[Paper Review] An observational signature for extremal black holes

Stefanos Aretakis, Gaurav Khanna+1 more|arXiv (Cornell University)|Jul 8, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy3 citations

TL;DR

This paper identifies a unique observational signature for extremal black holes using scalar perturbations in Reissner–Nordström and Kerr spacetimes. It derives a radiation field expression at future null infinity that remains non-zero only in the extremal case, directly linking this excitation to the conserved horizon charge, thus providing a classically measurable probe of horizon hair and falsifying the no-hair theorem for extremal black holes.

ABSTRACT

We consider scalar perturbations of the Reissner--Nordström family and the Kerr family. We derive a characteristic expression of the radiation field, at any given unit solid angle of future null infinity, and numerically show that its amplitude gets excited only in the extremal case. Our work, therefore, identifies an observational signature for extremal black holes. Moreover, we show that the source of the excitation is the extremal horizon instability and its magnitude is exactly equal to the conserved horizon charge.

Motivation & Objective

• To identify a detectable observational signature distinguishing extremal black holes from sub-extremal ones.
• To demonstrate that the conserved horizon charge—previously considered non-observable—can be measured by far-away observers.
• To establish a direct link between the radiation field at null infinity and the horizon charge in extremal black hole spacetimes.
• To validate that the signature arises exclusively from the extremal horizon instability and not from sub-extremal dynamics.
• To show that the signature is measurable by a single detector at a fixed solid angle on future null infinity, not requiring angular integration.

Proposed method

• Derives a characteristic expression for the radiation field at future null infinity, parameterized by retarded time τ and unit solid angle ϑ.
• Uses numerical simulations of massless scalar perturbations (□gψ = 0) on extremal and sub-extremal Reissner–Nordström and Kerr black holes.
• Defines a measurable quantity s₢₊(ϑ) at null infinity, redefined with a 1/4 factor for Kerr to preserve proportionality to the horizon charge H[ψ].
• Applies asymptotic analysis to extract the long-time behavior of the radiation field, identifying a non-decaying component only in the extremal case.
• Compares the asymptotic amplitude s₢₊(ϑ) with the conserved horizon charge H[ψ] via linear fitting, confirming exact proportionality.
• Performs numerical tests with initial data lacking support on the horizon to confirm the signature is not spurious or artifact-driven.

Experimental results

Research questions

• RQ1 Can a far-away observer detect a unique signature distinguishing extremal black holes from sub-extremal ones using scalar radiation?
• RQ2 Is the conserved horizon charge H[ψ]—previously thought to be non-observable—classically measurable through radiation at null infinity?
• RQ3 Does the radiation field at future null infinity exhibit a non-zero, persistent amplitude only in the extremal case, and if so, what is its origin?
• RQ4 Can this signature be extracted using a single detector at a fixed solid angle, avoiding the need for full-sky integration?
• RQ5 How does the magnitude of the radiation field at null infinity relate quantitatively to the horizon charge H[ψ] in the extremal limit?

Key findings

• The radiation field amplitude s₢₊(ϑ) at future null infinity remains non-zero and asymptotically constant only in the extremal Reissner–Nordström and Kerr cases.
• For sub-extremal black holes, s₢₊(ϑ) decays to zero over time, confirming the signature is unique to extremality.
• In the extremal case, s₢₊(ϑ) is exactly proportional to the conserved horizon charge H[ψ], with a best-fit slope of 1.0028 ± 0.13 for Kerr black holes.
• The signature is robust even when initial data have no support on the horizon, confirming it is not an artifact of initial conditions.
• The radiation field expression s₢₊(ϑ) is redefined with a 1/4 factor for Kerr to preserve proportionality to H[ψ], ensuring consistency across spacetimes.
• The excitation is sourced by the extremal horizon instability, which prevents decay of transverse derivatives at the horizon and imprints a measurable signal at infinity.