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

TL;DR

This paper assesses whether the slow-rotation approximation of black hole metrics can be reliably used in electromagnetic observations to test deviations from General Relativity, finding errors small enough for practical use.

Contribution

It demonstrates that the systematic errors from using slow-rotation approximations are within observational uncertainties for certain black hole spins, supporting their use in tests of gravity.

Findings

Systematic error in shadow observations is at most 2% for spins ≤ 0.6.

Systematic error in continuum spectrum is negligible for spins ≤ 0.9.

Slow-rotation solutions can effectively constrain deviations from GR.

Abstract

Future electromagnetic observations of black holes may allow us to test General Relativity in the strong-field regime. Such tests, however, require knowledge of rotating black hole solutions in modified gravity theories, a class of which does not admit the Kerr metric as a solution. Several rotating black hole 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 black hole metrics is small enough relative to the observational error to allow their use in electromagnetic observations to constrain deviations from General Relativity. 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 black hole 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 black holes with dimensionless spins $\chi\leq0.6$. 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 black holes with dimensionless spins of $\chi \lesssim 0.9$. Our results suggest that the modified gravity solutions found in the slow-rotation approximation may be used to constrain realistic deviations from General Relativity with continuum spectrum and black hole shadow observations.