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

TL;DR

The paper develops a finite-distance, polarization-resolved framework for vacuum birefringence in strong gravitational and electromagnetic fields within nonlinear electrodynamics. It introduces paired optical metrics for the two photon polarizations and uses a finite-distance Gauss-Bonnet construction to compute differential deflection while cleanly separating gravitational deflection from nonlinearity-induced effects via a weak-coupling expansion. Applying the method to Euler-Heisenberg QED and Born-Infeld theory, the authors derive explicit finite-distance corrections and magnetar-relevant suppression factors, showing that surface emission can substantially reduce observable birefringence relative to asymptotic scattering estimates. These results provide essential finite-distance calibrations for interpreting X-ray polarimetry and place robust, model-dependent constraints on strong-field QED and NLED parameters in astrophysical environments.

Abstract

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 nonlinear electrodynamics, 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 \emph{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 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 dipolar fields, this geometric effect yields a suppression factor that can diminish the predicted polarization-dependent deflection by as much as $\sim 50\%$ for limb emission relative to asymptotic scattering estimates. 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.

Figures (3)

• Figure 1: The normalized differential deflection angle $\Delta \alpha / \Delta \alpha_{\infty}$ as a function of the source distance $r_\text{S}$ scaled by the impact parameter $b$. The solid black line corresponds to the magnetar dipole model ($n=6$), while the dashed red line represents the magnetic monopole reference ($n=4$). The curves demonstrate that the dipole signal converges more rapidly to the asymptotic limit, indicating that the birefringent effect is more tightly confined to the strong-field region. For sources at $r_\text{S} \gtrsim 4b$, the finite-distance suppression becomes negligible ($<5\%$)
• Figure 2: The Finite-Distance Suppression Factor $\mathcal{F}(\psi)$ as a function of the emission angle $\psi$. The solid black line represents the physical model for a magnetar (dipole scaling, $n=6$), while the dashed red line shows the monopole approximation ($n=4$). For grazing emission ($\psi \to 90^\circ$), the observable birefringence is suppressed by exactly 50% ($c_\text{S}=1/2$) due to the truncation of the photon path relative to the asymptotic scattering limit.
• Figure 3: The cumulative fraction of the total birefringence signal accumulated along the photon path angle $\phi$, where $0^\circ$ corresponds to the observer and $180^\circ$ to a source at asymptotic infinity. The periastron (point of closest approach to the lens) is located at $90^\circ$. For the magnetar dipole field (solid black line), the signal accumulation is highly localized, with the majority of the phase shift generated in the immediate vicinity of the periastron. This localization explains why surface emission (which effectively starts the integration at $90^\circ$) suffers a substantial geometric suppression relative to the full scattering path.