  Gravitational parity violation is a possibility motivated by particle
physics, string theory and loop quantum gravity. One effect of it is amplitude
birefringence of gravitational waves, whereby left and right
circularly-polarized waves propagate at the same speed but with different
amplitude evolution. Here we propose a test of this effect through coincident
observations of gravitational waves and short gamma-ray bursts from binary
mergers involving neutron stars. Such gravitational waves are highly left or
right circularly-polarized due to the geometry of the merger. Using
localization information from the gamma-ray burst, ground-based gravitational
wave detectors can measure the distance to the source with reasonable accuracy.
An electromagnetic determination of the redshift from an afterglow or host
galaxy yields an independent measure of this distance. Gravitational parity
violation would manifest itself as a discrepancy between these two distance
measurements. We exemplify such a test by considering one specific effective
theory that leads to such gravitational parity-violation, Chern-Simons gravity.
We show that the advanced LIGO-Virgo network and all-sky gamma-ray telescopes
can be sensitive to the propagating sector of Chern-Simons gravitational parity
violation to a level roughly two orders of magnitude better than current
stationary constraints from the LAGEOS satellites.
