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Future electromagnetic observations of black holes may allow us to testGeneral Relativity in the strong-field regime. Such tests, however, requireknowledge of rotating black hole solutions in modified gravity theories, aclass of which does not admit the Kerr metric as a solution. Several rotatingblack hole solutions in modified theories have only been found in theslow-rotation approximation (i.e. assuming the spin angular momentum is muchsmaller than the mass squared). We here investigate whether the systematicerror due to the approximate nature of these black hole metrics is small enoughrelative to the observational error to allow their use in electromagneticobservations to constrain deviations from General Relativity. We address thisby considering whether electromagnetic observables constructed from aslow-rotation approximation to the Kerr metric can fit observables constructedfrom the full Kerr metric with systematic errors smaller than currentobservational errors. We focus on black hole shadow and continuum spectrumobservations, as these are the least influenced by accretion disk physics, withcurrent observational errors of about 10%. We find that the fractionalsystematic error introduced by using a second-order, slowly-rotating Kerrmetric is at most 2% for shadows created by black holes with dimensionlessspins $\chi\leq0.6$. We also find that the systematic error introduced by usingthe slowly-rotating Kerr metric as an exact metric when constructing continuumspectrum observables is negligible for black holes with dimensionless spins of$\chi \lesssim 0.9$. Our results suggest that the modified gravity solutionsfound in the slow-rotation approximation may be used to constrain realisticdeviations from General Relativity with continuum spectrum and black holeshadow observations.

Can the Slow-Rotation Approximation be used in Electromagnetic Observations of Black Holes?