  The gravitational wave observations GW150914 and GW151226 by Advanced LIGO
provide the first opportunity to learn about physics in the extreme gravity
environment of coalescing binary black holes. The LIGO Scientific Collaboration
and the Virgo Collaboration have verified that this observation is consistent
with General Relativity. This paper expands their analysis to a larger class of
anomalies, highlighting the inferences that can be drawn on non-standard
theoretical physics mechanisms. We find that these events constrain a plethora
of mechanisms associated with the generation and propagation of gravitational
waves, including the activation of scalar fields, gravitational leakage into
large extra dimensions, the variability of Newton's constant, a modified
dispersion relation, gravitational Lorentz violation and the strong equivalence
principle. Though other observations limit many of these mechanisms already,
GW150914 and GW151226 are unique in that they are direct probes of dynamical
strong-field gravity and of gravitational wave propagation. We also show that
GW150914 constrains inferred properties of exotic compact object alternatives
to Kerr black holes. We argue, however, that the true potential for GW150914 to
both rule out exotic objects and constrain physics beyond General Relativity is
severely limited by the lack of understanding of the merger regime in almost
all relevant modified gravity theories. This event thus significantly raises
the bar that these theories have to pass, both in terms of having a sound
theoretical underpinning, and being able to solve the equations of motion for
binary merger events. We conclude with a discussion of the additional
inferences that can be drawn if the lower-confidence observation of an
electromagnetic counterpart to GW150914 holds true; this would provide dramatic
constraints on the speed of gravity and gravitational Lorentz violation.
