Relativistic Effects and GRB Polarization in Power-Law Evolution

Despite decades of polarization observations and high-significance polarized $\gamma$-ray, X-ray, optical, and radio emissions in gamma-ray bursts (GRBs) have been accumulating in dozens of cases, people have yet to find a consistent scenario for understanding the globally observed timing properties of GRB polarization to date. Here, we report that the observed properties of GRB polarization exhibit a four-segment timing evolution at the cosmological distance: (I) an initial hump early on (within the first few seconds); (II) a later on power-law decay (from $\sim$10$^{1}$ to $\sim$10$^{4}$ s), which takes the form of $\pi_{\rm obs} \propto t^{-0.50 \pm 0.02}$; (III) afterwards a late-time rebrightening hump (from $\sim$10$^{4}$ to $\sim$10$^{5}$ s); and (IV) finally a flatting power-law decay (from $\sim$ 10$^{5}$ to $\sim$ 10$^{7}$ s), with the the form of $\pi_{\rm obs} \propto t^{-0.21 \pm 0.08}$. These findings may present a challenge to the mainstream of polarization models that assume the polarization time evolution change in different emission regions. We show that these results can be explained by relativistic and geometric effects of a highly relativistic and magnetized jet generated by the central engine, and "magnetic patches" distributed as a globally random but locally coherent form. Our analysis suggests that there is a single dominant mechanism that might account for the global observational properties of GRB polarization, and other emission mechanisms and effects might play a role in spatially local and temporally short effects on GRB polarization.