On generic parametrizations of spinning black-hole geometries

TL;DR

The paper examines the Johannsen–Psaltis metric as a framework to parametrize spinning black-hole geometries beyond Kerr in a model-independent way. It shows that the original JP form omits dominant corrections (notably those tied to $\epsilon_1$) and that a broader generalization with independent $h^t$ and $h^r$ perturbations introduces many more deformation parameters, substantially increasing degeneracies in the strong-field regime. The analysis reveals that even the most general JP-based parametrizations fail to reproduce several known spinning solutions in modified gravity (e.g., slowly rotating dynamical Chern–Simons and Einstein–Dilaton–Gauss–Bonnet black holes) and that regularity near extremality imposes delicate constraints among deformation parameters. The work therefore highlights significant limitations of current model-independent parametrizations for testing the Kerr hypothesis and motivates the search for a more comprehensive framework beyond the Newman–Janis-derived approaches. Overall, the results emphasize the need for careful treatment of higher-order multipoles and degeneracies when using parametric spacetimes to confront strong-field observations with theory.

Abstract

The construction of a generic parametrization of spinning geometries which can be matched continuously to the Kerr metric is an important open problem in General Relativity. Its resolution is more than of academic interest, as it allows 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 post-Newtonian and strong-field regimes; (iii) within this framework, we construct the most generic spinning black-hole geometry which contains twice as many the (infinite) parameters of the original metric; (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. Finally, we prove that even our generalization fails to describe the few known spinning black-hole metrics in modified gravity.

Figures (2)

• Figure 1: Relative corrections $\delta\Omega_k/\Omega_0$ to the ISCO frequency as a function of $J/{M^2_{\hbox{\tiny ADM}}}$ for the metric \ref{['metricJP']} to linear order in $\epsilon_k\ll1$ up to $k=9$. The ISCO frequency reads $\Omega=\Omega_0+\sum_k\delta \Omega_k\epsilon_k$, where $\Omega_0$ is the ISCO frequency of a Kerr geometry. The small-coupling approximation requires $(\delta \Omega_k/\Omega_0)\epsilon_k\ll1$ for consistency. Each $\epsilon_k-$line is built by setting to zero all other $\epsilon_i,\,i\neq k$.
• Figure 2: Same as Fig. \ref{['fig:deltaOmegaISCO']} for the generalized metric \ref{['eq:JP2_1']}--\ref{['eq:JP2_5']}. 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 $\Omega=\Omega_0+\sum_k\delta \Omega_k^t\epsilon_k^t+\sum_k\delta \Omega_k^r\epsilon_k^r$. The small-coupling approximation requires $(\delta \Omega_k^i/\Omega_0)\epsilon_k^i\ll1$ for consistency.