On generic parametrizations of spinning black-hole geometries

The construction of a generic parametrization of spinning geometries that can be matched continuously to the Kerr metric is an important open problem in general relativity. Its resolution is of more than academic interest, as it allows us to parametrize and quantify possible deviations from the no-hair theorem. Various approaches to the problem have been proposed, all with their own (severe) limitations. Here we discuss the metric recently proposed by Johannsen and Psaltis, showing that (i) the original metric describes only corrections that preserve the horizon area-mass relation of nonspinning geometries, (ii) this unnecessary restriction can be relaxed by introducing a new parameter that in fact dominates in both the weak-field and strong-field regimes, (iii) within this framework, we construct the most generic spinning black-hole geometry that contains twice as many (infinite) parameters as the original metric, and (iv) in the strong-field regime, all parameters are (roughly) equally important. This fact introduces a severe degeneracy problem in the case of highly spinning black holes. Our results suggest that using parametrizations that affect only the quadrupole moment of the Kerr geometry is problematic, because higher-order multipoles can be equally relevant for highly spinning objects. Finally, we prove that even our generalization fails to describe the few known spinning black-hole metrics in modified gravity.

Figure 1

Relative corrections $\delta {\mathrm{\Omega }}_k/{\mathrm{\Omega }}_0$ to the ISCO frequency as a function of $J/M_{\mathrm{ADM}}^2$ for the metric (3) to linear order in ${\epsilon }_k\ll 1$ up to $k=9$. The ISCO frequency reads $\mathrm{\Omega }={\mathrm{\Omega }}_0+\sum _k\delta {\mathrm{\Omega }}_k{\epsilon }_k$, where ${\mathrm{\Omega }}_0$ is the ISCO frequency of a Kerr geometry. The small-coupling approximation requires $(\delta {\mathrm{\Omega }}_k/{\mathrm{\Omega }}_0){\epsilon }_k\ll 1$ for consistency. Each ${\epsilon }_k-$line is built by setting to zero all other ${\epsilon }_i,i\ne k$.

Figure 2

Same as Fig. 1 for the generalized metric (20, 21, 22, 23, 24). The two panels refer to the corrections associated to ${\epsilon }_k^t$ (upper panel) and ${\epsilon }_k^r$ (lower panel), respectively. For ease of comparison, the range of the vertical axis is the same for both panels. In this case the total ISCO frequency reads $\mathrm{\Omega }={\mathrm{\Omega }}_0+\sum _k\delta {\mathrm{\Omega }}_k^t{\epsilon }_k^t+\sum _k\delta {\mathrm{\Omega }}_k^r{\epsilon }_k^r$. The small-coupling approximation requires $(\delta {\mathrm{\Omega }}_k^i/{\mathrm{\Omega }}_0){\epsilon }_k^i\ll 1$ for consistency.