In general relativity, astrophysical black holes are uniquely described by the Kerr metric. Observational tests of the Kerr nature of these compact objects and, hence, of general relativity,require a metric that encompasses a broader class of black holes as possible alternatives to the usual Kerr black holes. Several such Kerr-like metrics have been constructed to date, which depend on a set of free parameters and which reduce smoothly to the Kerr metric if all deviations vanish. Many of these metrics, however, are valid only for small values of the spin or small perturbations of the Kerr metric or contain regions of space where they are unphysical hampering their ability to properly model the accretions flows of black holes. In this paper, Tim Johannsen describe a Kerr-like black hole metric that is regular everywhere outside of the event horizon for black holes with arbitrary spins even for large deviations from the Kerr metric. This metric, therefore, provides an ideal framework for tests of the nature of black holes with observations of the emission from their accretion flows, and I give several examples of such tests across the electromagnetic spectrum with current and near-future instruments.
The Kerr metric is a stationary, axisymmetric, asymptotically flat, vacuum solution of the Einstein field equations and, due to the no-hair theorem, the only spacetime in general relativity that has all of these characteristics. Kerr-like metrics retain as many of these properties in order to closely mimic the observational appearance of Kerr black holes. All of them include the Kerr metric as the limiting case if the deviation parameters are set to zero. By construction, parametric deviations from the Kerr metric are usually stationary,
axisymmetric, and asymptotically flat. If such a metric is also a vacuum solution in general relativity, it either harbors a naked singularity or is plagued with pathological regions in the exterior domain where causality is violated (see Johannsen et al. 2012a). Some Kerr-like metrics are valid only for small or intermediate values of the black hole spin (e.g., Glampedakis & Babak 2006; Yunes & Pretorius 2009), while others are based on an expansion in the deviation parameters in order to remain a vacuum solution of the Einstein equations under certain conditions (e.g., Vigeland & Hughes 2010). Depending on the desired application, additional metric properties can be important, such as its Petrov type (Vigeland et al. 2011) or violations of parity (Yunes & Pretorius 2009; Yagi et al. 2012). The metric by Johannsen & Psaltis (2011b) in Boyer-Lindquist-like coordinates is given by
the line element
(ds^2=-[1+h(r,theta)](1-frac{2Mr}{Sigma})dt^2-frac{4aMrsin^2theta}{Sigma}[1+h(r, theta)]dtdphi+frac{Sigma[1+h(r,theta)]}{Delta+a^2sin^2theta h(r,theta)}dr^2)
[+Sigma d theta^2+[sin^2 theta ( r^2+a^2+frac{2a^2Mrsin^2theta}{Sigma}) +h(r,theta)frac{a^2(Sigma+2Mr) sin^4theta}{Sigma}]dphi^2]
where
(Sigma equiv r^2+a^2cos^2theta ), (Delta equiv r^2-2Mr+a^2), (h(r,theta) equiv sum_{k=0}^infty (epsilon_{2k}+epsilon_{2k+1}frac{Mr}{Sigma})(frac{M^2}{Sigma} )^k).
This metric is a stationary, axisymmetric, and asymptotically flat parametric deviation that describes an actual black hole for all values within the range of spins |a|(leq) M without relying on an expansion in the deviation parameters (epsilon_{k}) and that is regular, i.e., it is free of any pathologies outside of the event horizon. Possible nonzero lowest-order deviations (epsilon_{k}) , k = 0, 1, 2, of the Kerr metric are ruled out by current solar-system tests of gravity and by the requirement that the above metric reduces to Newtonian gravity far from the black hole. For simplicity, I will allow the leading-order coefficient (epsilon_{3}) to be the only nonvanishing deviation parameter.
