rays. The study establishes a foundational framework for understanding how spin precession modifies scattering cross-sections in curved spacetime.