HORIZON QUANTUM MECHANICS A hitchhiker’s guide to Quantum Black Holes

Horizon Quantum Mechanics
A hitchhiker’s guide to Quantum Black Holes

Roberto Casadio    Andrea Giugno    Octavian Micu
Abstract

It is congruous with the quantum nature of the world to view the space-time geometry as an emergent structure that shows classical features only at some observational level. One can thus conceive the space-time manifold as a purely theoretical arena, where quantum states are defined, with the additional freedom of changing coordinates like any other symmetry. Observables, including positions and distances, should then be described by suitable operators acting on such quantum states. In principle, the top-down (canonical) quantisation of Einstein-Hilbert gravity falls right into this picture, but is notoriously very involved. The complication stems from allowing all the classical canonical variables that appear in the (presumably) fundamental action to become quantum observables acting on the “superspace” of all metrics, regardless of whether they play any role in the description of a specific physical system. On can instead revisit the more humble “minisuperspace” approach and choose the gravitational observables not simply by imposing some symmetry, but motivated by their proven relevance in the (classical) description of a given system. In particular, this review focuses on compact, spherically symmetric, quantum mechanical sources, in order to determine the probability they are black holes rather than regular particles. The gravitational radius is therefore lifted to the status of a quantum mechanical operator acting on the “horizon wave-function”, the latter being determined by the quantum state of the source. This formalism is then applied to several sources with a mass around the fundamental scale, which are viewed as natural candidates of quantum black holes.

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Keywords: Horizon Wave-Function; Quantum Mechanics; Black Holes.

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1 Introduction

After Einstein introduced the theory of Special Relativity , we have grown accustomed to thinking of the space-time as the geometrical space where things happen. In this respect, Special Relativity just adds one dimension to the three-dimensional space of Newtonian physics, which is the natural arena for describing mathematically our intuitive notion of motion, or object displacements. However, we should not forget Einstein’s first great achievement came from a rethinking of the concept of time and length as being related to actual measurements, which in turn require synchronised clocks. Quantum physics emerged around the same time from the very same perspective: a proper description of atoms and elementary particles, and other phenomena mostly occurring at microscopic scales, required a more refined analysis of how variables involved in such phenomena are actually measured. Since measuring means interacting with the system under scrutiny, the uncertainty principle due to a finite Planck constant then came out as a fact of life, like the Lorentz transformations come out from the finite speed of light. This gave rise to the mathematical structure of the complex Hilbert space of states, on which observables are given by operators with suitable properties, and the outcome of any measurements could then be predicted with at best a certain probability. In Special Relativity one can nonetheless think of the space-time coordinates as being labels of actual space-time points, observables in principle, as they implicitly define an inertial observer.

Then came General Relativity , which allows for the use of any coordinates to identify space-time points, in a way that let us describe physics again much closer to what experimentalists do. The price to pay is that space-time correspondingly becomes a manifold endowed with a general Lorentzian metric, which acts as the “potential” for the universal gravitational force. This metric, in practice, determines the causal structure that was before given by the fixed Minkowski metric, and black holes (BHs) were found in this theory. The quantisation of matter fields on these metric manifolds led to the discovery of paradoxes and other difficulties, which are often pinpointed as the smoking gun that these two theories, of Quantum matter (fields) and General Relativity, are hard to unify. But if one looks back at how these two pillars of modern physics precisely emerged from the rethinking of the interplay between a physical system and the observer, the path to follow should become clear, at least ideally: one should give up as many assumptions as possible, and set up the stage for describing the most fundamental processes that involve both. In so doing, one preliminary question we can try to address is what are the best variables to use (for each specific system), regardless of what we have come to accept as “fundamental” or “elementary”. The very concept of space-time, as a “real” entity, should be put through this rethinking process. If the aim of our quantum theory is to describe the motion of objects, the space-time geometry is just an effective picture that we can conveniently employ in classical General Relativity, but which might be too difficult to describe fully in the quantum theory . In fact, the first step in this construction should be to give a clear modelling of the detection process by which we observe something somewhere: which observables should we employ then, and what are the physical restrictions we expect on them? All we wrote above is in fact nothing new. Any attempt at quantising canonically the Einstein-Hilbert action  falls into this scheme, in which the space-time is just a mathematical arena, and the metric becomes the basic observable, along with matter variables. Unfortunately, a mathematical treatment of the so called “superspace” of wave-functions describing all the possible states of the metric is extremely complicated. In fact, DeWitt himself, in his famous 1967 paper , immediately reverted to a simplified formulation in order to apply it to cosmology. His choice was based on preserving isotropy and homogeneity of the universe at the quantum level, which leads to the Friedman-Robertson-Walker family of metrics, with one degree of freedom, the scale factor. The corresponding space of quantum states is greatly simplified and referred to as the FRW “minisuperspace”.

On the other hand, one of the most relevant scenarios where we expect a quantum theory of gravitation could lead to strong predictions is the collapse of compact objects and the possible formation of BHs. This physical process cannot be realistically modelled as isotropic or homogeneous in all of its aspects, both because of the high non-linearity of the underlying relativistic dynamics and for the presence of many mechanisms, e.g. generating outgoing radiation . After the seminal papers of Oppenheimer and co-workers , the literature on the subject has grown immensely, but many issues are still open in General Relativity (see, e.g. Ref. , and references therein). This is not to mention the conceptual and technical difficulties one faces when the quantum nature of the collapsing matter is taken into account. Assuming quantum gravitational fluctuations are small, one can describe matter by means of Quantum Field Theory on the curved background space-time , an approach which has produced remarkable results, like the discovery of the Hawking evaporation . However, the use of a fixed background is directly incompatible with the description of a self-gravitating system representing a collapsing object, for which the evolution of the background and possible emergence of non-trivial causal structures cannot be reliably addressed perturbatively.

A general property of the Einstein theory is that the gravitational interaction is always attractive and we are thus not allowed to neglect its effect on the causal structure of space-time if we pack enough energy in a sufficiently small volume. This can occur, for example, if two particles (for simplicity, of negligible spatial extension and total angular momentum) collide with an impact parameter shorter than the Schwarzschild radius corresponding to the total center-of-mass energy of the system, that is aaaWe shall use units with , and always display the Newton constant , where and are the Planck length and mass, respectively, so that .

(1)

This hoop conjecture  has been checked and verified theoretically in a variety of situations, but it was initially formulated for BHs of (at least) astrophysical sizes , for which the very concept of a classical background metric and related horizon structure should be reasonably safe (for a review of some problems, see the bibliography in Ref. ). Whether the concepts involved in the above conjecture can also be trusted for masses approaching the Planck size, however, is definitely more challenging. In fact, for masses in that range, quantum effects may hardly be neglected (for a recent discussion, see, e.g., Ref. ) and it is reasonable that the picture arising from General Relativistic BHs must be replaced in order to include the possible existence of “quantum BHs”. Although a clear definition of such objects is still missing, most would probably agree that their production cross-section should (approximately) comply with the hoop conjecture, and that they do not decay thermally (see, e.g., Refs. ).

The main complication in studying the Planck regime is that we do not have any experimental insight thereof, which makes it very difficult to tell whether any theory we could come up with is physically relevant. We might instead start from our established concepts and knowledge of nature, and push them beyond the present experimental limits. If we set out to do so, we immediately meet with a conceptual challenge: how can we describe a system containing both Quantum Mechanical objects (such as the elementary particles of the Standard Model) and classically defined horizons? The aim of this review is precisely to show how one can introduce an operator (observable) for the gravitational radius, and define a corresponding horizon wave-function (HWF) , which can be associated with any localised Quantum Mechanical particle or source . This horizon quantum mechanics (HQM) then provides a quantitative (albeit probabilistic) condition that distinguishes a BH from a regular particle. Since this “transition” occurs around the Planck scale, the HQM represents a simple tool to investigate properties of (any models of) quantum BHs in great generality. We shall also review how the HQM naturally leads to an effective Generalised Uncertainty Principle (GUP)  for the particle position, a decay rate for microscopic BHs , and a variety of other results for BHs with mass around the fundamental Planck scale  (for a review of the results obtained from the HWF for Bose-Einstein condensate models of astrophysical size BHs, see Ref. ).

The paper is organised as follows: in the next Section, we first recall a few relevant notions about horizons in General Relativity and then illustrate the main ideas that define the HQM  and how it differs from other attempts at quantising horizon degrees of freedom; in Section 3, we apply the general HQM to the particularly simple cases of a particle described by a Gaussian wave-function at rest, electrically neutral in four  and in dimensions (with and , and with electric charge in four dimensions ; we also consider collisions of two such particles in one spatial dimension and extend the hoop conjecture into the quantum realm ; in Section 4, we recall a proposal for including the time evolution in the HQM  and, finally, in Section 5, we comment on such findings and outline future applications.

2 Horizon Quantum Mechanics

The very first attempt at solving Einstein’s field equations resulted in the discovery of the Schwarzschild metric 

(2)

with

(3)

and the appearance of the characteristic length associated to the source. In fact, given a spherically symmetric matter source, the Schwarzschild radius measures the area of the event horizon, which makes the interior of the sphere causally disconnected from the outer portion of space-time. At the same time, Quantum Mechanics (QM) naturally associates a Compton-de Broglie wavelength to a particle. This is the minimum resolvable length scale, according to the Heisenberg uncertainty principle, and it can be roughly understood as the threshold below which quantum effects cannot be neglected. It is clear that any attempt at quantising gravity should regard those two lengths on somewhat equal grounds. We therefore start with a brief review of these concepts before discussing how to deal with them consistently in the quantum theory.

2.1 Gravitational radius and trapping surfaces

In order to introduce the relevant properties of a classical horizon, we start by writing down the most general metric for a spherically symmetric space-time as 

(4)

where is the areal coordinate and are coordinates on surfaces where the angles and are constant. It is clear that all the relevant physics takes place on the radial-temporal plane and we can safely set and from now on. Heuristically, we can think of a (local) “apparent horizon” as the place where the escape velocity equals the speed of light, and we expect its location be connected to the energy in its interior by simple Newtonian reasoning. More technically, in General Relativity, an apparent horizon occurs where the divergence of outgoing null congruences vanishes , and the radius of this trapping surface in a spherically symmetric space-time is thus determined by

(5)

where is the covector perpendicular to surfaces of constant area . But then General Relativity makes it very hard to come up with a sensible definition of the amount of energy inside a generic closed surface. Moreover, even if several proposals of mass functions are available , there is then no simple relation between these mass functions and the location of trapping surfaces. Accidentally, spherical symmetry is powerful enough to overcome all of these difficulties, in that it allows to uniquely define the total Misner-Sharp mass as the integral of the classical matter density weighted by the flat metric volume measure,

(6)

as if the space inside the sphere were flat. This Misner-Sharp function represents the active gravitational mass bbbRoughly speaking, it is the sum of both matter energy and its gravitational potential energy. inside each sphere of radius and also determines the location of trapping surfaces, since Einstein equations imply that

(7)

where . Due to the high non-linearity of gravitational dynamics, it is still very difficult to determine how a matter distribution evolves in time and forms surfaces obeying Eq. (2.1), but we can claim that a classical trapping surface is found where the gravitational radius equals the areal radius , that is

(8)

which is nothing but a generalisation of the hoop conjecture (1) to continuous energy densities. Of course, if the system is static, the above radius will not change in time and the rapping surface becomes a permanent proper horizon (which is the case we shall mostly consider in the following).

It stands out that the above picture lacks of any mass threshold, since the classical theory does not yield a lower limit for the function . Therefore, it seems that one can set the area of the trapping surface to be arbitrarily small and eventually have BHs of vanishingly small mass.

2.2 Compton length and BH mass threshold

As we mentioned above, quantum mechanics provides a length cut-off through the uncertainty in the spatial localisation of a particle. It is roughly given by the Compton length

(9)

if, for the sake of simplicity, we consider a spin-less point-like source of mass . It is a well-established fact that quantum physics is a more fundamental description of the laws of nature than classical physics. This means that only makes sense when it is not “screened” by , that is

(10)

and, equivalently, the BH mass must satisfy

(11)

or . We want to remark that the Compton length (2.2) can also be thought of as a quantity which rules the quantum interaction of with the local geometry. Although it is likely that the particle’s self-gravity will affect it, we still safely assume the flat space condition (2.2) as a reasonable order of magnitude estimate.

In light of recent developments, the common argument that quantum gravity effects should become relevant only at scales of order or higher appears to be somewhat questionable, since the condition (2.2) implies that a classical description of a gravitational system with should be fairly accurate (whereas for the judge remains out). This is indeed the idea of “classicalization” in a nutshell, as it was presented in Refs.  and, before that, of models with a minimum length and gravitationally inspired GUPs . The latter are usually presented as fundamental principles for the reformulation of quantum mechanics in the presence of gravity, following the canonical steps that allow to bring a theory to the quantum level. In this picture, gravity would then reduce to a “kinematic effect” encoded by the modified commutators for the canonical variables. In this review, we shall instead follow a different line of reasoning: we will start from the introduction of an auxiliary wave-function that describes the horizon associated with a given localised particle, and retrieve a modified uncertainty relation as a consistent result .

2.3 Horizon Wave-Function

We are now ready to formulate the quantum mechanical description of the gravitational radius in three spatial dimensions in a general fashion . For the reasons listed above, we shall only consider quantum mechanical states representing spherically symmetric objects, which are localised in space. Since we want to put aside a possible time evolution for the moment (see Section 4), we also choose states at rest in the given reference frame or, equivalently, we suppose that every function is only taken at a fixed instant of time. According to the standard procedure, the particle is consequently described by a wave-function , which we assume can be decomposed into energy eigenstates,

(12)

As usual, the sum over the variable represents the decomposition on the spectrum of the Hamiltonian,

(13)

regardless of the specific form of the actual Hamiltonian operator . Note though that the relevant Hamiltonian here should be the analogue of the flat space energy that defines the Misner-Sharp mass (2.1). Once the energy spectrum is known, we can invert the expression of the Schwarzschild radius in Eq. (1) in order to get

(14)

We then define the (unnormalised) HWF as

(15)

whose normalisation is fixed by means of the Schrödinger scalar product in spherical symmetry,

(16)

In this conceptual framework, we could naively say that the normalised wave-function yields the probability for an observer to detect a gravitational radius of areal radius associated with the particle in the quantum state . The sharply defined classical radius is thus replaced by the expectation value of the operator . Since the related uncertainty is in general not zero, this gravitational quantity will necessarily be “fuzzy”, like the position of the source itself. In any case, we stress that the observational meaning of the HQM will appear only after we introduce a few derived quantities.

In fact, we recall that we aimed at introducing a quantitative way of telling whether the source is a BH or a regular particle. Given the wave-function associated with the quantum state of the source, the probability density for the source to lie inside its own horizon of radius will be the product of two factors, namely

(17)

The first term,

(18)

is the probability that the particle resides inside the sphere of radius , while the second term,

(19)

is the probability density that the value of the gravitational radius is . Finally, it seems natural to consider the source is a BH if it lies inside its horizon, regardless of the size of the latter. The probability that the particle described by the wave-function is a BH will then be given by the integral of (2.3) over all possible values of the horizon radius , namely

(20)

which is the main outcome of the HQM.

In the following, we shall review the application of this construction to some simple, yet intriguing examples, in which the source is represented by Gaussian wave-functions. We anticipate that such states show very large horizon fluctuations and are not good candidates for describing astrophysical BHs  (for which extended models instead provide a better semiclassical limit ), but appear well-suited for investigating BHs around the fundamental Planck scale as unstable bound states .

2.4 Alternative horizon quantizations

Fig. 1: Pictorial view of the HQM radial fluctuations (dashed lines) and quantum field theoretic fluctuations (dotted line) around the classical horizon radius (solid line).

It is important to remark the differences of the HQM with respect to other approaches in which the gravitational degrees of freedom of (or on) the horizon are quantised according to the background field method  (see, e.g. Refs. ). In general, such attempts consider linear perturbations of the metric on this surface , and apply the standard quantum field construction , which is what one would do with free gravitons propagating on a fixed background. Of course, the fact that the horizon is a null surface implies that these perturbative modes enjoy several peculiar properties. For instance, they can be described by a conformal field theory , which one can view as the origin of the idea of BHs as holograms .

In the HQM, one instead only describes those spherical fluctuations of the horizon (or, more, precisely, of the gravitational radius) which are determined by the quantum state of the source. These fluctuations therefore do not represent independent gravitational degrees of freedom, although one could suggest that they be viewed as collective perturbations in the zero point energy of the above-mentioned perturbative modes (see Fig. 1). In this respect, the HWF would be analogous to the quantum mechanical state of a hydrogen atom, whereas the perturbative degrees of freedom would be the quantum field corrections that lead to the Lamb shift.

Let us finally point out that the HQM also differs from other quantisations of the canonical degrees of freedom associated with the Schwarzschild BH metric , in that the quantum state for the matter source plays a crucial role in defining the HWF. The HQM is therefore complementary to most of the approaches one usually encounters in the literature. In fact, it can be combined with perturbative approaches, like it was done in Ref. , to show that the poles in the dressed graviton propagator  can indeed be viewed as (unstable) quantum BHs.

3 Spherically symmetric Gaussian sources

We can make the previous formal construction more explicit by describing the massive particle at rest in the origin of the reference frame with the spherically symmetric Gaussian wave-function 

(21)

We shall often consider the particular case when the width (related to the uncertainty in the spatial size of the particle) is roughly given by the Compton length (2.2) of the particle,

(22)

Even though our analysis holds for independent values of and , one expects that and Eq. (3) is therefore a limiting case of maximum localisation for the source. It is also useful to recall that the corresponding wave-function in momentum space is given by

(23)

with being the square modulus of the spatial momentum, and the width

(24)

Note that the mass is not the total energy of the particle, and if the spectrum of is positive definite.

3.1 Neutral spherically symmetric BHs

In order to relate the momentum to the total energy , the latter being the analogue of the Misner-Sharp mass (2.1), we simply and consistently assume the relativistic mass-shell equation in flat space-time,

(25)

From Eq. (2.3), and fixing the normalisation in the inner product (2.3), we then obtain the HWF 

(26)

where we defined and the Heaviside step function appears in the above equation because . Finally,

(27)

is the upper incomplete Gamma function. In general, one has two parameters, the particle mass and the Gaussian width . The HWF will therefore depend on both and so will the probability , which can be computed only numerically  (see also section 4).


(A)

(B)

Fig. 2: Probability densities in Eq. (2.3) (solid line) and (dashed line) for (upper panel) and (lower panel), assuming .

Fig. 3: Probability density in Eq. (3.1) that particle is inside its horizon of radius , for (solid line) and for (dashed line), assuming .

Fig. 4: Probability in Eq. (29) that particle of width is a BH.

Fig. 5: Probability in Eq. (29) that particle of mass is a BH.

As we mentioned previously, it seems sensible to assume . In particular, the condition in Eq. (3) precisely leads to a BH mass threshold of the form given in Eq. (2.2). We indeed expect that the particle will be inside its own horizon if , and Eq. (2.2) then follows straightforwardly from and . For example, this conclusion is illustrated in Fig. 2, where the density is plotted along with the probability density for and . In the former case, the horizon is more likely found within a smaller radius than the particle’s, with the opposite situation occurring in the latter. As a matter of fact, the probability density (2.3) can be explicitly computed,

(28)

where is the lower incomplete Gamma function. One can integrate the density (3.1) for from to infinity and the probability (2.3) for the particle to be a BH is finally given by

(29)

where is the Owen’s function (A.7cccMore detailed calculations of cumbersome integrals are given in Appendix A. In this particular case, the variable , and we made use of Eq. (A.8) with .. Since we are assuming that , this probability can also be written as a function of the mass as

(30)

In Fig. 3, we plot the probability density (3.1), for different values of the Gaussian width . It is already clear that such a probability decreases with (eventually vanishing below the Planck mass). In fact, in Fig. 4, we show the probability (29) that the particle is a BH as a function of the width , and in Fig. 5 the same probability as a function of the particle mass . From these plots of , we can immediately infer that the particle is most likely a BH, namely , for or - equivalently - . We have therefore derived the condition (2.2) from a totally Quantum Mechanical picture.

We conclude by recalling that a simple analytic approximation is obtained by taking the limit in Eq. (3.1), namely 

(31)

from which follows the approximate probability

(32)

Fig. 6 shows graphically that this approximation slightly underestimates the exact probability in Eq. (29).

Fig. 6: Plot of exact in Eq. (29) (straight line) and its approximation in Eq. (32) (dashed line).

3.1.1 Effective GUP and horizon fluctuations

From the Gaussian wave-function (3), we easily find that the uncertainty in the particle’s size is given by

(33)

where

(34)

Analogously, the uncertainty in the horizon radius results in

(35)

where

(36)

is the generalised exponential integral. Since

(37)

we can write the width of the Gaussian as , and, finally, assume the total radial uncertainty is a linear combination of Eqs. (33) and (35), thus obtaining 

(38)

where is an arbitrary coefficient (presumably of order one), and

(39)

This GUP is plotted in Fig. 7 (for ), and is precisely of the kind considered in Ref. , leading to a minimum measurable length

(40)

obtained for

(41)

Of course, this is not the only possible way to define a combined uncertainty, but nothing forces us to consider a GUP instead of making direct use of the HWF.

Fig. 7: Uncertainty relation (38) (solid line) as a combination of the Quantum Mechanical uncertainty (dashed line) and the uncertainty in horizon radius (dotted line).

One of the main conclusions for the HQM of Gaussian states can now be drawn from Eq. (35), that is

(42)

which means the size of the corresponding horizon shows fluctuations of magnitude . This is clearly not acceptable for BHs with mass , which we expect to behave (semi)classically. In other words, the classical picture of a BH as the vacuum geometry generated by a (infinitely) thin matter source does not seem to survive in the quantum description, and one is led to consider alternative models for astrophysical size BHs .

3.1.2 Quantum BH evaporation

One of the milestones of contemporary theoretical physics is the discovery that BHs radiate thermally at a characteristic temperature 

(43)

However, if we try to extrapolate this temperature to vanishingly small mass , we see that diverges.

One can derive improved BH temperatures for from the GUP (see Refs.  for detailed computations). Here, we just recall that one obtains dddThe parameter here is analogue, but not necessarily equal, to the parameter in Eq. (38).

(44)

with the condition , which is necessary for the existence of a minimum BH mass (see Fig. 8). We remark that this is consistent with our previous analysis, since we stated repeatedly that a particle with a mass significantly smaller than should not be a BH, i.e.  whenever . It is straightforward to extremise (44) and get

(45)

Moreover, we can invert (44) in order to obtain and consider the “physical” branch, which reproduces the Hawking behaviour for . When we can expand the result for around , hence

(46)

We note that such an expansion for is well-defined even for , suggesting that the microscopic structure of the space-time may be arranged as a lattice . In the same approximation, we can also expand the canonical decay rate

(47)
(48)

where when  .


                                                    

Fig. 8: Temperature vs. mass according to Eq. (44) with : solid line reproduces the Hawking behaviour for large ; dotted line is the unphysical branch, and their meeting point represents the BH with minimum mass.

The reader may deem unlikely that an object with a mass of the order of can be faithfully described by the same standard thermodynamics which arises from a (semi-)classical description of BHs. On the other hand, the HQM is specifically designed to hold in a quantum regime. We can therefore guess that the decay of a Planck size BH will be related to the probability that the particle is found outside its own horizon eeeThe subscript T stands for tunnelling, which alludes to the understanding of the Hawking emission as a tunnelling process through the horizon . . Of course, if the mass , the HWF tells us the particle is most likely not a BH to begin with, so the above interpretation must be restricted to (see again Fig. 5). We first define the complementary probability density

(49)

where now

(50)

Upon integrating the above probability density over all values of , we then obtain

(51)

where and are positive constants. We can accordingly estimate the amount of particle’s energy outside the horizon as

(52)

On the other hand, from the time-energy uncertainty relation, , one gets the typical emission time

(53)

employing (1) and (35). Putting the two pieces together, we find that the flux emitted by a Planck size black hole would satisfy 

(54)

whose functional behaviour agrees with the result (48) obtained from a GUP.

There is a large discrepancy between the numerical coefficients in Eq. (48) and those in Eq. (54). First, we note that Eq. (47) holds in the canonical ensemble of statistical mechanics, and the disparity may therefore arise because a Planck mass particle cannot be consistently described by standard thermodynamics, which in turn requires the BH is in quasi-equilibrium with its own radiation . In fact, the canonical picture does not even enforce energy conservation, which is instead granted in the microcanonical formalism . However, the HQM is insensitive to thermodynamics and it is therefore remarkable that the HQM and the GUP yield qualitatively similar results. In any case, the above analysis of BH evaporation is very preliminary and significant changes are to be expected when considering a better description of the microscopic structure of quantum BHs .

3.2 Electrically charged sources

An extension of the original HQM regards the case of electrically charged massive sources , and was obtained in Refs.  from the Reissner-Nordström (RN) metric . The latter is of the form (2) with

(55)

where is again the ADM mass and is the charge of the source. In the following, it will be convenient to employ the specific charge

(56)

The case reduces to the neutral Schwarzschild metric. For , the above function has two zeroes, namely

(57)

and the RN metric therefore describes a BH. Moreover, the two horizons coincide for and the BH is said to be extremal, while the singularity is naked, i.e. accessible to an external observer, for .

3.2.1 Inner Horizon

The case was considered in Ref. , where the HQM was extended for the presence of more than one trapping surface. A procedure similar to the neutral case was followed for each of the two horizon radii (57): one initially determines the HWFs and then uses them to compute the probability for each horizon to exist. Eqs. (57) is lifted to the quantum level by introducing the operators and , which replace their classical counterparts and . Moreover, these operators are chosen to act multiplicatively on the respective wave-functions, whereas the specific charge remains a simple parameter (c-number) fffAs usual, going from the classical to the quantum realm is affected by ambiguities, and this choice is not unique..

First we note the total energy can be expressed in terms of the horizon radii as

(58)

and one also has

(59)

We then obtain the HWFs for and by expressing from the mass-shell relation (3.1) in terms of the eigenvalue of in Eq. (58), and then replacing one of the relations (59) into the wave-function representing the source in momentum space, as in Eq. (3). For the usual limiting case (3), , it is straightforward to obtain

(60)

where the minimum radii are given by

(61)

The probability densities for the source to be found inside each of the two horizons turn out to be

(62)

In the neutral case , is of course ill-defined, while equals the probability density (3.1), which means that becomes the Schwarzschild radius .

Fig. 9: Probability density in Eq. (62) that the particle is inside its outer horizon , for (thick lines) and (thin lines) with (continuous lines), (dotted lines) and (dashed lines). For , the two horizons coincide and .

Fig. 10: Probability density in Eq. (62) that particle is inside its inner horizon , for (thick lines) and (thin lines) with (continuous lines), (dotted lines) and (dashed lines). For , the two horizons coincide and .

Fig. 9 shows the probability density for the massive source to reside inside the external horizon for two values of the width (above and below the Planck scale) and three values of the specific charge . The maximum of this function clearly decreases when increases above or, equivalently, when gets smaller than the Planck mass. Fig. 10 shows the analogous probability densities for the inner horizon . Obviously, the smaller the smaller is the probability that a trapping surface occurs. Moreover, as we expected from the start, the density profiles coincide in the extremal case (thick and thin dashed lines), because the two horizons merge.


                                                            

Fig. 11: Probability in Eq. (63) for the particle to be a BH (thick lines) and in Eq. (63) for the particle to be inside its inner horizon (thin lines) as functions of for (continuous line), (dotted line) and (dashed line). For the two probabilities merge.

Integrating over , we obtain the probabilities ggg It is convenient to define , and use again Eq. (A.8), with .

(63)

where is again the Owen’s function (A.7).

Fig. 11 shows how these probabilities vary with the parameter for values of above or below the Planck scale. For the outer horizon, it is clear that for widths (mass larger than ). On the contrary, when (or ), the probability sensibly decreases as the specific charge approaches from below. We see that this probability is does not exactly vanish even when exceeds the Planck length . As an example, for , corresponding to , we find for a large interval of values of the specific charge. only falls below right before the BH becomes maximally charged (). As far as the inner horizon is concerned, the scenario is profoundly different. The same plot shows that the probability for small values of and increases with this parameter. However, the role of is prominent because the sharper the Gaussian packet is localised in space (or the more massive it is), the smaller the value of for which this probability becomes significant. To summarise, there is an appreciable range of values of the specific charge for which the inner horizon is not likely to exist (), while the system is a BH ().

Fig. 12: Probability for the particle to be a BH (thick lines) and for the particle to be inside its inner horizon (thin lines), in Eq. (63), as functions of , for (continuous line), (dotted line) and (dashed line). For thick and thin dashed lines overlap.

Fig. 13: Probability for the particle to be a BH (thick lines) and for the particle to be inside its inner horizon (thin lines), in Eq. (63), as functions of , for (continuous line), (dotted line) and (dashed line). For thick and thin dashed lines overlap.

The probabilities as functions of the width are shown in Fig. 12 and as functions of the mass in Fig. 13, for , and . It is evident that smaller values of allow for to approach for smaller masses . The specular situation happens when studying the inner probability . If we focus on the smallest specific charge considered here, , we notice that both probabilities are close to only around , and not at the naively expected scale . Hence, there exists a non-negligible interval in the possible values of (around the Planck scale) for in which

(64)

In this interval, the system is most likely a BH, because it is the outer horizon which dictates this property, while the inner horizon is still not very likely to exist. Lowering the value of this range grows larger, while it narrows and eventually vanishes when approaching the maximally charged limit .

We conclude by remarking that we could have guessed this result. In fact, the smaller , the more the system looks like a neutral (Schwarzschild) BH, since the mass becomes the dominant parameter and the presence of charge is (at most) a small perturbation. However, the existence of an inner horizon at is phenomenologically very important, because of the possible instability known as mass inflation  related to the specific features of such a Cauchy horizon. Eq. (64) suggests that this instability should not always occur for , even when the particle is (most likely) a BH.

3.2.2 Quantum Cosmic Censorship

Overcharged sources with were analysed in Ref.