Black Hole Production in the Presence of a Maximal Momentum in Horizon Wave Function Formalism
aa r X i v : . [ phy s i c s . g e n - p h ] A p r Black Hole Production in the Presence of a MaximalMomentum in Horizon Wave Function Formalism
S. Saghafi a, ∗ , K. Nozari a,b, † , A. D. Kamali c, ‡ a Department of Physics, Faculty of Basic Sciences, University of Mazandaran,P. O. Box 47416-95447, Babolsar, IRAN b Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),P. O. Box 55134-441, Maragha, Iran c Department of Physics, Science and Research Branch,Islamic Azad University, Tehran, Iran
Abstract
We study the Horizon Wave Function (HWF) description of a generalized uncertaintyprinciple (GUP) black hole in the presence of two natural cutoffs as a minimal length anda maximal momentum. This is motivated by a metric which allows the existence of sub-Planckian black holes, where the black hole mass m is replaced by M = m (cid:16) β M pl m − β M pl m (cid:17) .Considering a wave-packet with a Gaussian profile, we evaluate the HWF and the probabilitythat the source might be a (quantum) black hole. By decreasing the free parameter thegeneral form of probability distribution, P BH , is preserved , but this resulted in reducing theprobability for the particle to be a black hole accordingly. The probability for the particleto be a black hole grows when the mass is increasing slowly for larger positive β , and fora minimum mass value it reaches to 0. In effect, for larger β the magnitude of M and r H increases, matching with our intuition that either the particle ought to be more localizedor more massive to be a black hole. The scenario undergoes a change for some values of β significantly, where there is a minimum in P BH , so this expresses that every particle canhave some probability of decaying to a black hole. In addition, for sufficiently large β wefind that every particle could be fundamentally a quantum black hole. PACS : 04.70.-s, 04.70.Dy
Key Words : Quantum Gravity, Generalized Uncertainty Principle, Black Holes, HorizonWave Function ∗ s.saghafi@stu.umz.ac.ir † [email protected] ‡ [email protected] Introduction
Our present understanding of the Universe relies on two theories: the theory of General Relativ-ity which is applicable at very large scales, and Quantum Mechanics which provides a very gooddescription of the microscopic universe. While these two theories work very well in their ownregimes, the regime in which they collide is near the Planck scale, when quantum mechanicalwave-packets can turn into Black Holes. Another interesting problem is the description of thegravitational collapse which again leads to the formation of black holes (firstly investigated inthe seminal papers of Oppenheimer et al. [1,2]). A lot of work has been done on this subject,but a good description of the physics of such processes is still challenging. A vast amount ofresearch has been done on this subject (see, e.g. Ref. [3]), but many conceptual problems remainunsolved, such as accounting for the quantum mechanical nature of collapsing matter.Until now, the only unanimously accepted idea is that gravitation becomes important when-ever a large enough amount of matter is compacted within a sufficiently small volume. K. Thornewas the first one who formulated this idea in the hoop conjecture [4], which remarks that a BHforms when two objects which collide together, collapse within their black disk. presumingthe final structure is (approximately) spherically symmetric, this happens when the collapsingobjects occupy a sphere with radius r and this radius is smaller than the Schwarzschild radius, r ≤ R H ≡ ℓ pl Em pl (1)where the Planck length is denoted by ℓ pl (defined as ℓ pl = q G ¯ hc ) and m pl represents the Planckmass.One can find many attempts at quantizing BH metrics in the literature, which focus onthe purely gravitational degrees of freedom, and result in a description of the horizon whichis unrelated to the matter state that sourced the geometry to begin with [5]. The approachdiscussed in this paper, is one of the approaches in comprehending the essence of quantum blackholes by considering the quantum mechanical conditions for their creation according to theirhorizon wave function. In this formalism which is called the horizon wave function (HWF)approach, one can suppose that the black hole is a quantum mechanical particle, so a spatialwave function can be attributed to this black hole which is placed in its classical event horizon[6,7]. If these particles can be created in high energy collisions, so the probability for creating ablack hole can be assessed by the corresponding collisions probability. Furthermore, the HWFhas been studied for understanding the quantum nature of black hole thermodynamics [6-8].Also in [7], HWF has been applied to the lower and higher dimensional models and the end2f black hole evaporation in a lower dimensional model is studied in [9]. The most appealingfeature of the HWF approach is the aspect of generalized uncertainty principle in which thewave and particle gravitational length scales will affect the quantum uncertainties. In fact itis shown that as a property of string theory, generalized uncertainty principle (GUP) is one ofthe proposed outcomes of the final quantum gravity theory which essentially has no dependenceon models. That is, although the functional form of GUP (and therefore modified dispersionrelations) are model dependent, existence and the very nature of these uncertainties are modelindependent. These theories are loop quantum gravity [10], non commutative geometry [11] andother minimal length scale scenarios such as the doubly special relativity. Even though most ofminimal length scale approaches consider a minimal limit for black hole mass, recently in Ref.[12] the authors found a new generalized uncertainty principle that modifies the Schwarzschildmetric and admits the existence of sub-Planckian black holes. In this paper we consider the effectof adding an additional term to this GUP because of the existence of maximal momentum. Thisextra feature is a result of the doubly special relativity theories [13]. As the HWF approach canpredict the probability of black hole formation for arbitrary masses and also the origin of thishorizon wave function should be limited due to the uncertainty principle, we are looking for theinformation that how the probability of black hole formation can be affected by the generalizeduncertainty principle in the black hole metric. We focus also on the role of maximal momentumon the probability distribution, P BH . When one approaches very small length scales (towards the Planck length), the standard Heisen-berg’s uncertainty relation gets modified to, for instance, the following generalized uncertaintyprinciple [14,15,16] ∆ x ∆ p ≥ ¯ h h − β ∆ p + β (∆ p ) i which gives ∆ x ≥ ¯ h h p − β + β ∆ p i . (2)In this relation β = β ℓ pl ¯ h and β is a dimensionless constant parameter. This GUP admits twonatural cutoffs as a minimal length and a maximal momentum and can cause a similar relationfor momentum and mass in the domain of characteristic length of the system. In Ref. [12]authors discussed that the black hole uncertainty principle correspondence propose that blackholes with mass smaller than the Planck scale but radius of order the Compton scale insteadof the Schwarzschild scale, could exist. They introduce a modified self-dual Schwarzschild likemetric that though in the large mass limit it remains Schwarzschild , reproduces acceptable3spects of a variety of several models in sub-Planckian limit. They considered a GeneralizedUncertainty Principle which admits the existence of a minimal length. In this paper we alsoconsidered the effect of maximal momentum. The main task of this case is that the generalizeduncertainty principle is situated in the very notion of the geometry of spacetime, so the metricwhich is used for describing this spacetime must have a specific relation for mass. In the limitof large masses, M > M pl , where quantum effects are insignificant, the Schwarzschild solutioncan be recovered. In this case the black hole mass can be defined in terms of energy-momentumtensor. When M < M pl , the exact meaning of mass parameter becomes ambiguous [12]. Onecan consider both a black hole and particle. As the size of horizon of such sub-Planckian blackholes is smaller than the Planck length, relativistic description becomes deficient. Hence, in thiscase there would be a particle of mass M ∼ ¯ hλ c where λ c is the Compton wavelength. Even thismass can be represented as a kind of Komar mass as M ≡ Z Σ d x γ n µ κ ν T µν ∼ = − π Z c dr r T , (3)in which γ is the deformed spatial determinant of γ ij , T µν is energy-momentum tensor and T measures the energy density on the length scale λ c . In the absence of a full quantum gravitytheory, the exact form of energy-momentum tensor is ambiguous, but one can consider that T shows the quantum mechanical distribution of matter [12]. So the exact definition of massboth contains the large scale mass (ADM) and also the mass of particles in small scale. As in[12] the dual role of M has been inspired in the GUP , now we can suggest that the Arnowitt-Deser-Misner (ADM) mass which coincides with the Komar energy in the stationary case, shouldbe M ADM = M (cid:16) β M pl M − β M pl M (cid:17) (4)On the basis of the above discussion and according to what is done in [12], the quantumdeformed form of the Schwarzschild metric is as follows ds = F ( r ) dt − F − ( r ) dr − r d Ω , (5)where F ( r ) ≡ − m pl Mr (cid:16) β m pl M − β m pl M (cid:17) . (6)In fact, this metric enfolds all the characteristics of the Schwarzschild spacetime as this modifiedterm is a coordinate-independent relation. In this metric the event horizon can be written as r H = M + ( M + βm pl ) M m pl (7)4hich leads to M ≫ m pl −→ r H ≃ Mm pl , (8) M = m pl −→ r H ≃ m pl (1 + β ) m pl , (9) M ≪ m pl −→ r H ≃ βM , (10)for black holes which have the mass of super-Planckian, Planckian and sub-Planckian respec-tively. Figure 1 shows the behavior of event horizon as a function of mass for different values of β .In Ref. [12] it is shown that although the singularity is not removed, it can never be reached.This can be proved by inverting the relation (7) according to two masses relating to the givenhorizon which leads to a minimal horizon radius as r min = 2 βm pl , (11)and also a minimum mass as M min = βm pl . (12)Figure 1: Event horizon as a function of mass.
As it was mentioned in Refs. [6,7], the horizon is a classical concept in general relativity and itcannot be defined clearly when the source is not described by classical physics. One can considera horizon wave function for any localized quantum mechanical wave-function that can lead tocompute the probability of finding a horizon of a given radius centralized around the source.5o a probability can be associated to each quantum particle that is a black hole and also tothe existence of minimum black hole mass. This agrees with the results of the hoop conjecture[17] and the Heisenberg Uncertainty Principle: the black hole can form whenever the impactparameter b of two objects which are colliding is shorter than the Schwarzschild radius of thesystem, it means: b ≤ ℓ pl Em pl ≡ r H (13)where E is total energy in the centre of mass frame. This conjecture has been checkedin a variety of situations. For understanding a classical horizon in a spherically symmetricspace-time,a general spherically symmetric metric g µν can be written as ds = g ij dx i dx j + r ( x i )( dθ + sin θdφ ) (14)By considering this classical metric, we can find out exactly the location of a trapping horizon,a surface where the escape velocity is equal to the speed of light, as [18] :0 = g ij ∇ i r ∇ j r = 1 − ℓ pl ( mm pl ) r . (15)where ∇ i r is defined as the covector perpendicular to surfaces of constant area A = 4 πr .The function M = ℓ pl mm pl is the active gravitational (or Misner-Sharp) mass, which shows the totalenergy which is surrounded by a sphere of radius r . It is obvious that in general following thedynamics of a given matter distribution and verifying the existence of surfaces satisfying Eq.15 isso complex, but an horizon can be found if there exists values of r such that r H = 2 M ( t, r ) > r ,which is reexpressing of the hoop conjecture mathematically (13).According to what is done in [6], now we consider a point-like mass m which is also a spin-less particle, and it’s Schwarzschild radius is given by r H in Eq. (13) with E = m . For such aparticle, the Heisenberg principle of quantum mechanics introduces an uncertainty in its spatiallocalisation, typically of the order of the Compton-de Broglie length λ m ≃ ℓ pl m pl m . (16)As quantum physics is a more processed description of reality, the conflict of the two lengths, r H and λ m , points that it only makes sense if: r H ≥ λ m ⇒ m ≥ m pl . (17)Now the spread in localization can be represented by the wavefunction6 Ψ S i = X E C ( E ) | Ψ E i , (18)As usual, the sum over the variable E represents the decomposition on the spectrum of theHamiltonian, ˆ H | Ψ E i = E | Ψ E i (19)Once the energy spectrum is known, we can use (13) to get E = m pl r H ℓp (20)Now the HWF can be defined as Ψ H ( r H ) = C ( m pl r H ℓ pl ) (21)and can be normalized as h ψ H | φ H i = 4 π Z ∞ ψ ∗ H ( r H ) φ H ( r H ) r H dr H . (22)The concept of normalized HWF | ψ H i is that it is the probability that an observer measures theparticle at the quantum state | ψ S i and associates it to an event horizon with the radius r = r H .As a result, the defined classical horizon is replaced by the expectation value of operator r H .The probability of the gravitational source to be a black hole is that it should be located totallyin its horizon P BH = Z ∞ P < ( r < r H ) dr H , (23)in which the probability density P < ( r < r H ) = P S ( r < r H ) P H ( r H ) , (24)is a combination of the probability that a particle be at rest in a sphere with the radius r = r H and also the probability that r H is gravitational radius. These quantities can be calculated as P S ( r < r H ) = Z r H P S ( r ) dr = 4 π Z r H | ψ S ( r ) | r dr , (25) P H ( r H ) = 4 πr H | ψ H ( r H ) | . (26)According to what is done in Refs. [7,19,20,21,22], a massive particle at rest in the referenceframe can be characterized by the following Gaussian profile ψ S ( r ) = e − r ℓ ( ℓ √ π ) . (27)7ow with the inspiration by twofold role of mass m in the generalized uncertainty principle, onecan investigate the existence of sub-Planckian black holes meaning that they are both particleand black hole. In this framework the usual m is replaced by the modified M of the generalizeduncertainty principle as M = m (cid:16) β m p m − β m p m (cid:17) . (28)In the next step we consider a case that the length ℓ is related to the uncertainty in the sizeof the particle. It can be equal to the Compton length as ℓ = λ m ≃ ℓ p m p M . (29)Remembering that these analysis are appropriate for independent ℓ and m , this case is re-lated to the maximum localization for the gravitational source as one expects, ℓ ≥ λ m .By taking the Fourier Transform, the corresponding wavefunction in momentum space gives˜ ψ S ( p ) = e − p (∆ √ π ) (30)Assuming the relativistic mass-shell equation in flat space-time to account for high-energyparticle collisions, we relate the momentum p to the total energy EE = p + M (31)From the Schwarzschild relation (7) and by maintaining the normalization according to (22),we derive the HWF as ψ H ( r H ) ≃ Θ( r H − r min ) exp " − ℓ ℓ p (cid:16) r H − r min (cid:17) , (32)in which r H = Mm p and the Heaviside step function results in from of the fact that E ≥ M .Now the density probability can be explicitly computed as follows P < = ℓ r H e − ℓ r H √ π Erf (cid:18) r H ℓ (cid:19) − r H e − r Hℓ √ πℓ (33)and 8 < = 2Erf (cid:18) m (cid:16) β m − βm (cid:17) (cid:19) e √ πm (cid:16) β m − βm (cid:17) − π exp − m β m − βm ! − , (34)from which by integrating the density from r H to infinity, the probability (23) for a particle tobe a black hole is obtained as P BH ( ℓ ) = 2 π " arctan (cid:18) ℓ (cid:19) − ℓ ( ℓ − ℓ + 4) . (35)By writing P BH as a function of m , this relation can be rephrased as follows P BH ( m ) = 2 π arctan m β m − βm ! + 16 − m (cid:16) β m − βm (cid:17) πm (cid:16) β m − βm (cid:17) − m (cid:16) β m − βm (cid:17) . (36)Figure 2: Probability density as a function of Gaussian width ℓ for some values of β . Figures (2) and (3) show the probability densities as a function of the size and mass ofthe particle for different values of β , that is, the probability for the particle to be inside itsown horizon. Also the probability that the particle is a black hole is plotted as a functionof mass m in Fig. (4). This scenario has some limitation because of the cutting mass andas it was discussed previously, this cutting mass is related to the minimum radius r min whichis M rmin = βm P . The masses which are smaller than this cutoff are just used in numericalcalculation. It should be mentioned that for very small β the probability that a particle to be ablack hole exactly resembles the standard Schwarzschild metric [7]. This result is in agreement9igure 3: Probability density as function of mass for some values of β . Figure 4:
Probability that a particle to be a black hole versus its mass for some values of β . β = 0 , M → m and the horizon radius equals to theclassical Schwarzschild radius.On the other hand, for some values of β there would be a minimum in their probability accordingto the Fig. 4; it means that for any amount of particle’s mass there would be a certain probabilityto be a black hole (when β = 1 this case is more obvious). Also, for enough large β , everythingcan be a black hole, in accommodation with the fact that the GUP introduced here does’nt exertany minimal mass directly. So the effect of increasing the free parameter β would be gatheringlarger masses in a larger horizon radius. Therefore, it is more probable that a particle to bea black hole. An important property in this regard is that m ≥ M pl will result in P BH ≃ − β m p m in equation (18), reduces the value of M for positive β . This reduction of M leads to a less probability of forming black hole. So, maximal momentum cutoff reduces theblack hole production probability in essence. In this paper the probability of black hole formation in a gravitational field and within thehorizon wave function approach is studied in the presence of natural cutoffs as a minimal lengthand maximal momentum. Our focus is mainly on the role of maximal momentum in this setup.For this goal, the gravitational source is considered to be immersed in a modified metric, themodification of which are coming from the GUP consisting a minimal measurable length and amaximal momentum. By considering a wave packet with a Gaussian profile, the correspondinghorizon wave function is computed and the probability that the gravitational source is a blackhole (a quantum black hole), i.e. situating in its horizon radius, is derived. We treated the prob-lem by some numerical analysis on the probability as a function of the mass for different valuesof β . First of all we observed that when increasing the positive free parameter β , a minimum in P BH would occur which means that essentially every particle can collapse into a black hole inthis situation. More ever, for large β , every particle is essentially a quantum black hole becauseby increasing β , M and R H would become larger and due to this fact it would be more probablefor a particle to lie in its event horizon and therefore to be a black hole. An important resultof inclusion of maximal momentum cutoff in our setup is that for positive beta, the resultingmodified mass, M , reduces (by the factor − β m p m ) which reduces probability of a particle to bea quantum black hole in this framework. It should be mentioned that even though we restrictthe sub-Planckian black holes to a mass-cutoff, but the exact essence of black holes especially inextremely small masses depends on which theory of quantum gravity would be authentic finally.11 cknowledgement The work of K. Nozari has been supported financially by Research Institute for Astronomy andAstrophysics of Maragha (RIAAM) under research project number 1/6025-00.
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