Holographic screens in ultraviolet self-complete quantum gravity
aa r X i v : . [ h e p - t h ] M a r Holographic screens in ultravioletself-complete quantum gravity
Piero Nicolini ∗ a,b and Euro Spallucci † ca Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, D-60438Frankfurt am Main, Germany b Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany c Dipartimento di Fisica, Sezione di Fisica Teorica, Universit`a degli Studi di Triesteand INFN, Sezione di Trieste, Strada Costiera 11, I-34151 Trieste, Italy
July 5, 2018
In this paper we study the geometry and the thermodynamics of a holo-graphic screen in the framework of the ultraviolet self-complete quantumgravity. To achieve this goal we construct a new static, neutral, non-rotatingblack hole metric, whose outer (event) horizon coincides with the surface ofthe screen. The space-time admits an extremal configuration correspond-ing to the minimal holographic screen and having both mass and radiusequalling the Planck units. We identify this object as the space-time funda-mental building block, whose interior is physically unaccessible and cannotbe probed even during the Hawking evaporation terminal phase. In agree-ment with the holographic principle, relevant processes take place on thescreen surface. The area quantization leads to a discrete mass spectrum. Ananalysis of the entropy shows that the minimal holographic screen can storeonly one byte of information while in the thermodynamic limit the area lawis corrected by a logarithmic term. “Quantum gravity” is the common tag for any attempt to reconcile gravity and quantummechanics. Since the early proposals by Wheeler [1, 2] and deWitt [3], up to the recentultraviolet (UV) self-complete scenario [4], the diverse formulations of a would be quan-tum theory of gravity have shown a common feature, i.e. a fundamental length/energy ∗ E-mail: [email protected] † E-mail: [email protected] GeV or, 10 − TeV. Thevery concept of distance becomes physically meaningless at the Planck scale and space-time “evaporates” into something different, a sort of “foamy” structure, a spin network,a fractal dust, etc. , according with the chosen model [5]. As a matter of fact, one ofthe most powerful frameworks for describing the Planckian phase of gravity is definitely(Super)String Theory. The price to pay to have a perturbatively finite, anomaly-freequantum theory is to give up the very idea of point-like building blocks of matter, andreplace them with one-dimensional vibrating strings. As there does not exist any phys-ical object smaller than a string, there is no physical ways to probe distances smallerthan the length of the string itself. In this regard two properties of fundamental stringsare worth mentioning: • string excitations correspond to different mass and spin “particle” states; • highly excited strings share various physical properties with black holes.Thus, we infer that string theory provides a bridge between particle-like objects andblack holes (see for instance [6]). However, it is important to remark that while theCompton wavelength of a particle-type excitation decreases by increasing the mass, theSchwarzschild radius of a black hole increases with its mass. Thus, the first tenet of highenergy particle physics, which is “higher the energy shorter the distance”, breaks downwhen gravity comes into play and turns a “particle” into a black hole. The above remarkis the foundation of the UV self-complete quantum gravity scenario, where the Planck-ian and sub-Planckian length scales are permanently shielded from observation due tothe production of black hole excitations at Planck energy scattering [7]. Accordinglythe Planck scale assumes the additional meaning of scale at which matter undergoes atransition between its two admissible “phases”, i.e. , the particle phase and the blackhole phase [8, 9, 10]. From this perspective, trans-Planckian physics is dominated bylarger and larger black hole configurations. It follows that only black holes larger, or atmost equal to Planck size objects, can self-consistently fit into this scheme. However,classical black hole solutions do not fulfill this requirement, i.e. the existence of a lowerbound for their mass and size (see Fig. 1).A first attempt to overcome this limitation is offered by the noncommutative geometryinspired solutions of the Einstein equations [11]. The latter are a family of regular blackholes which span all possible combinations of parameters, such mass [12], charge [13],angular momentum [14, 15]. In addition such regular geometries admits a variety of com-plementary gravitational configurations such as traversable wormholes [16], dirty blackholes [17], dilaton gravity black holes [18] and collapsing matter shells [19]. Recently thisfamily of black holes has been recognized as viable solutions of non-local gravity [20, 21], i.e. , a set of theories exhibiting an infinite number of derivative terms of the curvaturescalar [22, 23, 24] in place of the mere Ricci scalar as in the standard Einstein-Hilbertaction. More importantly extensions of noncommutative geometry inspired metrics tothe higher dimensional scenario [25, 26] are currently under scrutiny at the LHC for theirunconventional phenomenology [27]: specifically the terascale black holes described by2 igure 1: The dotted and the solid curves represent the particle Compton wave length λ C andthe Schwarzschild radius as a function of the energy M in Planck units (quantities arerescaled). The squared bullet is the Planck scale. The grey area of the diagram isactually excluded, meaning that a particle cannot be compressed at distances smallerthan the Planck length: at trans-Planckian energy only black hole form. The arrowshows the inadequacy of the Schwarzschild metric: black holes have no lower massbounds, can have size smaller than the Planck length and can expose the curvaturesingularity by decaying through the Hawking process. such regular metrics tend to have a slower evaporation rate [28] and emit only soft par-ticles mainly on the four dimensional brane [29]. A characteristic feature of this typeof solutions is that the minimum size configuration is given by the extremal black holeconfiguration which exists even in the neutral non-spinning case [30, 31, 32]. This factautomatically implies a minimum energy for black hole production in particle collisions[33] without any further need of correcting formulas of cross sections with ad hoc thresh-old functions. Extremal configurations play a crucial role in the physics of the decayingdeSitter universe via the nucleation of microscopic black holes. It has been shown thatPlanck size noncommutative inspired black holes might have been copiously producedduring inflationary epochs [34]. This fact has further phenomenological repercussions:being stable, non-interacting objects, extremal black holes turn out to be a reliable can-didate for dark matter component. On the theoretical side, extremal configurations inthe presence of a negative cosmological term can provide a short scale completion of theHawking-Page diagram which switches to a more realistic Van der Waals phase diagram[35]. 3xtremal configurations can be either descend from the introduction of a fundamentallength in the line element and can alternatively be interpreted as a phenomenologicalinput from quantum gravity: in the latter case it has been shown that such extremalblack holes fit pretty well in the UV self-complete scenario providing a stable, minimumsize, probe at the transition point between particles and black holes [36].In this paper we want to take a step further in the realization of this program by avoidingthe introduction of an additional principle to justify the presence of a minimal length,rather we demand the radius of a Planck size extremal black hole to provide the naturalUV cut-off of a quantum spacetime. In this framework gravity is expected to be self-regular in the sense that the actual regulator cutting off sub-Planckian length scalesis given in terms of the gravitational coupling constant, i.e. √ G = L P . The paper isorganized as follows. In Section II we derive a black hole metric, consistent with theabove discussion and the concept of holographic screen. The latter coincides with theouter horizon of the black hole whose mass spectrum is bounded from below by the massof the extremal configuration equalling the Planck mass. Once trans-Planckian lengthscales are cut-off, the “interior” of the black hole loses its physical meaning in the sensethat all the relevant degrees of freedom are necessarily located on the horizon itself. InSection III we discuss the thermodynamics of the screen. We find that the area law ismodified by logarithmic corrections and that there exists a minimal holographic screenwith zero thermodynamic entropy. Finally we propose an “holographic quantization”scheme where the area of the extremal configuration provides the quantum of surface.In Section IV we offer to the reader a brief summary of the main results of this work. A simple but intriguing model of singularity-free black hole has been “guessed” in [37], inthe sense that the metric was assigned as an in-put for the Einstein equations. Sometimesthis inverted procedure is called “engineering” because the actual source term of fieldequations is not known a priori. The distinctive feature of the solution is the presencein the line element of a free parameter with dimension of a length, acting as a shortdistance regulator for the spacetime curvature, allowing a safe investigation of back-reaction effects of the Hawking radiation. In [38] an higher dimensional extension ofthis model has been proposed; it was also shown that, by a numerical rescaling of theshort-distance regulator, it is possible to identify this fundamental length scale withthe radius of the extremal configuration. With hindsight, we are going to take a stepforward to improve this inverse procedure. Specifically, we want to follow the “ directway ” by building up a consistent source for Einstein equations: we introduce a physicallymotivated energy momentum tensor which allows for transitions between particle-likeobjects and black holes as consistently required by UV self-complete quantum gravity.4e start from the energy density for a point-particle in spherical coordinates as ρ p ( r ) = M πr δ ( r ) , (1)where δ ( r ) is the Dirac delta. The energy distribution (1) implies a black hole for anyvalue of mass M even for sub-planckian values where one expects just particles. Beforeproceeding, we would like to recall that a Dirac delta function can be represented as thederivative of a Heaviside step-function Θ δ ( r ) = ddr Θ ( r ) . (2)Against this background, we want to accommodate both particles and black holes by asuitable modification of the energy distribution in order to overcome the ambiguities ofthe Schwarzschild metric in the sub/trans-Planckian regimes (see also Fig. 1). This canbe done by considering a “ smooth ” function h ( r ) in place of the Heaviside stepΘ ( r ) −→ h ( r ) . (3)The new profile ρ ( r ) of the energy density is defined through h ( r ) by the relation ρ ( r ) = M πr ddr h ( r ) ≡ T . (4)By means of the conservation equation ∇ µ T µν = 0 one can determine the remainingcomponents of the stress tensor, which turns to be out of the form T νµ = diag ( − ρ, p r , p ⊥ , p ⊥ ) (5)The condition for the metric coefficients g = − g − determines the equation of state,namely the relation between the energy density and the radial pressure, p r = − ρ . Theangular pressure is specified by the conservation of the stress tensor and reads p ⊥ = p r + r ∂ r p r .By plugging the tensor (5) in Einstein equations, one finds that the metric reads ( G = 1) ds = − (cid:18) − m ( r ) r (cid:19) dt + (cid:18) − m ( r ) r (cid:19) − dr + r d Ω , (6)with m ( r ) = 4 π Z r dr ′ ( r ′ ) ρ (cid:0) r ′ (cid:1) (7)At large distances r ≫ L P , the above energy density has to quickly vanish, i.e. ρ ( r ) → r & L P , the density ρ ( r ) (and accordingly h ( r )) has to depart from the point-particleprofile in order to fulfil the following requirements:5 igure 2: The plot shows a length/energy relation consistent with the self-complete quantumgravity arguments in Planck units. Particles (dotted line) and black holes (solid line)cannot probe length shorter than the Planck length. The grey area is permanently inac-cessible and accordingly represents the minimal spacetime time region or fundamentalconstituent i.e. the “atom” the spacetime is supposed to be made of. i) no curvature singularity in the origin;ii) self-implementation of a characteristic scale l in the spacetime geometry by meansof the radius of the extremal configuration r , i.e. , r = l .The latter condition is crucial. For instance noncommutative geometry inspired blackholes [11] are derived by the direct way, they enjoy i), but fail to fulfill the conditionii). This means that the characteristic length scale of the system l and the extremalconfiguration radius r are independent quantities. Indeed noncommutative geometry isthe underlying theory which provides the scale l in terms of an “external” parameter,namely the noncommutative parameter θ . In other words one needs to invoke a prin-ciple, like a modification of commutators in quantum mechanics, or the emergence of aquantum gravity induced fundamental length to achieve the regularity of the geometryat short scales. Against this background, we want just to use r as fundamental scale,getting rid of any l as emerging from any theory or principle not included in Einsteinfield equations. This is a step forward since it opens the possibility for Einstein gravityto be self-protected in the ultraviolet regime. To emphasize this point, we introducedthe word “ self -implementation” in ii). Since there exists actually only one additionalscale beyond r , i.e. the Planck length L P = √ G , or the Planck mass M P = 1 / √ G ,6e can implement the condition ii) in the most natural way by setting r = L P andaccordingly M = M P , where M ≡ M ( r ) is the extremal black hole mass.Despite the virtues of the above line of reasoning, we feel that the set of conditions i)and ii) can be relaxed and a further simplification is possible. Having in mind thatfor extremal black hole configurations the Hawking emission stops we just need to finda metric for which only the condition ii) holds. This would be enough for completingthe program of the UV self-complete quantum gravity by protecting the short distancebehavior of gravity during the final stages of the evaporation process. In this regard,the resulting extremal black hole is just the smallest object one can use to probe short-distance physics, In other words, in the framework of UV self-complete quantum gravity, it is not physically meaningful to ask about curvature singularity inside the horizon asthe very concept of spacetime is no longer defined below this length scale .According with such a line of reasoning, we can determine the function h ( r ) by droppingthe condition i) and keeping just the condition ii). Inside the class of all admissibleprofiles for h ( r ), the most natural and algebraically compact choice is given by h ( r ) = 1 − L r + L (8)A similar procedure has been already used in [33] and accounts for the fact that in thepresence of L P the step cannot be any longer sharp. Thus, the smeared energy density ρ ( r ) turns out to be ρ ( r ) = M πr L (cid:0) r + L (cid:1) (9)As a result we find the following metric which is derived from a stress tensor modelinga particle-black hole system (5) ds = − (cid:18) − M L rr + L (cid:19) dt + (cid:18) − M L rr + L (cid:19) − dr + r d Ω , (10)where the arbitrary constant M is defined as follows: M ≡ L r h (cid:0) r + L (cid:1) . (11)We give M the physical meaning of mass for a spherical, holographic screen with radius r h . The basic idea is that gravitational phenomena taking place in three-dimensionalspace can be projected on a two-dimensional “ viewing screen ” with no loss of in-formation [39]. The idea of holographic screen has been proposed in [40] and it hasmathematically been formulated in [41]: the holographic screen plays the role of “ basicconstituent ” of space where the Newton potential is constant.” Along this line of reason-ing, the idea of holographic screen has been used also in the context of noncommutativeinspired metric to derive compelling deviations to Newton’s law [42]. For what concerns7he current discussion, however, we just need to recall that a special case of holographicscreen is given by an event horizon where the entropy is maximized.Several remarks are in order: • It is easy to show that M ≥ M P and equals the Planck mass only for r h = L P . • The line element (10) admits a pair of horizons provided M ≥ M P . The radii r ± of the horizons are given by r ± = L (cid:18) M ± q M − M (cid:19) (12)For M = M P the two horizons merge into a single (degenerate) null surface at r ± = r = L P . For M ≫ M P the outer horizon approaches the conventional valueof the Schwarzschild geometry, i.e. , r + ≃ M L . • By inserting (11) into (12) one finds r + = r h , r − = L r h . We see that the holo-graphic screen surface coincides with the (outer) black hole horizon r + , while theinner Cauchy horizon has a radius which is always smaller or equal to the Plancklength. This fact lets us circumvent the issue of potential blue shift instabili-ties [43, 44] (see for instance recent analyses for noncommutative inspired [45, 46]and other quantum gravity corrected metrics [47, 48]) because r − simply loses itsphysical meaning being not accessible to any sort of measurement process. In whatfollows we can identify the holographic screen with the black hole outer horizonwithout distinguishing between the two surfaces any longer. • “Light” objects, with M < M P , are “particles” rather than holographic screens.By particles we mean localized lumps of energy of linear size given by the Comptonwavelength λ C = 1 /M , that can never collapse into a black hole. Rather they giverise to horizon-less metrics (see Fig. 2) and cannot probe distances smaller than λ C . The “transition” particle −→ black holes is discussed below in terms of critical surface density .As a further analysis of this result, it is interesting to consider the surface energy density of the holographic screen which is defined as σ h ≡ M πr = 18 πL r + L r . (13)From the above relation we see that σ h is a monotonically decreasing function of thescreen radius We notice that there exists a minimal screen encoding the physically max-imum attainable energy density, i.e. the Planck (surface) density: σ h ( r + = L P ) = 14 πL = M P πL . (14)We stress that there is no physically meaningful “ interior ” for the minimal screen, i.e. the “volume” of such an object is not even defined, in the sense that it can never be8robed. Thus, we can only consider energy per unit area, rather than per unit volume.If we, formally, define a surface energy for a particle as σ p ≡ M πλ C = 14 πλ C (15)we see that the two curves (13) and (15) cross at λ C = L P = r + . This result offersan additional interpretation for the Planck length which consistently turns to be theminimal size for a particle as well for a black hole (see Fig. 2). Accordingly, the Planckdensity (14) is the critical density for a particle to collapse into a black hole. Thisargument is usually formulated in terms of volume energy density having in mind thepicture of macroscopic body gravitationally collapsing under their own weight. Fromour holographic vantage point, where “surfaces” are the basic dynamical objects, itis natural to reformulate this reasoning in terms of areal densities [39]. In additionholography offers a way to circumvent potential conflicts between the mechanism ofspontaneous dimensional reduction [49, 50] and the UV self complete paradigm. If weperform the limit for r → ds −→ − ( 1 − M r ) dt + ( 1 − M r ) − dr + O (cid:0) r /L (cid:1) . (16)As explained in [51], this mechanism would lead the formation of lower dimensional blackholes for length scales below the Planck length, in contrast with the predicted semi-classical regime of trans-Planckian black holes in four dimensions. However, contrary tothe Schwarzschild metric that eventually reduces into dilaton gravity black holes when r ≃ L P (for reviews of the mechanism see [52, 53]), the presence of the holographicscreen forbids the access to length scales r < L P and safely protects the arguments atthe basis of the UV self complete quantum gravity. In this section we would like to investigate the thermodynamics of the black hole de-scribed by (10) and determine the relation between entropy and area of the event horizon.It is customary to consider the area law for granted in any case, but this assumptionleads to an inconsistency with the Third Law of thermodynamics: extremal black holeshave zero temperature but non vanishing area. Here, we stick to the textbook definitionof thermodynamical entropy and not to more exotic quantity like R´enyi, or entanglemententropy. To cure this flaw, we shall derive the relation between entropy and area fromthe First Law, rather than assuming it. The Hawking temperature associated to themetric (10) can be calculated by evaluating the surface gravity κT H = κ π = 14 π (cid:18) dg dr (cid:19) r = r + = 14 πr + (cid:18) − L r + L (cid:19) (17)9 igure 3: The solid curve represents the Hawking temperature T H and as a function of thehorizon radius r + in Planck units. The dotted curve represents the correspondingclassical result in terms of the Schwarzschild metric. while the heat capacity C ≡ ∂U/∂T H is C ≡ ∂M∂T H = − πr + (cid:18) r − L L (cid:19) ( r + L ) r − L r − L . (18)One can check that for large distances, i.e. , r + ≫ L P both (17) and (18) coincide withthe conventional results of the Schwarzschild metric, i.e. , T H ≈ πr + and C ≈ − πr (see Fig. 3 and Fig. 4). On the other hand at Planckian scales, contrary to the standardresult for which a Planckian black hole has a temperature T H = M P / π , we have that T H −→ r + → r = L P as expected for any extremal configurations. This discrepancywith the classical picture is consistent with the genuine quantum gravitational characterof the black hole and is reminiscent of the modified thermodynamics of noncommutativeinspired black holes [54, 55].The Hawking emission is a semi-classical decay where gravity is considered just in termsof a classical spacetime background. Such a semi-classical approximation conventionallybreaks down as the Planck scale is approached. On the other hand for our metric, at r + = r M = p √ L P ≃ . L P the temperature admits a maximum corresponding to apole in the heat capacity. In the final stage of the evaporation, i.e. L P < r + < r M , theheat capacity is positive, the Hawking emission slows down and switches off at r + = L P .From a numerical estimate of the maximum temperature one finds T H ( r M ) = 0 . M P .10 igure 4: The solid curve represents the black hole heat capacity C as a function of the horizonradius r + in Planck units. The dotted curve represents the corresponding classicalresults in terms of the Schwarzschild metric. This implies that the ratio temperature/mass is T H /M < T H ( r M ) /M ≃ . r + ≥ L P .We can summarize the process with the following scheme: • “ large ” , far-from-extremality, black holes are semi-classical objects which radiatesthermally; • “ small ” , quasi-extremal, black holes are quantum objects; • r = r M is “ critical point ” where the heat capacity diverges (see Fig. 4). Since C > r < r + < r M and C < r M < r + , we conclude that a phasetransition takes place from large thermodynamically unstable black holes to smallstable black holes.As a matter of fact, the black hole emission preceding the evaporation switching off(often called “SCRAM phase” [11]) might not be thermal. It has been argued that sucha quantum regime might be characterized by discrete jumps towards the ground state[7, 56]. To clarify the nature of this mechanism we proceed by studying the black holeentropy profile and the related area quantization. By integrating the First Law, taking11nto account that no black hole can have a radius smaller than r = L P , i.e. , S ( r + ) = Z r + r dMT H = πL (cid:0) r − L (cid:1) + 2 π ln (cid:18) r + L P (cid:19) . (19)We can cast the entropy in terms of the area of the event horizon A + ≡ π r as S ( A + ) = π A ( A + − A ) + π ln ( A + / A ) (20)where A = 4 πL is the area of the extremal event horizon. We remark that themodifications to the Schwarzschild metric, encoded in our model, are in agreement withall the major approaches to quantum gravity, which universally foresee a logarithmicterm as a correction to the classical area law. For brevity we recall that this is thecase for string theory [57, 58], loop quantum gravity [59, 60, 61] and other results basedon generic arguments [62, 63], on the Cardy’s formula [64], conformal properties ofspacetimes [65] and other mechanism for counting microstates [66, 67, 68]. We cancheck that this is the case for the metric (10) by performing the limit r + ≫ L P for (20)to obtain S ( A + ) ≈ A + L + π ln (cid:18) A + πL (cid:19) . (21)Conversely for r + → L P the entropy vanishes, i.e. , S ( A + ) ≈ πL P ( r + − L P ) + O (cid:16) ( r + − L P ) (cid:17) . (22)This result is consistent both with the Third Law of thermodynamics and the entropystatistical meaning. The Planck size, zero temperature, black hole configuration is theunique ground state for holographic screens. Thus, it is a zero entropy state as thereis only one way to realize this configuration. To see this we promote the extremalconfiguration area to the fundamental quantum of area. A + ≡ A n − = n A = 4 πnL , (23)where L represents the basic information pixel and n = 1 , , . . . is the number ofbytes. From the above condition one obtains r n − ≡ n / L P , (24) M n − ≡ (cid:16) n / + n − / (cid:17) M P . (25) We borrow here the names of some units of digital information. In the present context, each byteconsists of 4 π bits. Each bit, represented by L is the basic capacity of information of the holographicscreen. In the analogy with the theory of information for which a byte represents the minimumamount of bits for encoding a single character of text, here the byte represents the minimum numberof basic pixel L for encoding the smallest holographic screen. r = L P and M = M P , while for n ≫ M n ≡ M n − M n − ∼ n − / M P . (26)We notice that for n ≤ C > n >
C < i.e. , ∆ M n /M n ≤ / r + = r M , the system undergoes a phase transition from a semi-classical regime to a genuine quantum gravity regime. As a conclusion we have thatlarge black holes decay thermally, while small objects decay quantum mechanically, byemitting quanta of energy (for a recent phenomenological analysis of such kind of decaysee [69]). The end-point of the decay is a Planck mass, holographic screen.The quantization of the area of the holographic screen lets us disclose further featuresof the informational content of the holographic screen. We have that the surface densitycan be written as σ h ( n ) = 12 (cid:18) n / + 1 n / (cid:19) M P πL (27)while the entropy reads S ( n ) = π ( n + ln( n ) − n of bytes, the surface density decreases.This confirms that the extremal configuration is nothing but a single byte, zero-entropy,Planckian density holographic screen. In this paper we have presented a neutral non-spinning black hole geometry admitting anextremal configuration whose mass and radius coincide with the Planck units. We havereached this goal by suitably modelling a stress tensor able to accommodate both theparticle and black hole configurations, undergoing a transition at the Planck scale. Weshowed that the horizon of the degenerate black hole represents the minimal holographicscreen, within which we cannot access to any information about the matter-energy con-tent of spacetime.We showed that a generic holographic screen is described in terms of the outer horizon ofthe metric (10), while the inner horizon lies within the prohibited region, i.e. , inside theminimal holographic screen. The whole scheme fits into the gravity self-completenessscenario. For sub-Planckian energy scales one has just a quantum particle able to probeat the most distances of the order of its Compton wavelength. By increasing the degreeof compression of the particle, one traverses the Planck scale where a collapse into ablack hole occurs, before probing a semi-classical regime at trans-Planckian energies.The virtual curvature singularity of the geometry in r = 0 is therefore wiped out sincein such a context sub-Planckian lengths have no physical meaning. From this vantage13oint spacetime stops to exists beyond the Planck scale as there is no physical way toaccess this regime. Thus, the curvature singularity problem is ultimately resolved bygiving up the very concept of spacetime at sub-Planckian length scales.The study of the associated thermodynamic quantities confirmed that at trans-Planckianenergies black holes radiate thermally before undergoing a phase transition to smaller,quantum black holes. The latter decay by emitting a discrete spectrum of quanta ofenergy and reach the ground state of the evaporation corresponding to the minimalholographic screen. We came to this conclusion by quantizing the black hole horizonarea in terms of the minimal holographic screen which actually plays the role of a basicinformation byte. We showed that in the thermodynamic limit, the area law for theblack hole entropy acquires a logarithmic correction in agreement with all the majorquantum gravity formulations.In conclusion, we stress that the line element (10) not only captures the basic features ofmore “ sophisticated ” models of quantum gravity improved black holes ( e.g. noncom-mutative geometry inspired black holes [11], loop quantum gravity black holes [70, 71],asymptotically safe gravity black holes [72, 73] and other studies about collapses inquantum gravity [74, 75]), but overcomes some of their current weak points: specificallythere is no longer any concern for potential Cauchy instabilities or for conflicts betweenthe gravity self-completeness and the Planck scale spontaneous dimensional reductionmechanism, as well as, the scenario of the terminal phase of the evaporation for static,non-rotating, neutral black holes. In addition, for its compact form the new metricallows straightforward analytic calculations and opens the route to testable predictions. References [1] J. A. Wheeler, Annals Phys. , 604 (1957).[2] J. A. Wheeler, Int. J. Mod. Phys. A , 4013 (1993).[3] B. S. DeWitt, Phys. Rev. , 1113 (1967).[4] G. Dvali and C. Gomez, “ Self-Completeness of Einstein Gravity ,” arXiv:1005.3497[hep-th].[5] L. J. Garay, Int. J. Mod. Phys. A , 145 (1995).[6] G. T. Horowitz and J. Polchinski, Phys. Rev. D , 6189 (1997)[7] G. Dvali, S. Folkerts and C. Germani, Phys. Rev. D , 024039 (2011).[8] G. Dvali, G. F. Giudice, C. Gomez and A. Kehagias, JHEP , 108 (2011).[9] G. Dvali and D. Pirtskhalava, Phys. Lett. B , 78 (2011).[10] G. Dvali and C. Gomez, JCAP , 015 (2012).1411] P. Nicolini, Int. J. Mod. Phys. A , 1229 (2009).[12] P. Nicolini, J. Phys. A A , L631 (2005).[13] S. Ansoldi, P. Nicolini, A. Smailagic, E. Spallucci, Phys. Lett. B645 , 261-266 (2007).[14] A. Smailagic, E. Spallucci, Phys. Lett.
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