Paper detail

Finite Distance Corrections to Vacuum Birefringence in Strong Gravitational and Electromagnetic Fields

We study polarization dependent photon propagation in static, spherically symmetric spacetimes permeated by strong magnetic fields, with the aim of quantifying how finite emission and detection radii modify vacuum birefringence signals. Working in the geometric optics limit of NLED, we formulate the two polarization modes as null geodesics of distinct effective (optical) metrics. We then develop a controlled weak-coupling expansion that cleanly separates the standard gravitational deflection from the birefringent contribution induced by the electromagnetic nonlinearity. Using a finite distance Gauss-Bonnet construction on the associated optical manifolds, we derive a general expression for the differential bending angle in which the source and observer are kept at arbitrary radii, thereby extending the usual scattering-at-infinity treatment. As benchmark applications, we specialize our results to the Euler-Heisenberg effective action of quantum electrodynamics (QED) and to Born-Infeld electrodynamics. We find that the observable birefringence is generically reduced by finite-distance truncation of the curvature flux, and we provide explicit correction series suitable for data analysis. For magnetar-motivated dipole-like falloff, the same geometric truncation can reduce the predicted polarization-dependent deflection at the order-tens-of-percent level for surface/near-surface emission; in a simple one-sided (outward-only) benchmark the suppression can approach 1/2 for near-limb rays. Since realistic dipole magnetospheres are axisymmetric rather than spherically symmetric, we present this as an illustrative scaling estimate and leave a fully axisymmetric treatment to future work. Our results furnish a necessary finite-distance calibration for interpreting current and future X-ray polarimetry measurements and for placing unbiased constraints on strong-field QED and broader NLED parameters.