Horizon Wave-Function and the Quantum Cosmic Censorship

We investigate the Cosmic Censorship Conjecture by means of the horizon wave-function (HWF) formalism. We consider a charged massive particle whose quantum mechanical state is represented by a spherically symmetric Gaussian wave-function, and restrict our attention to the superxtremal case (with charge-to-mass ratio $\alpha>1$), which is the prototype of a naked singularity in the classical theory. We find that one can still obtain a normalisable HWF for $\alpha^2<{2}$, and this configuration has a non-vanishing probability of being a black hole, thus extending the classically allowed region for a charged black hole. However, the HWF is not normalisable for $\alpha^2>2$, and the uncertainty in the location of the horizon blows up at $\alpha^2=2$, signalling that such an object is no more well-defined. This perhaps implies that a quantum Cosmic Censorhip might be conjectured by stating that no black holes with charge-to-mass ratio greater than a critical value (of the order of $\sqrt{2}$) can exist.

We investigate the Cosmic Censorship Conjecture by means of the horizon wave-function (HWF) formalism. We consider a charged massive particle whose quantum mechanical state is represented by a spherically symmetric Gaussian wave-function, and restrict our attention to the superxtremal case (with charge-to-mass ratio α > 1), which is the prototype of a naked singularity in the classical theory. We find that one can still obtain a normalisable HWF for α 2 < 2, and this configuration has a non-vanishing probability of being a black hole, thus extending the classically allowed region for a charged black hole. However, the HWF is not normalisable for α 2 > 2, and the uncertainty in the location of the horizon blows up at α 2 = 2, signalling that such an object is no more well-defined. This perhaps implies that a quantum Cosmic Censorhip might be conjectured by stating that no black holes with charge-to-mass ratio greater than a critical value (of the order of √ 2) can exist.

I. INTRODUCTION
A complete understanding of the gravitational collapse of a compact object remains one of the most challenging issues in contemporary theoretical physics. The general relativistic (GR) description, resulting in the formation of a black hole (BH) or naked singularity (NS), was first investigated in the papers of Oppenheimer and coworkers [1]. Although the literature on the subject has grown immensely (see, e.g. Ref. [2]), many technical and conceptual issues remain. One of these is the famous Cosmic Censorship Conjecture (CCC), proposed by Penrose in 1969 [3], which states that no singularities will ever become visible to an outer observer in a generic gravitational collapse starting from reasonable nonsingular initial states. To date, the conjecture remains unproved, and it is considered one of the most important open problems in gravitational physics. Another great open issue in GR is the problem of considering the quantum mechanical (QM) nature of the collapsing matter [4]. We will here address both issues for the Reissner-Nordström (RN) geometry, which describes charged BHs, a subject of many theoretical investigations in the past (see, e.g. Ref. [5]).
Most attempts at quantising BH metrics consider the gravitational degrees of freedom unrelated to the matter state that sources the geometry. More recently, the Horizon Wave Function (HWF) formalism was proposed [6], as a way of quantising the Einstein equation that determines the gravitational radius of a spherically symmetric matter source and its time evolution [7], which instead relates the quantum state of the horizon to the quantum state of matter. This formalism was then applied to a few different case studies [8,9], yielding apparently sensible results in agreement with (semi)classical expectations, and there is therefore hope that it will facilitate our understanding of the formation of BHs from QM particles. In particular, it seems natural to extend this formalism beyond Schwarzschild BHs and tackle the CCC from a quantum perspective by considering an electrically charged particle represented by a Gaussian wave-packet in the classical regime in which it would be a NS.

II. ELECTRICALLY CHARGED SPHERICAL SOURCES
We start by recalling the classical RN metric can be written as where M and Q respectively represent the ADM mass and charge of the source, p is the Planck length and m p the Planck mass 1 . For |Q| < p M/m p , the above metric contains two horizons, namely and represents a BH. The two horizons overlap for |Q| = p M/m p , the so-called extremal BH case, while for |Q| > p M/m p no horizon exists and the central singularity is therefore accessible to outer observers. This is the prototype of a NS, which we will refer to as the "superextremal geometry". It is in fact more convenient to express all relevant quantities in terms of the mass M and the (positive definite) specific charge Using this parameter, the above expression (3) becomes and the three regimes mentioned above are then explicitly parametrised as i) 0 < α < 1 for the BH with two horizons 2 , ii) α = 1 for the extremal BH, and iii) α > 1 for the superextremal geometry.
We shall now investigate the superextremal geometry from a quantum mechanical perspective by first determining the HWF for α < 1 and then extending it continuously into the regime α > 1. This procedure is based on lifting the classical relation (5) to the rank of an equation for the operatorsR ± andM , which are chosen to act multiplicatively on the horizon wave-function (the ratio α will instead be considered as a simple parameter).

A. HWF for Gaussian sources
Let us consider as a source of the RN space-time an electrically charged massive particle at rest in the origin of the reference frame, represented by a spherically symmetric Gaussian wave-function We shall always assume that the width of the Gaussian is the minimum compatible with the Heisenberg uncertainty principle, that is where λ m is the Compton length of the particle of rest mass m.
In momentum space, the wave-function of the particle described above is where p 2 = p · p is the square modulus of the spatial momentum, and the width ∆ = m p p / m. For the energy of the particle, we assume the relativistic massshell relation in flat space, For α < 1, it is clear that one can now write a HWF for each of the two horizons. In fact, from the quantum version of Eq. (3), the total energy M can be expressed in terms of the two horizon radii as andR Note that we promoted M , R + , and R − into operatorŝ M ,R + , andR − , which are related to the corresponding observables. Our specific choice is not unique, and it is associated with usual ambiguities when going from a classical to quantum formalism. The unnormalised HWFs for R + and R − are then obtained by expressing p from the mass-shell relation (9) in terms of the energy M in Eq. (10), and then replacing one of the relations in Eq. (11) into Eq. (8). The two HWFs are then given by where the Heaviside function arises from the minimum energy in the spectral decomposition of the wavefunction (6) being M = m, which corresponds to Finally, the normalizations N ± are fixed by assuming the scalar product 3 where, like in the previous equation, the upper signs are used for the normalization of ψ H (R + ), while the lower signs are used when normalizing ψ H (R − ). The probability density that the particle lies inside its horizon of radius r = R ± can now be calculated starting from the wave-functions (12) associated with (6) as where is the probability that the particle is inside a sphere of radius r = R ± , and is the probability that the sphere of radius r = R ± is a horizon. Finally, one can integrate (15) over all possible values of the horizon radius R + to find the probability for the particle described by the wave-function (6) to be a BH, namely The analogous quantity for R − , will instead be the probability that the particle lies further inside its inner horizon, and both R − and R + are therefore realized (for more details about this case, see Ref. [10]).

B. Superextremal geometry
We will now focus on studying overcharged sources, represented by the range of specific charge α > 1. It is well known that in the classical theory of gravity, the CCC a priori forbids the existence of NSs. In the case of the classical charged BHs, this would precisely correspond to α > 1, so it is interesting to investigate whether quantum physics leads to any modifications or predictions therein. Our guiding principle will be to assume that the quantum states in the regime α > 1 can be obtained by extending continuously the HWF from the case α < 1.
The first problem that we encounter for α > 1 is that the operatorsR ± are not Hermitian. We could stop right there and say that there are no observables which correspond toR ± in the regime α > 1. However, we will make a simple (and perhaps non-unique) choice, which will allow us to proceed and probe the classically forbidden region. We will take the real parts of the multiplicative operatorsR ± , which are certainly Hermitian, to correspond to quantum observables. With this choice, the modulus squared of the two HWFs are given from Eq. (12), for R ± > R min± , by which, for α > 1, becomes one expression where R now replaces both R + and R − . This HWF is still normalizable in the scalar product (14) if R belongs to the real axis and the specific charge lies in the range We could therefore infer that no normalizable quantum state with α 2 > 2 is allowed, or that there is an obstruction that prevents the system from crossing α 2 = 2. This point will be further clarified after we have fully determined the HWF. The fact that the HWF (21) is the same for R + and R − reflects the classical behaviour according to which the two real horizons merge at the critical value of α = 1, and then mathematically extend as one into the complex realm for α > 1. We then need to address what happens to the Heaviside function in Eq. (12) when we extend it into the superextremal regime. First of all, we note that although Eq. (13) becomes complex for α > 1, its real part is again the same for R + and R − , namely We can then show that the same continuity principle, which led us to Eq. (21), requires that R be bounded from below by this R min . In fact, the expectation value forR is, in this case, and it matches exactly the corresponding expressions for α < 1, namely lim α 1 One can likewise show that the uncertainty matches the corresponding uncertainties at the specific charge α = 1, but we omit the explicit expressions since they are rather cumbersome. We just note that, for α = 1, the width of the Gaussian > R for m < 2 + e √ π erf (1) m p /2 0.8 m p , so that quantum fluctuations in the source's position will dominate for masses significantly smaller than the Planck scale (in qualitative agreement with the neutral case [6,7,10]). It is now interesting to analyse the limit α 2 → 2. One may have already noticed that so that the ratio R / blows up at α 2 = 2 for any values of the mass m = m p p / . The same indeed occurs to the uncertainty, since for α 2 → 2 (see also Fig. 1). Cases with m mp are not plotted since they behave the same as m = 2 mp, i.e. an object with 1 < α 2 < 2 must be a BH.
Using Eq. (18), one can also calculate the probability P BH that the particle is a BH for α in the allowed superextremal range (22). This probability is displayed in Fig. 2. One notices that, for a particle mass above the Planck scale, P BH is practically one throughout the entire range of α (thus extending a similar result that holds for α < 1 [10]). Moreover, even for m significantly less than m p , P BH approaches one in the limit α 2 → 2. We recall here that P BH 1 for small m is essentially related to R , and the system is thus dominated by quantum fluctuations in the source's position well below the Planck scale. On the other end, since both R and ∆R blow up on approaching α 2 = 2, the superextremal configurations with a significant probability of being BHs contain strong quantum fluctuations in the horizon's position.

III. CONCLUSIONS
From the above analysis we can learn two important things. First, quantum mechanical effects are perhaps able to continuously take us into the classically forbidden region of α > 1. This means that even an overcharged object, with a charge-to-mass ratio greater than unity, can still make a quantum BH. The basic reason for this is that in our formalism the location of the horizon is not given by a sharp classical value, instead it is described by a quantum wave function with associated uncertainties. Second, the charge-to-mass ratio α cannot be arbitrarily large, even in the context of QM. We found that for α 2 > 2 the HWF cannot be normalised, and thus it is not describing a well defined physical object. Moreover, at the same value of α 2 = 2, the uncertainties in the location of the horizon become infinite, signalling again that such an object stops being well defined. We should warn the reader that the specific value of the upper limit α 2 = 2 in Eq. (22) might simply be a consequence of describing the source as the Gaussian function (6), and should not be taken literally. However, it is likely that the overall qualitative picture remains in a more general context, and our results imply that perhaps a quantum version of the CCC might be formulated by stating that no BHs with the charge-to-mass ratio greater than a critical value (of order √ 2) can exist. We should here recall that for the charged Reissner-Nordström metric with α ≤ 1 analysed in Ref. [10], as well for neutral sources [7,9], the single Gaussian constituent (6) leads to unacceptably large uncertainties in the horizon size of large astrophysical BHs. In fact, one has ∆R ∼ R , even for very large mass m, for which one expects a semiclassical behaviour for the horizon size R. The above quantum CCC will therefore have to be tested further, by considering models of BHs that allow for a semiclassical limit ∆R R . An example of such models is given by those in Refs. [9,11], which contain a very large number N of light constituents, whose wave-functions span the entire region inside R, and ∆R/ R ∼ N −1 . The emerging picture is that BHs of any size should be treated as macroscopic quantum objects (just like superconductivity and superfluidity are macroscopic quantum phenomena at scales where one expects classical physics to be a good description).
Finally, let us point out that the analysis performed in this work, and the above quantum CCC, should hold for sources with mass m within a few orders of magnitude of the Planck mass. Primordial BHs formed in the early universe by large density fluctuations could have masses in this range. Moreover, it is also plausible that overcharged configurations with such small masses emerge from the gravitational collapse of astrophysical objects, acting as seeds for much larger BHs. Our results should then apply straightforwardly to these two cases.