Advances in High Energy Physics

Volume 2016 (2016), Article ID 1543741, 8 pages

http://dx.doi.org/10.1155/2016/1543741

## Horizon Wavefunction of Generalized Uncertainty Principle Black Holes

Department of Physics, Loyola Marymount University, Los Angeles, CA 90045-2659, USA

Received 27 September 2016; Revised 14 November 2016; Accepted 17 November 2016

Academic Editor: Elias C. Vagenas

Copyright © 2016 Luciano Manfredi and Jonas Mureika. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the Horizon Wavefunction (HWF) description of a Generalized Uncertainty Principle inspired metric that admits sub-Planckian black holes, where the black hole mass is replaced by . Considering the case of a wave-packet shaped by a Gaussian distribution, we compute the HWF and the probability that the source is a (quantum) black hole, that is, that it lies within its horizon radius. The case is qualitatively similar to the standard Schwarzschild case, and the general shape of is maintained when decreasing the free parameter but shifted to reduce the probability for the particle to be a black hole accordingly. The probability grows with increasing mass slowly for more negative and drops to 0 for a minimum mass value. The scenario differs significantly for increasing , where a minimum in is encountered, thus meaning that every particle has some probability of decaying to a black hole. Furthermore, for sufficiently large we find that every particle is a quantum black hole, in agreement with the intuitive effect of increasing , which creates larger and terms. This is likely due to a “dimensional reduction” feature of the model, where the black hole characteristics for sub-Planckian black holes mimic those in dimensions and the horizon size grows as .

#### 1. Introduction

Black holes are special objects in gravitational physics because they are expected to reveal features of both classical and quantum gravitation. Indeed, one can note this connection in the simple fact that the defining parameter of the quantum gravity scale, the Planck mass , simultaneously sets the strength of classical gravitation . A complete understanding of black hole physics will thus help shed light on this elusive theory. Although large black holes may reveal hints of quantum effects through, for example, the morphology of their shadows [1], it is anticipated that eventual observation of quantum scale black holes formed in high-energy collisions will provide direct evidence. In this regime, these objects transcend classical and quantum gravitation, and thus forming reliable predictions of their physics becomes tenuous in the absence of a complete theory of quantum gravity.

Although such a formulation is still incomplete, the literature is replete with first steps beyond the classical regime and into the quantum realm. Such semiclassical approaches generally rely on a classical framework, extended to include quantum effects at the appropriate energy or length scale that tame or remove the singularity. Examples include noncommutative geometry inspired models ([2]; also see [3] for an overview and additional references therein), the Generalized Uncertainty Principle [4, 5], and asymptotic safety [6]. Other approaches stem from quantum mechanical first principles and introduce gravitation as an energy or potential constraint. The Schrodinger-Newton equation [7, 8] can be derived from the weak-field Einstein’s equations with the stress-energy tensor replaced by the expectation of a quantum operator, from which potential table-top quantum gravity measurements may be possible [9]. General aspects of quantum field theories in curved space-time are also well-known modifications of quantum theories to include gravitation [10–12].

A more recent approach in understanding the nature of quantum black holes is to consider the quantum mechanical conditions for their creation in terms of a wavefunction description. Using a framework known as the “Horizon Wavefunction” (HWF), the black hole is treated as a quantum particle whose spatial wavefunction is contained within its classical horizon radius [13, 14]. If such particles are created in high-energy collisions, then the chance of creating a black hole can be assessed by evaluating the associated probability. Particularly, the HWF has been used to understand aspects of quantum black hole thermodynamics, including evaporation signatures in four-dimensional space-time [15–19], as well as extra- and -dimensional scenarios [20]. Potential experimentally detectable signatures that arise from such a description have been discussed in [21].

A particularly interesting feature of the HWF formalism is the appearance of a Generalized Uncertainty Principle (GUP), in which quantum uncertainties are simultaneously influenced by the wave and gravitational length scales of a particle. Originally noted as a feature of string theory [22], the GUP has been shown to be a largely model-independent prediction of quantum gravity theories, including loop quantum gravity [23], noncommutative quantum mechanics [24], gravity ultraviolet self-completeness [25], and other minimum length scenarios [26, 27]. While most minimal length approaches yield a lower bound to black hole masses, a recently derived GUP-modified Schwarzschild metric was shown to allow the existence of sub-Planckian black holes [4]. A special feature of these sub-Planckian black holes is that their physical and thermodynamic characteristics mimic those of -dimensional black holes—that is, the horizon size varies inversely with the black hole mass, , and the Hawking temperature linear in the mass, .

Since the HWF can predict both the probability of black hole formation for arbitrary masses and also source constraints on a GUP from the quantum mechanical side, we seek to understand how encoding the GUP in the metric will influence the probability of black hole formation. In this paper, we apply the HWF prescription to the GUP-inspired metric of [4]. After first reviewing the formalism for both HWF and the GUP metric in Sections 2 and 3, we derive expressions for the HWF and black hole probabilities in both the super- and sub-Planckian mass regimes for varying GUP model parameters. In the former case, we find the results to be in agreement with those of the Schwarzschild HWF. In the sub-Planckian regime, we show that the probability of a particle of arbitrarily small mass becoming a black hole tends to unity. We discuss the results in the Conclusions.

#### 2. The Horizon Wavefunction Formalism

In order to follow the prescription outlined above, we review the first ingredient of the approach, namely, the HWF framework. The following arguments reproduce the standard approach detailed in, for example, [13] and similar references. We start from the definition of the trapping surface: where is normal to spherical surfaces of area . From this, one can derive the metric function if one assigns coordinates . The quantities and are the Planck mass and length, respectively. Assuming a rough flat space, the Misner-Sharp mass can be calculated as where is the local matter density. The condition for a trapping surface to be formed follows from the constraint that the gravitational radius is for a given value of coordinates . If the source is completely contained within this region, then is identified with the usual Schwarzschild radius. More generally, condition (4) gives a more rigorous representation of the hoop conjecture [28], which allows for the formation of a black hole in the collision of two masses if their impact parameter is contained within the Schwarzschild radius. From the above definitions, this can be reexpressed as the condition where is the total energy in the centre-of-mass frame.

Since such an object would be manifestly quantum mechanical, one must also introduce an uncertainty in its position. This will be on the order of the system’s Compton wavelength, , providing the additional constraint on the gravitational radius This spread in localization can be represented by the wavefunction As usual, the sum over the variable represents the decomposition on the spectrum of the Hamiltonian: Once the energy spectrum is known, we can use (5) to get One can now define the HWF as which can be normalized as Conceptually, the normalized HWF yields the probability for an observer to measure particle in the quantum state and associate with it a horizon of radius . Consequently, the sharply defined classical radius is replaced by the expectation radius of the operator .

The probability for the source to be a black hole is that it lies completely within its horizon: where the density is a combination of requiring the particle to rest within a sphere of radius and the probability that is the gravitational radius. These are, respectively, calculated as

#### 3. Generalized Uncertainty Principle Black Holes

As one approaches the Planck scale, it has been argued [22, 29–31] that the Heisenberg Uncertainty Principle (HUP) should be replaced by a Generalized Uncertainty Principle (GUP) of the form where is a dimensionless constant that depends on the particular model of interest. This introduces a duality in the momentum uncertainty of the form , and, assuming the correspondence , one can determine a similar mass duality to be present in the characteristic length scale of the system.

The Planck scale is generally regarded as the transition point between classical and quantum regimes. Rewriting (15) using the substitution and , one can define both a generalized Compton radius (approaching from the sub-Planckian regime) and a gravitational radius (approaching from the super-Planckian regime) [4]: Note the right-hand side of the inequality in (17) becomes a GUP perturbation to the Schwarzschild radius if . There is no reason that should possess such a generic constraint, however. Thus, the above expression is further generalized to a new parameter , giving The latter expression is the true GUP-modified horizon scale. The crossover of solutions to (16) and (18) can potentially allow for the simultaneous classification of particles as black holes. In fact, the expressions for and coincide when where is the true fundamental mass scale at which the transition between quantum mechanics and classical gravity takes place. Proper choice of and can thus keep particles and black holes separate. In this paper we will not deal any further with or the generalized Compton scale; the interested reader is referred to a more detailed discussion in [4].

Although this characteristic was not derived from a general relativistic approach, one can posit that it should be. That is, the GUP itself is encoded in the space-time geometry and, as argued in [4], a metric description of space-time must incorporate such a mass duality. In the large mass limit , where quantum effects are negligible, one should recover the Schwarzschild solution. In this case, the black hole mass is well-defined in terms of the stress-energy tensor. When , however, the exact meaning of the mass parameter becomes ambiguous, referring to both a particle and a black hole. Since the horizon radius of sub-Planckian mass black holes would be shorter than the Planck length itself, the relativistic description becomes unreliable. Consequently, one takes the object to be a particle of mass , with being the Compton wavelength. This can also be expressed as a variant of the Komar mass: where is the determinant of the spatially induced metric , is a time-like unit vector, is a killing vector describing time-translation symmetry, is the stress-energy tensor, and quantifies the energy density over the length scale .

Since we lack a full quantum theory of gravity, however, the exact form of the stress-energy tensor above is nebulous, and so one can assume the value of to represent some quantum mechanical distribution of matter [4]. Consequently, the full definition of the mass will include both the large scale (e.g., ADM) mass and the short scale particle mass.

Incorporating this new relation for the mass, one arrives at a quantum corrected form of the Schwarzschild metric [4]: In essence, this metric encapsulates all features of the Schwarzschild solution by virtue of the fact that the modification in the mass term is coordinate-independent. Furthermore, natural dimensional reduction features are demonstrated in the gravitational radius and thermodynamics of sub-Planckian objects () that resemble that of ()-D gravity.

Specifically, the horizon is which yields for super-Planckian, Planckian, and sub-Planckian mass black holes. The horizon behaviour as a function of black hole mass is shown in Figure 1.