A Metric for Rapidly Spinning Black Holes Suitable for Strong-Field Tests of the No-Hair Theorem (1105.3191v2)

Abstract: According to the no-hair theorem, astrophysical black holes are uniquely characterized by their masses and spins and are described by the Kerr metric. Several parametric deviations from the Kerr metric have been suggested to study observational signatures in both the electromagnetic and gravitational-wave spectra that differ from the expected Kerr signals. Due to the no-hair theorem, however, such spacetimes cannot be regular everywhere outside the event horizons, if they are solutions to the Einstein field equations; they are often characterized by naked singularities or closed time-like loops in the regions of the spacetime that are accessible to an external observer. For observational tests of the no-hair theorem that involve phenomena in the vicinity of the circular photon orbit or the innermost stable circular orbit around a black hole, these pathologies limit the applicability of the metrics only to compact objects that do not spin rapidly. In this paper, we construct a Kerr-like metric which depends on a set of free parameters in addition to its mass and spin and which is regular everywhere outside of the event horizon. We derive expressions for the energy and angular momentum of a particle on a circular equatorial orbit around the black hole and compute the locations of the innermost stable circular orbit and the circular photon orbit. We demonstrate that these orbits change significantly for even moderate deviations from the Kerr metric. The properties of our metric make it an ideally suited spacetime to carry out strong-field tests of the no-hair theorem in the electromagnetic spectrum using the properties of accretion flows around astrophysical black holes of arbitrary spin.

• The paper introduces a novel Kerr-like metric with additional deviation parameters to test the no-hair theorem under extreme gravity.
• It employs the Newman-Janis algorithm and parametric deformation to construct an axisymmetric, asymptotically flat spacetime free from common pathologies.
• Results reveal that deviations alter ISCO and photon orbits, offering measurable predictions for probing alternative theories of gravity.

An Analytical Metric for Rapidly Rotating Black Holes to Test the No-Hair Theorem

The no-hair theorem is a cornerstone in the paper of astrophysical black holes, asserting that such entities are fully characterized by only their mass and spin, yielding a spacetime described by the Kerr metric. However, exploring deviations from this scenario presents a fertile ground for testing the limits of general relativity, especially through observational signals like gravitational waves and electromagnetic spectra. Tim Johannsen and Dimitrios Psaltis propose a novel Kerr-like metric, encapsulating additional parameters that allow for potential deviations from the standard Kerr scenario while remaining regular everywhere outside the event horizon. This paper explores a mathematical framework that stabilizes these deviations even for rapidly spinning black holes and scrutinizes its consequences for a broad spectrum of astrophysical observations.

Construction and Properties of the Metric

The authors begin by constructing a metric that incorporates free parameters alongside mass and spin to account for deviations from the Kerr metric. A challenge in such constructions lies in avoiding irregularities like naked singularities and closed timelike curves, which typically mar spacetimes in which the no-hair theorem is violated. By introducing a parametric deviation to the Schwarzschild metric and extending it via the Newman-Janis algorithm, Johannsen and Psaltis achieve a Kerr-like spacetime that is axisymmetric and asymptotically flat.

The choice of the functional parameterization $h(r,\theta)$ ensures that the spacetime avoids pathologies up to the maximum value of spin. Consequently, this metric allows researchers to perform strong-field tests of the no-hair theorem even near the circular photon orbits or the innermost stable circular orbits (ISCOs).

Experimental and Observational Relevance

To align with observational constraints, the metric maintains compliance with the parameterized post-Newtonian (PPN) framework, particularly adhering to the strong limits on deviations from the expected behavior in weak-field regimes. Investigations of the ISCO and photon orbit reveal that these orbits vary substantially from Kerr predictions under moderate deviations, opening avenues to discern potential deviations from Kerr black holes in astronomical observations.

A significant result is that the radius of the ISCO decreases with increasing deviation parameters, which profoundly affects the accretion disk dynamics and radiation signatures that are often scrutinized in observational astrophysics, such as continuum spectra and quasi-periodic oscillations.

Implications for Future Research

The implications of this research are manifold. The parameterization offers a versatile tool for the astrophysical community, aiming to detect and measure deviations from Kerr behavior in strong-field regions. This becomes particularly pertinent in the context of upcoming observational missions focused on precisely measuring the properties of black holes through gravitational waveforms and electromagnetic signals.

Further, the framework could serve to test alternative theories of gravity, especially those predicting deviations in the strong-field regime. The adaptability of Johannsen and Psaltis's metric allows for easy integration into existing ray-tracing codes, enhancing its utility for theoretical and computational studies aimed at matching observational data with theoretical models.

Conclusions

In constructing a metric that remains robust up to maximal spins while allowing for parametric deviations, Johannsen and Psaltis provide an analytical tool capable of pushing the boundaries of current theoretical models against empirical data from existing and forthcoming telescopic and gravitational wave observations. Although aligned with general relativity's core principles, their metric invites us to test whether nature abides strictly by these principles in regions of extreme gravity or harbors deviations yet to be observed. As high-precision measurements become more feasible, frameworks like this one are crucial to expanding our understanding of black holes and the fundamental laws governing our universe.