Quantifying General and Special Relativistic Contributions to Observed Polarization in Black Hole Accretion Disks Using Exact Analytical Formulae

Abstract

Accretion disk polarimetry provides a valuable probe of the strong gravity region surrounding black holes. However, the physical origin and geometric dependence of X-ray polarization signatures in black hole X-ray binaries are still not fully understood. In this work, an analytical study of polarization from accretion disks around a Schwarzschild black hole is presented, with application to the well-studied system Cygnus X-1. The analysis adopts the analytical framework developed by Loktev et al.[27], which yields explicit expressions for the rotation of the polarization plane and accurately incorporates gravitational light bending and relativistic aberration without relying on computationally expensive ray-tracing techniques. Polarimetric maps are produced for observer inclinations of 30◦, 50◦, and 75◦ to illustrate how strong gravitational effects modify the observed polarization patterns. The assumptions of the model, including optically thick, plane-parallel disk emission and purely azimuthal gas motion, are discussed in detail. While real accretion flows may exhibit optically thin regions, radial velocity components, non-aligned magnetic fields, and localized emission from hot spots, the results demonstrate that polarization signatures remain highly sensitive to viewing geometry. This study provides useful insight for interpreting current and future X-ray polarimetric observations of black hole systems, particularly from missions like IXPE (Imaging X-ray Polarimetry Explorer). The methodology presented here offers an efficient computational approach that can be applied to analyze polarimetric data from various black hole systems, contributing to our understanding of accretion physics and strong-field general relativity.