Asymptotically flat, parametrized black hole metric preserving Kerr symmetries

Abstract

Recently the Event Horizon Telescope Collaboration, with very-long baseline interferometric observations, resolved structure at the scale of ∼ 5 Schwarzschild radii about the center of M87 * , the supermassive black hole resident at the center of Messier 87. This important observation has paved the way for testing what is known as the "no-hair" theorem, stating that isolated black holes are described by the Kerr metric, parameterized only by their mass and spin. Generic, parameterized spacetimes beyond Kerr allow one to arbitrarily test the no-hair theorem for deviations from the Kerr result with no prior theoretical knowledge or motivation. In this paper, we present such a new general, stationary, axisymmetric and asymptotically flat black hole solution with separable geodesic equations (thus preserving symmetries of a Kerr black hole), extending the previous work of Johannsen. In this new metric, five free non-linear functions parameterically deviate from the Kerr result, allowing one to effectively transform to many alternative black hole solutions present in the literature. We then derive analytic expressions for the Keplerian and epicyclic frequencies, the orbital energy and angular momentum, and the location of the innermost stable orbit of circular equatorial particle orbits. We also compute the image of the photon rings in the new spacetime, which correspond to the boundary of the black hole shadow image taken by the Event Horizon Telescope. We finally compare each quantity for the Kerr result against various parameterizations of the metric, finding that, especially for highly rotating black holes, the two solutions disagree significantly. Such a metric parameterization allows one to perform the no-hair tests in a modelindependent way, and finally map constraints to specific alternative theories of gravity.

Probing the black hole metric: Black hole shadows and binary black-hole inspirals

“…The only way to deform the Kerr metric while removing any pathologies outside its horizon (at least for a broad range of deformation parameters) is to ignore explicitly the requirement that the metric be Ricci flat. In recent years, this has led to a number of parametrically deformed metrics that are free of pathologies but allow for deformations to be dialed in with different phenomenological parameters: the Johannsen-Psaltis metric (hereafter JP) [20,22] and its extensions by Cardoso, Pani, & Rico [56] and Carson & Yagi [57], the modified-gravity bumpy Kerr metric of Vigeland, Yunes, and Stein [21] (hereafter MGBK), etc. In a different approach, a general metric can be written in terms of polynomial or rational functions with free coefficients such that, when a particular discrete set of coefficients is chosen, the metric approximates non-Kerr solutions to various modified-gravity field equations; if the non-Kerr solution is free of pathologies, so is the polynomial or rational expansion [58].…”

Testing general relativity with x-ray reflection spectroscopy: The Konoplya-Rezzolla-Zhidenko parametrization

“…The top-down metrics are those which are obtained as a solution of an alternative theory of gravity, e.g., the Einstein-dilaton-Gauss-Bonnet BHs [37][38][39][40][41][42], the Chern-Simons BHs [43][44][45][46], and the Kerr-Sen BHs [47][48][49][50]. The bottom-up metrics on the other hand are obtained not from a specific theory of gravity but by generalizing the Kerr metric [28,29,31,36,51]. Each approach has its advantages and disadvantages.…”

Quantifying closeness between black hole spacetimes: a superspace approach

“…For this reason, there is interest in developing generic, parameterised BH spacetimes [22][23][24], with the goal of using these to test for a variety of hypothetical deviations from the Kerr metric in a theory-agnostic way. On the other hand, a high level of generality implies that many of the proposed metrics can be mapped to one another [26,27], though it is not always obvious how this can be achieved in practice. It is therefore useful to have a tool which can quantitatively distinguish between various parameterised metrics, or indeed between non-Kerr metrics arising as exact solutions in beyond-Einstein theories of gravity.…”

“…To address the dearth of exact solutions, various approaches to constructing metrics representing generic BHs in a theory-agnostic manner have been developed [22][23][24][25]. These metrics are designed to represent parameterised departures from a Kerr description in some particular way, such as including deformations that still preserve the Killing tensor symmetry [26,27]. Recently, Konoplya and Zhidenko (KZ hereafter) [28] derived a metric which contains "the only parameters that matter", in the sense that they found including higher-order terms beyond those in their parameterised metric affected electromagnetic observables very little.…”