Detection of gravitational waves (GW) provides us an opportunity to test
general relativity in strong and dynamical regimes of gravity. One of the tests
is checking whether GW propagates with the speed of light or not. This test is
crucial because the velocity of GW has not ever been directly measured.
Propagation speed of a GW can deviate from the speed of light due to the
modification of gravity, graviton mass, and the nontrivial spacetime structure
such as extra dimensions and quantum gravity effects. Here we report a simple
method to measure the propagation speed of a GW by directly comparing arrival
times between gravitational waves, and neutrinos from supernovae or photons
from short gamma-ray bursts. As a result, we found that the future
multimessenger observations of a GW, neutrinos, and photons can test the GW
propagation speed with the precision of ~10^{-16} improving the previous
suggestions by 8-10 orders of magnitude. We also propose a novel method that
distinguishes the true signal due to the deviation of GW propagation speed from
the speed of light and the intrinsic time delay of the emission at a source by
looking at the redshift dependence.Comment: 8 pages, 6 figure
