  In General Relativity, the spacetimes of black holes have three fundamental
properties: (i) they are the same, to lowest order in spin, as the metrics of
stellar objects; (ii) they are independent of mass, when expressed in geometric
units; and (iii) they are described by the Kerr metric. In this paper, we
quantify the upper bounds on potential black-hole metric deviations imposed by
observations of black-hole shadows and of binary black-hole inspirals in order
to explore the current experimental limits on possible violations of the last
two predictions. We find that both types of experiments provide correlated
constraints on deviation parameters that are primarily in the tt-components of
the spacetimes, when expressed in areal coordinates. We conclude that,
currently, there is no evidence for a deviations from the Kerr metric across
the 8 orders of magnitudes in masses and 16 orders in curvatures spanned by the
two types of black holes. Moreover, because of the particular masses of black
holes in the current sample of gravitational-wave sources, the correlations
imposed by the two experiments are aligned and of similar magnitudes when
expressed in terms of the far field, post-Newtonian predictions of the metrics.
If a future coalescing black-hole binary with two low-mass (e.g., ~3 Msun)
components is discovered, the degeneracy between the deviation parameters can
be broken by combining the inspiral constraints with those from the black-hole
shadow measurements.
