  The Laser Interferometer Gravitational-Wave Observatory (LIGO) has recently
discovered gravitational waves (GWs) from its first neutron star-neutron star
merger at a distance of $\sim 40$~Mpc from the Earth. The associated
electromagnetic (EM) detection of the event, including the short gamma-ray
burst within $\Delta t \sim 2$~s after the GW arrival, can be used to test
various aspects of sources physics and GW propagation. Using GW170817 as the
first GW-EM example, we show that this event provides a stringent direct test
that GWs travel at the speed of light. The gravitational potential of the Milky
Way provides a potential source of Shapiro time delay difference between the
arrival of photons and GWs, and we demonstrate that the nearly coincident
detection of the GW and EM signals can yield strong limits on anomalous
gravitational time delay, through updating the previous limits taking into
account details of Milky Way's gravitational potential. Finally, we also obtain
an intriguing limit on the size of the prompt emission region of GRB 170817A,
and discuss implications for the emission mechanism of short gamma-ray bursts.
