(Transient) Scalar Hair for (Nearly) Extreme Black Holes

It has been shown recently that extreme Reissner-Nordstr\"{o}m black holes perturbed by a minimally coupled, free, massless scalar field have permanent scalar hair. The hair - a conserved charge calculated at the black hole's event horizon - can be measured by a certain expression at future null infinity: the latter approaches the hair inversely in time. We generalize this newly discovered hair also for extreme Kerr black holes. We study the behavior of nearly extreme black hole hair and its measurement at future null infinity as a transient phenomenon. For nearly extreme black holes the measurement at future null infinity of the length of the newly grown hair decreases quadratically in time at intermediate times until its length becomes short and the rate at which the length shortens further slows down. Eventually, the nearly extreme BH becomes bald again like non-extreme BHs.

It has been shown recently that extreme Reissner-Nordström black holes perturbed by a minimally coupled, free, massless scalar field have permanent scalar hair. The hair -a conserved charge calculated at the black hole's event horizon -can be measured by a certain expression at future null infinity: the latter approaches the hair inversely in time. We generalize this newly discovered hair also for extreme Kerr black holes. We study the behavior of nearly extreme black hole hair and its measurement at future null infinity as a transient phenomenon. For nearly extreme black holes the measurement at future null infinity of the length of the newly grown hair decreases quadratically in time at intermediate times until its length becomes short and the rate at which the length shortens further slows down. Eventually, the nearly extreme BH becomes bald again like non-extreme BHs.

I. INTRODUCTION
Scalar fields, which are ubiquitous in theoretical physics (e.g, the Higgs field) and in astrophysics (e.g., the inflaton, certain dark matter and dark energy models), have been proposed as candidates for black hole (BH) hair [1], in possible violation of the no-hair conjecture. The latter states that all BH solutions of the Einstein-Maxwell equations of general relativity can be completely characterized by three and only three externally observable classical parameters, specifically the BH's mass M , charge q, and spin angular momentum a. Bekenstein provided a proof for the nonexistence of scalar hair given a set of assumptions [2][3][4]. A number of scalar field hair models have been found, where one or more of the assumptions underlying Bekenstein's theorem are violated. Those include scalar fields with non-strictly positive potentials, scalar fields which are non-canonical or nonminimally coupled to gravity, bound states of "bald" BHs and solitons [1], or in spacetimes with more than four dimensions [5]. Also, non-scalar field hair models have been suggested, including non-abelian Yang-Mills [6] or Proca fields [7]. In all these examples it is the field itself that constitutes the BH's hair. In addition, when quantum mechanical effects are included, BHs can carry quantum numbers [8] and have soft hair [9].
More recently, a different kind of scalar hair for extreme Reissner-Nordström (ERN) BHs was found by Angelopoulos, Aretakis, and Gajic [10] (AAG), where a certain quantity s[ψ] evaluated at future null infinity (I + ) ("measurement at I + of AAG hair") equals a non-vanishing quantity H[ψ] ("Aretakis charge," "AAG hair," "horizon integral") calculated on the BH's event horizon (EH), but vanishes if the BH is non-extreme. Since H[ψ] is a conserved charge for ERN [14], it would naturally be related with a candidate for BH hair. Indeed, in [10] it was shown that s[ψ] equals H[ψ]. The AAG hair may be construed as a different class of BH hair than the types of hair discussed above, as it is made of minimally-coupled, free, massless scalar field. However, it is not the scalar field itself which constitutes the measurement at I + of the AAG hair, but a functional of the scalar field ψ which is calculated by adding two terms evaluated at I + , an "asymptotic term" s I [ψ] and a "global term", s II [ψ]:  In what follows we first verify numerically the occurrence of AAG hair for ERN. We then generalize the AAG hair also for extreme Kerr (EK) BHs. We next consider nearly extreme BHs (NERN or NEK, respectively), and show the AAG hair as a transient behavior, including observational features from far away.

II. NUMERICAL METHOD
Our numerical simulations begin with writing the 2+1 dimensional scalar wave equation in RN or Kerr spacetime backgrounds (Teukolsky equation) for azimuthal (m = 0) modes in compactified hyperboloidal coordinates, which allow us to access I + at a finite radial coordinate [11]. The resulting second-order hyperbolic partial differential equation is then re-written as a coupled system of two first-order hyperbolic equations. We then solve this system by implementing a second-order Richtmeyer-Lax-Wendroff iterative evolution scheme [12,13]. The initial data are a "truncated" Gaussian (to ensure compact support) with non-zero initial field values on the EH. Specifically, in hyperboloidal coordinates (ρ, τ ) (see [12] for definitions), the initially spherical (ℓ = 0) Gaussian pulse is centered at ρ = 1.0M with a width of 0.22M , so that we have horizon penetrating initial data that lead to H[ψ] = 0 on the initial data surface [10]. (For example, the horizon is at ρ = 0.95M for ERN and EK in these coordinates.) The Gaussian is truncated beyond ρ = 8.0M and the outer boundary is located at S = ρ(I + ) = 19.0M .
In practice, we approximate . At finite times the difference between τ and v (see Fig. 1 in [13]) is manifested in an apparent variation in H[ψ](v) which is a numerical artifact resulting from this approximation. For that reason, the physically relevant value which we use is We then apply s[ψ] and H[ψ] also for EK, and present our results in Fig. 2. Accurate numerical calculation of H[ψ] is more challenging for EK than for ERN, and requires us to increase the numerical grid density substantially. Figure 2 is the first evidence for AAG hair for EK. We find also for EK that s[ψ](u) approached

IV. NEARLY AND NON-EXTREME RN/KERR: NUMERICAL RESULTS
Next, we consider NERN and NEK. The AAG hair Fig. 3 for a number of q/M and a/M values for RN and Kerr BHs, respectively. For the extreme cases Fig. 3 shows the respective Aretakis charges [14].  for nearly extreme BHs are close at early times to those of their extreme counterparts, but at late times approach those of non-extreme BHs (i.e., vanishing values). The closer the BH is to extremality, the longer it takes to lose its grown hair and achieve baldness. We examine the rate at which this behavior occurs below. The behaviors shown above allow us to distinguish qualitatively between extreme, non-extreme, and nearlyextreme BHs, where the third exhibits transient behaviors between the first and the second.       ter the QNR phase (∼ 100M ), and then lasts for several hundred to thousands of M depending on a/M . We can now combine the previous results, and suggest that for NEK and for NERN

V. DISTINGUISHING EXTREME, NEAR-EXTREME AND NON-EXTREME RN/KERR
This deviation of nearly-extreme BHs from their extremal counterparts allows for their observational identification by distant observers. Specifically, measurements at I + of a newly perturbed nearly extreme BH shows initial growth of AAG hair. But whereas for EK or ERN where this hair is permanent, for nearly extreme BH the length of the newly grown hair decreases initially quadratically in time until its length becomes short and the rate at which the length shortens further slows down. Eventually the nearly extreme BH becomes bald again like non-extreme BHs. The nearly extreme BH may repeat its hair regrowth attempts when it is perturbed again, but will never succeed for long: It is to eventually lose its regrown hair and become bald again.
We can gain additional insight into the transient behavior of NEK and NERN by considering the relative contributions of the two terms in Eq. (1) Fig. 8 shows that at early times the ratio s I [ψ]/s II [ψ] is close to its extreme BH counterpart, but at late times it approaches negative unity, as for non-extreme BHs. We again find that the closer the BH to extremality, the longer it takes the ratio to get close to −1.
Our analysis provides an answer to the question of when a BH is considered nearly extreme. As implied by Figs. 3, 4, 7, and 8, when 1 − q/M = 5.0 × 10 −5 the transient scalar hair of the BH behaves as for non-extreme BHs. For 1 − q/M = 4.5 × 10 −6 we already see typical transient behavior, the hallmark of nearly extreme BHs. This effect complements the signature that can be detected by the emission of gravitational waves from a plunge into a nearly extreme BH [15].