Probing the black hole metric: Black hole shadows and binary black-hole inspirals

In general relativity, the spacetimes of black holes have three fundamental properties: (i) they are the same, to the 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 deviations from the Kerr metric across the 8 orders of magnitude in mass and 16 orders in curvature 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., $\ensuremath{\sim}3\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$) 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.

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