  On 2017 August 17, the gravitational-wave event GW170817 was observed by the
Advanced LIGO and Virgo detectors, and the gamma-ray burst (GRB) GRB 170817A
was observed independently by the Fermi Gamma-ray Burst Monitor, and the
Anticoincidence Shield for the Spectrometer for the International Gamma-Ray
Astrophysics Laboratory. The probability of the near-simultaneous temporal and
spatial observation of GRB 170817A and GW170817 occurring by chance is
$5.0\times 10^{-8}$. We therefore confirm binary neutron star mergers as a
progenitor of short GRBs. The association of GW170817 and GRB 170817A provides
new insight into fundamental physics and the origin of short gamma-ray bursts.
We use the observed time delay of $(+1.74 \pm 0.05)\,$s between GRB 170817A and
GW170817 to: (i) constrain the difference between the speed of gravity and the
speed of light to be between $-3\times 10^{-15}$ and $+7\times 10^{-16}$ times
the speed of light, (ii) place new bounds on the violation of Lorentz
invariance, (iii) present a new test of the equivalence principle by
constraining the Shapiro delay between gravitational and electromagnetic
radiation. We also use the time delay to constrain the size and bulk Lorentz
factor of the region emitting the gamma rays. GRB 170817A is the closest short
GRB with a known distance, but is between 2 and 6 orders of magnitude less
energetic than other bursts with measured redshift. A new generation of
gamma-ray detectors, and subthreshold searches in existing detectors, will be
essential to detect similar short bursts at greater distances. Finally, we
predict a joint detection rate for the Fermi Gamma-ray Burst Monitor and the
Advanced LIGO and Virgo detectors of 0.1--1.4 per year during the 2018-2019
observing run and 0.3--1.7 per year at design sensitivity.
