Probing shock acceleration in BL Lac jets through X-ray polarimetry: the time-dependent view

Polarimetric measurements, especially if extended at high energy, are expected to provide important insights into the mechanisms underlying the acceleration of relativistic particles in jets. In a previous work we have shown that the polarization of the synchrotron X-ray emission produced by highly energetic electrons accelerated by a mildly relativistic shock carries essential imprints of the geometry and the structure of the magnetic fields in the downstream region. Here we present the extension of our analysis to the non-stationary case, especially suitable to model the highly variable emission of high-energy emitting BL Lacs. We anticipate a large ($\Pi \approx 40\%$), almost time-independent degree of polarization in the hard/medium X-ray band, a prediction soon testable with the upcoming mission {\it IXPE}. The situation in other bands, in particular in the optical, is more complex. A monotonic decrease of the optical degree of polarization is observed during the development of a flare. At later stages $\Pi$ reaches zero and then it starts to increase, recovering large values at late times. The instant at which $\Pi=0$ is marked by a rotation of the polarization angle by $90$ degrees. However, at optical frequencies it is likely that more than one region contributes to the observed emission, potentially making it difficult to detect the predicted behavior.