Can the slow-rotation approximation be used in electromagnetic observations of black holes?

Future electromagnetic observations of black holes (BHs) may allow us to test general relativity (GR) in the strong-field regime. Such tests, however, require knowledge of rotating BH solutions in modified gravity theories, a class of which does not admit the Kerr metric as a solution. Several rotating BH solutions in modified theories have only been found in the slow-rotation approximation (i.e. assuming the spin angular momentum is much smaller than the mass squared). We here investigate whether the systematic error due to the approximate nature of these BH metrics is small enough relative to the observational error to allow their use in electromagnetic observations to constrain deviations from GR. We address this by considering whether electromagnetic observables constructed from a slow-rotation approximation to the Kerr metric can fit observables constructed from the full Kerr metric with systematic errors smaller than current observational errors. We focus on BH shadow and continuum spectrum observations, as these are the least influenced by accretion disk physics, with current observational errors of about 10%. We find that the fractional systematic error introduced by using a second- order, slowly rotating Kerr metric is at most 2% for shadows created by BHs with dimensionless spins c ≤ 0.6. We also find that the systematic error introduced by using the slowly rotating Kerr metric as an exact metric when constructing continuum spectrum observables is negligible for BHs with dimensionless spins of c ≲ 0.9. Our results suggest that the modified gravity solutions found in the slow-rotation approximation may be used to constrain realistic deviations from GR with continuum spectrum and BH shadow observations.