  In the next decade, x-ray polarimetry will open a new window on the
high-energy Universe, as several missions that include an x-ray polarimeter are
currently under development. Observations of the polarization of x-rays coming
from the accretion disks of stellar-mass and supermassive black holes are among
the new polarimeters' major objectives. In this paper, we show that these
observations can be affected by the quantum electrodynamic (QED) effect of
vacuum birefringence: after an x-ray photon is emitted from the accretion disk,
its polarization changes as the photon travels through the accretion disk's
magnetosphere, as a result of the vacuum becoming birefringent in presence of a
magnetic field. We show that this effect can be important for black holes in
the energy band of the upcoming polarimeters, and has to be taken into account
in a complete model of the x-ray polarization that we expect to detect from
black-hole accretion disks, both for stellar mass and for supermassive black
holes. We find that, for a chaotic magnetic field in the disk, QED can
significantly decrease the linear polarization fraction of edge-on photons,
depending on the spin of the hole and on the strength of the magnetic field.
This effect can provide, for the first time, a direct way to probe the magnetic
field strength close to the innermost stable orbit of black-hole accretion
disks and to study the role of magnetic fields in astrophysical accretion in
general.
