  About 0.4s after the Laser Interferometer Gravitational-Wave Observatory
(LIGO) detected a transient gravitational-wave (GW) signal GW150914, the Fermi
Gamma-ray Burst Monitor (GBM) also found a weak electromagnetic transient (GBM
transient 150914). Time and location coincidences favor a possible association
between GW150904 and GBM transient 150914. Under this possible association, we
adopt Fermi's electromagnetic (EM) localization and derive constraints on
possible violations of the Weak Equivalence Principle (WEP) from the
observations of two events. Our calculations are based on four comparisons:
(1)The first is the comparison of the initial GWs detected at the two LIGO
sites. From the different polarizations of these initial GWs, we obtain a limit
on any difference in the parametrized post-Newtonian (PPN) parameter
$\Delta\gamma\lesssim 10^{-10}$. (2) The second is a comparison of GWs and
possible EM waves. Using a traditional super-Eddington accretion model for GBM
transient 150914, we again obtain an upper limit $\Delta\gamma\lesssim
10^{-10}$. Compared with previous results for photons and neutrinos, our limits
are five orders of magnitude stronger than those from PeV neutrinos in blazar
flares, and seven orders stronger than those from MeV neutrinos in SN1987A. (3)
The third is a comparison of GWs with different frequencies in the range [35
Hz, 250 Hz]. (4) The fourth is a comparison of EM waves with different energies
in the range [1 keV, 10 MeV]. These last two comparisons lead to an even
stronger limit, $\Delta\gamma\lesssim 10^{-8}$. Our results highlight the
potential of multi-messenger signals exploiting different emission channels to
strengthen existing tests of the WEP.
