  The coincident detection of a gravitational-wave (GW) event GW170817 with
electromagnetic (EM) signals (e.g., a short gamma-ray burst SGRB 170817A or a
macronova) from a binary neutron star merger within the nearby galaxy NGC 4933
provides a new, multimessenger test of the weak equivalence principle (WEP),
extending the WEP test with GWs and photons. Assuming that the arrival time
delay between the GW signals from GW170817 and the photons from SGRB 170817A or
the macronova is mainly attributed to the gravitational potential of the Milky
Way, we demonstrate that the strict upper limits on the deviation from the WEP
are $\Delta \gamma<1.4\times10^{-3}$ for GW170817/macronova and $\Delta \gamma
<5.9\times10^{-8}$ for GW170817/SGRB 170817A. A much more severe constraint on
the WEP accuracy can be achieved ($\sim0.9\times10^{-10}$) for GW170817/SGRB
170817A when we consider the gravitational potential of the Virgo Cluster,
rather than the Milky Way's gravity. This provides the tightest limit to date
on the WEP through the relative differential variations of the $\gamma$
parameter for two different species of particles. Compared with other
multimessenger (photons and neutrinos) results, our limit is 7 orders of
magnitude tighter than that placed by the neutrinos and photons from supernova
1987A, and is almost as good as or is an improvement of 6 orders of magnitude
over the limits obtained by the low-significance neutrinos correlated with GRBs
and a blazar flare.
