  The LIGO detection of GW150914 provides an unprecedented opportunity to study
the two-body motion of a compact-object binary in the large velocity, highly
nonlinear regime, and to witness the final merger of the binary and the
excitation of uniquely relativistic modes of the gravitational field. We carry
out several investigations to determine whether GW150914 is consistent with a
binary black-hole merger in general relativity. We find that the final
remnant's mass and spin, as determined from the low-frequency (inspiral) and
high-frequency (post-inspiral) phases of the signal, are mutually consistent
with the binary black-hole solution in general relativity. Furthermore, the
data following the peak of GW150914 are consistent with the least-damped
quasi-normal mode inferred from the mass and spin of the remnant black hole. By
using waveform models that allow for parameterized general-relativity
violations during the inspiral and merger phases, we perform quantitative tests
on the gravitational-wave phase in the dynamical regime and we determine the
first empirical bounds on several high-order post-Newtonian coefficients. We
constrain the graviton Compton wavelength, assuming that gravitons are
dispersed in vacuum in the same way as particles with mass, obtaining a
$90\%$-confidence lower bound of $10^{13}$ km. In conclusion, within our
statistical uncertainties, we find no evidence for violations of general
relativity in the genuinely strong-field regime of gravity.
