Fundamentals of Planckian physics

Transcription

1 Fundamentals of Planckian physics Antonio Aurilia 1 Department of Physics and Astronomy, California State Polytechnic University, Pomona, USA Euro Spallucci 2 Dipartimento di Fisica, Gruppo Teorico, Università di Trieste, and INFN, Sezione di Trieste, Italy Abstract We wish to draw attention to a chain of physical identities that lie at the core of Planckian physics but also have important ramifications in quantum and classical physics. First in the chain is a somewhat surprising property of black holes, namely: the linear energy density of Schwarzschild black holes, unlike the conventional volume density, has the universal value: F s = c 4 /4G N, regardless of their mass and size; second, and equally surprising, this new black hole constant F s coincides with the Planckian limit of both Coulomb s electrostatic force and Newton s gravitational force, that is, it represents the unification point of the static electro-gravitational force at the Planck scale of energy where Planck mass and Planck charge become interchangeable; third, F s is identical with the value of the maximal tension conjectured by Gibbons in the framework of General Relativity, regardless of Planckian effects. We often refer to this maximal linear energy density, or tension as superforce, or Planck force, in view of the identities noted above and here we discuss some of the far reaching consequences of this superforce. Among them: i) the existence of an unsurpassable concentration of energy at the core of singularity-free black holes, ii) the vanishing of quantum effects beyond the threshold of Planckian physics, an idea also known as classicalization of trans-planckian physics and, iii) the Planckian saturation of a Lorentz boost and the concomitant threshold energy for the production of micro black holes in the so called T ev quantum gravity scenario. 1 address: aaurilia@csupomona.edu 2 address: spallucci@ts.infn.it Preprint submitted to Elsevier Preprint 13 May 2013

2 1 Introduction and background The purpose of this paper is to discuss some little-known aspects of ultrarelativistic physics, down to a scale of distance that is comparable with the Planck length. At this scale of distance, it is generally agreed that quantum gravitational effects come into play so that the structure of the spacetime manifold is anything but Lorentzian. As a matter of fact, in the old days the Planck world was envisaged as the arena of violent quantum gravity fluctuations that disrupt the very fabric of space and time [1] so that the notion of Lorentz-symmetry becomes meaningless. Eventually, the notion of spacetime foam evolved into a Planckian phase with a different description according to String/M-Theory, Loop Quantum gravity, Non-Commutative geometry, Fractal space-time, etc. The implicit common denominator of such radically new ideas is the set of fundamental units that define the domain of quantum gravity, and it is here that we begin our discussion, primarily for the essential purpose of fixing the normalization factors of those units and thus provide a quantitative framework for what follows. The appropriate standards of length, mass and time were originally introduced by Max Planck on a purely dimensional basis by combining the speed of light c, the Gravitational coupling constant G 3 and the Planck constant h. In other words, Planck recognized that it is possible to combine Special Relativity, Quantum Mechanics and Gravity in the following universal dimensional package, L P hg hc, M c 3 P (1) G Planck Time is simply the unit of length divided by c. Clearly, this dimensional approach defines the Planck units up to a numerical factor providing only an orders of magnitude estimate of Planckian physics. In order to determine the numerical constants in (1) some extra argument is due. For instance, one may declare that the Planck Mass is defined by the equality between the quantum mechanical wavelength of a particle and its gravitational critical radius: h M c = 2M G c 2 (2) 3 G can be either the Newton constant, or the higher-dimension gravitational coupling of T ev quantum gravity. 2

3 Thus, L = 2 hg hc, M c 3 = (3) 2G Interestingly enough, an alternative, but consistent definition of (3), is the following: L is the geometric mean of the quantum mechanical wavelength λ C = h/mc of a particle and its critical gravitational radius R s = 2mG /c 2 L 2 hgn λ C R s = (4) c 3 Further insight into the physical meaning of L can be obtained from the Generalized Uncertainty Principle (GUP)[2,3], where L is often identified with the string length, i.e., L = α x h p + L2 4 p h (5) By minimizing the uncertainties, one finds p = 2 h, L (6) x = L (7) From equation (7) we see that L represents the minimal uncertainty in the particle/string localization. From this point of view, L is the minimal length which is physically meaningful since, for a shorter one, the uncertainty is larger than the length itself. In contrast to this, it seems worth observing that the Planck mass is neither an absolute minimum nor an absolute maximum. It is, rather, an extremal value, or turning point, in the sense that, as implied by its definition (2), it represents the largest mass that an elementary particle may possess, or the smallest mass attributable to a micro-black hole [4]. Once the Planck units are precisely defined, as Planck himself recognized, one may assign a universal Planckian value to every other physical quantity and then consider the remarkable possibility that all of physics is limited, in the sense that all physical quantities are bounded from above, or below, or at least have an extremal value expressible in terms of Planck units. As a matter of fact, these are the central properties of the Planck units that we are going to exploit in the following discussion. Thus, in the next section we argue that there exists in nature a universal, unsurpassable linear energy density that lies at the core of every Schwarzschild black hole, regardless of their mass or size. 3

4 This surprising new property of black holes, in turn, has several far reaching consequences, especially in the context of Planckian physics. Among them, in order of discussion: The existence of a maximal tension in nature, a suggestion first made by Gibbons in the framework of General Relativity [5]. We often refer to this maximal tension as superforce, or Planck force in order to underscore its manifest connection with the unified, static electro-gravitational force at the Planck scale of energy. The existence of singularity-free black holes, an idea originally proposed and developed by several researchers, including one of the authors (E.S.) in the context of non-commutative geometry [7 13,30]. The saturation of a Lorentz boost by the formation of black holes. This is perhaps the most direct consequence of the existence of a minimal length and a maximal tension; the existence of a terminal boost signals the end of the familiar Lorentzian, length-contracting phase and the onset of a trans-planckian length-expanding phase. During this transition, quantum effects seem to vanish and physics is dominated by classical fieldconfigurations as the would be trans-planckian region is shielded by the creation of black holes whose linear dimension, i.e., the gravitational radius, increases with energy. With the above classicalization of quantum gravity in mind, we comment on the remarkable identity between three seemingly independent universal quantities, namely, a) the string tension, b) the linear energy density of all black holes, and c) the superforce, i.e., the presumed unification point of all fundamental forces. The intriguing possibility of detecting quantum gravity signals at LHC if the Planck scale is lowered down to the T ev scale as contemplated in some string-inspired unified models with large extra-dimensions. 2 Is there a limiting force in Nature? 2.1 The linear energy density of a black hole is a universal constant To go directly to the point: consider the conventional volume density of a body. In the case of a black hole this leads to the well known, somewhat counterintuitive result that the density is inversely proportional to the square of the mass ρ BH = M 4πR 3 s/3 = 3c6 32πG 3 N 1 M 2 (8) 4

5 so that, while mini black holes may possess a nuclear density, galactic black holes can be less dense than water. On the other hand, by considering the linear energy density, the dependence on the mass disappears and one obtains the remarkable expression ρ = Mc2 2R s = c4 4G N (9) which represents a limiting, universal characteristic of all black holes. 4 At first sight this simple result may seem disconnected from our present narrative about physics at the Planck scale. In particular, a) the qualifier limiting requires an explanation, and b) we note that the expression (9) is purely classical and has no apparent connection with the Planck units; even the numerical factor 1/4 seems fortuitous. However, it is readily verified that the expression (9) holds true for a Planckian black hole by substituting the expressions (3) for the mass and the Schwarzschild radius. This is a legitimate substitution since the gravitational radius associated with the Planck mass is the Planck length itself. Thus, the expression c4 4G N represents the Planckian value of the linear energy density. Dimensionally, however, this is the same as a force or tension and this observation, in turn, leads us to another elementary, but rather instructive calculation. 2.2 A glimpse of super-unification at the core of singularity-free black holes With the Planck units at our disposal, it is clear that we can assign a universal Planckian value to every physical quantity we care to consider. We 4 The result may be extended to a large class of black holes for which the mass M is related to the radius of the event horizon, ( which for the sake of simplicity we continue to call R s even in the case of more general black hole geometries ) through a relation of the form M = R sc 2 2G N [ 1 + H ( R s ) ]. (10) Here, the function H ( R s ) usually contains the dependence on the electric/magnetic charge, angular momentum, cosmological constant. By considering the linear energy density, the dependence on the various parameters once again disappears and one obtains the same remarkable expression ρ = Mc2 2R s H ( R s ) = c4 4G N (11) which represents the same universal characteristic of all Schwarzschild black holes. 5

6 saw this in the case of the linear energy density of a body. Another important example discussed in the following is that of Planck charge and the associated limiting electrostatic force. More specifically, in the following we argue that there is a converging Planckian limit, say, a superforce F s, of the two long-range forces of Nature. Consider first the Coulomb force, in e.s.u. F C = q2 r 2 (12) In the Planckian limit, this fundamental law becomes F C = q2 L 2 (13) where q is the Planck charge normalized by the condition that the Coulomb and Newton forces coincide at the Planck scale of distance, or energy q 2 L 2 = G M 2 L 2 q 2 = hc 2 (14) Inserting L from (3) into (13) one finds again the remarkable classical expression, F C = c4 4G = F N F s (15) in spite of the explicit presence of h in the expression of Planck s units. Thus, we conclude that the Planckian value of the linear energy density discussed in the previous subsection can be interpreted as the expression of the unified, static electro-gravitational force, or maximal tension that exists at the core of every black hole. Interestingly enough, some time ago Gibbons suggested to incorporate a Principle of Maximal Tension in General Relativity and gave arguments, different from ours, in support of the expression (15), including the numerical factor 1/4 [5]. Apart from that numerical factor, the expression (15) coincides, as surmised earlier, with the expression of the Planck force obtained on purely dimensional grounds using the Planck units (1). Note, also, that Eq. (14) implies that mass and charge are interchangeable at the Planck scale of energy as one would expect in a super-unified theory of fundamental interactions. Quantum effects, however, seem to have vanished from the expression of the super-force and one must account for that. In the concluding section of this paper, we provide our own interpretation of the physical meaning of F s on the basis of the duality between fundamental strings and black holes. 6

7 2.3 Singularity-free black holes One concise way to summarize the chain of arguments presented in the previous sections is this: the core of a black hole is a hard, incompressible, core in the sense that, because of quantum gravitational effects, it cannot be crushed into a singular point; rather, it is characterized by an irreducible linear extension and an unsurpassable linear energy density barrier. In other words, the core of a black hole is the domain of the Planck force that prevents its implosion into a singular point by making it impossible to increase the concentration of energy/unit length beyond the limiting barrier of c4. 4G We can think of at least two important consequences of this upper bound in the linear energy density of a body: i) the existence of singularity- free, or regular Schwarzschild solutions in General Relativity, and ii) the existence of a critical boost in ultra-relativistic particle physics that marks the transition from the quantum, or Lorentzian particle-phase characterized by the conventional Lorentz-Fitzgerald contraction, to the trans-planckian phase characterized by the dilation of physical length. On this inversion, or transition from the contracting phase to an expanding phase, and the concomitant absence of quantum effects displayed in the very expression of the Planck force, or Gibbons maximal tension, rests the notion of classicalization of trans-planckian physics that is the central idea of the so called UV self-complete theory of quantum gravity([15 20]) The issue of a critical, or terminal Lorentz boost is considered in some detail in the next section. 3 Quantum gravity modification of the Lorentz-Fitzgerald lengthcontraction rule and classicalization of trans-planckian physics From a phenomenological point of view, this section is especially important as we discuss the likely impact that the Planckian universal constants, i.e., minimal length, maximal tension, may have on the physics of elementary particles. A preliminary account of the scenario described here was reported by the authors, many years ago, in an unpublished essay [34]. Thus, to go directly to the point of interest, recall that high-energy particle physics is based on the notion that smaller and smaller distance scales can be investigated by increasing the energy of the probe particle. Thus, in a collision experiment, elementary particles can resolve distances comparable with their quantum mechanical wavelength. The more is the energy, the shorter is the wavelength in agreement with the relativistic rule of length contraction. Quantum mechanics and Special Relativity work together to open a window on the microscopic world. Gravity plays no role here, as is commonly accepted, at least up to distance scales that are directly accessible by current particle 7

8 physics experiments. This is the Lorentzian phase in elementary particle physics where the quantum de Broglie wavelength is the only characteristic length that sets the observable scale of distance. This simple picture becomes less clear when we imagine to approach the Planck scale of distance, or energy, and consider the concomitant quantum gravity effects discussed in the previous section, most notably, the production of Planckian black holes in high energy scattering experiments characterized by an irreducible gravitational radius, namely, the Planck length, and a maximal tension, namely, the unified Planck force. This problem has long been ignored on the basis that the Planck world is altogether so outlandish that no particle accelerator will ever be able to probe it. However, the picture is completely different, as is now widely accepted, when we consider the string-inspired unified models with large extra-dimensions, where the unification scale can be as low as some T ev. In this kind of scenario, quantum gravity effects, including micro black hole production in partonic hard scattering, have been suggested to occur near the LHC peak energy, i.e., 14 T ev [21 28], [29,30]. In this new physics, presumably characterized by quantum-gravitational fluctuations, the distinction between point-like elementary particles and extended quantum gravity excitations, whatever they may be, i.e., black holes, D-branes, string balls, etc., turns out to be fuzzy: recall, for instance, that according to the fundamental equality (2) the Planck mass is the largest mass that an elementary particle may possess, or, the smallest mass attributable to a black hole. This remarkable turnaround, from elementary particles to black holes, is visualized in the adjoining figure (1) [34] l l P m P m Fig. 1. Plotted in the figure is the (qualitative) mass dependence of the quantum Compton wavelength of a point particle (the hyperbolic curve) as opposed to the linearly increasing classical radius of a black hole. 8

9 Interestingly enough, the intersection point corresponds to the geometric mean of the two characteristic lengths, Eq.( 4), and signals the onset of the Planck scale of mass and length and the concomitant classicalization of trans-planckian physics. Put differently, if the above reasoning is correct, then some conventional notions of particle physics based on Special Relativity and Quantum Mechanics, require a quantum-gravitational revision. For instance, Heisenberg s Uncertainty Relation must be extended in the form of the Generalized Uncertainty Principle (GUP) as noted in the Introduction, Eq.(5). Then, one may argue that a similar extension is required for the Lorentz-Fitzgerald length contraction rule derived in Special Relativity. The specific reason, in this case, may be stated as follows: The size of a (micro) black hole, i.e., its critical gravitational radius, cannot be smaller than the Planck length and, if the graph in figure (1) is any guide, then the length contraction effect is reversed when the Planck threshold is reached. In a nutshell, the quantum of length L [31,32] is a new universal constant on the same footing as c and h, and as such it must be observer independent. It follows that L must act as an unbreakable barrier for the special relativistic length-contraction rule. How can we account for that? 3.1 Terminal boost and the onset of gravitational effects Before we tackle the question posed at the end of the previous section, we must emphasize that what we are seeking here is not a modification of Special Relativity itself as advocated in some multi scale versions of the theory based, for example, on mathematical deformations of the Lorentz group. To the extent that we are considering some new aspects of Planckian physics, it is implied that Special Relativity ends and Quantum Gravity begins. Somehow, however, this switch-over must be reflected in a modified form of the Lorentz-Fitzgerald formula, at least as a clue concerning a fundamental aspect of Quantum Gravity. Thus, following a time-honored procedure that Planck himself adopted when he introduced a quantum of action in physics, we are looking for an empirical formula that, through a new quantum of length, signals a structural change in the fabric of spacetime, namely, a change from the familiar Lorentzian phase characterized by flat spacetime where lengths contract, to a trans-planckian phase, where lengths expand with increasing energy. We propose to account for this transition by redefining the relationship between quantum wavelength and energy-momentum of a particle in the presence of a short-distance Planck barrier. This new relationship is the crux of the following discussion. 9

10 In Special Relativity a rod of length L 0 in its rest frame is seen to be contracted in the direction of motion according to the Lorentz-Fitzgerald rule L ( β ) = L 0 1 β 2, β v/c (16) An immediate consequence of (16) is that L can contract to an arbitrarily small length as β 1. Usually, the story of Special Relativity ends here. However, pushing this contraction rule to its physical and mathematical limit, one may argue that the Lorentz-Fitzgerald length-contraction ends at a singular point, not unlike the end-point of the gravitational collapse of a massive body in General Relativity. More specifically, there are at least two types of objections to the vanishing limit of the Lorentz-Fitzgerald length contraction rule: Quantum objection, or, the absence of h: even though equation (16) is routinely applied to the world of particle physics, it was conceived with macroscopic, i.e. classical, objects in mind. Stated otherwise, the quantum of action h seemingly has no effect in the length-contraction rule, but we expect this to change in the ultra-relativistic regime of particle physics. Gravitational objection, or, the absence of G N : equation (16) ignores the fact that any physical object produces its own gravitational field, and thus introduces a critical gravitational length scale, that is, the Schwarzschild radius R s = 2MG N /c 2.However, according to the so-called hoop-conjecture, any physical object which can be encircled along any direction by a circular hoop of radius less than its Schwarzschild radius, collapses into a black hole [33]. By combining the two arguments, both extraneous to Special Relativity, one concludes that Quantum Mechanics plus Gravity impose intrinsic limits to the relativistic contraction of physical objects. Thus, we are left with the empirical question: is there any way to modify the law (16) on a quantum gravitational basis? Any such modification ought to contain both Newton and Planck constant, G N and h, and reproduce (16) when G N or h is switched-off. Let us consider the first argument. As the nature of spacetime at the Planck scale is still under debate we take a phenomenological approach and consider a common feature of various models, which is the presence of a minimal length, say L, characterizing the transition from (semi)classical spacetime to a new quantum state. If L represents an observer-independent quantity we need to introduce a modified contraction law which we conjecture to be as 10

11 follows, L ( β ) = L 0 1 β 2 + L2 Θ H ( β ) 4L 0 1 β 2 (17) where, Θ H, is the Heaviside step function which guarantees that deviation from the classical form of the Lorentz contraction law does not affect the measure of L at rest 5. Moreover, since any macroscopic length is tens of orders of magnitude larger than L, the second term in (17) gives a relevant contribution only in the ultra-relativistic regime β 1. The minimum of the function L ( β ) is d L dβ = 0 γ = 2L 0 L, L ( β ) = L (18) For γ > γ the function L ( β ) bounces back and the expanding phase begins. A similar behavior is shown by a fundamental string which cannot shrink below its minimal length l s = α. Providing more and more energy, higher and higher vibration modes are excited forcing the string to elongate. Thus, a natural choice for L is L = l s = α. In this connection, it may be worth to recall that a highly excited string looks very much like a black hole. With this identification, (18) says that in any inertial reference frame no physical length can be smaller that the string length : L ( β ) α (19) and the critical boost representing the turning point between contraction and dilatation turns out to be γ = 2L 0 / α. 3.2 A couple of remarkable examples Let us take a closer look at (17). 5 We define Θ H (x) as Θ H (x) = 1 x > 0, Θ H (x) = 0 x 0, Sometimes, it is conventionally chosen Θ H (0) 1/2. In this case, a β-independent quantity L 2 /8L 0 must be subtracted in (17) 11

12 Take for L 0 the Compton wavelength λ C = 1/m of a particle. In analogy with the string improved GUP, we find the following modified de Broglie formula λ ( β ) = λ C 1 β 2 + α Θ H ( β ) 4λ C 1 β 2 (20) As λ ( β ) cannot be smaller than α it follows that the mass spectrum of an elementary particle is bounded from above by the limiting mass λ α m 1 α (21) Next, let us consider the case of a Schwarzschild black hole, L 0 = R s. Equation (17) tells us how the horizon radius will appear from a Lorentz boosted frame R s ( β ) = 2MG N 1 β 2 + α Θ H ( β ) 8MG N 1 β 2 (22) The first term shows how the Schwarzschild radius of a moving mass appears contracted as any other physical length. The second term in (22) takes into account the existence of a hard core, that is, a universal, unsurpassable linear energy density (tension) that prevents further contraction, or collapse into a point singularity. The critical boost is γ = 4MG N α (23) For γ = γ the Schwarzschild radius reaches its minimal value R H ( β ) = α. A snapshot of a black hole at this minimal size will show an object with an effective mass M defined as R H ( β ) 2M G N M = α 2G N (24) In a string theoretical formulation of quantum gravity, the Regge slope is related to Newton s constant: α = 2G N. Thus, we find M = 1/ 2G N = M P and R H ( β ) = L P. 6 At this point we have: 6 It seems worth mentioning that regular black holes admit an extremal configuration representing the lowest mass state of the system [7 12]. These objects have the smallest radius for the event horizon which equals few times the minimal length. In some simple models, it is possible to choose the free length scale regularizing the short distance behavior in such a way that the radius of the extremal configuration is exactly the Planck length [17,29,20]. Without introducing the modified Lorentz law the very idea of a minimal size object would become observer-dependent. 12

13 (1) equation (17) for a (semi)classical length L 0 with string corrections; (2) equation (20) for the debroglie wave length with string corrections; (3) equation (22) for the Schwarzschild radius with string corrections. With the above results in hands, it is time to consider the hoop conjecture and check the self-consistency of our formula. Let us begin with case (1). Can a boosted object be seen contracted below its Schwarzschild radius? If so, the hoop conjecture would imply that a physical object may be turned into a black hole by a mere coordinate transformation between two inertial reference frames. L 0 1 β 2 + α Θ H ( β ) 4L 0 1 β 2 R s 1 β 2 + α Θ H ( β ) 4R s 1 β 2 (25) We can read this relation either as an equation for the radius L 0 below which the object is shielded by an horizon, or as the equation that defines the terminal speed β that, once surpassed, will make the object to appear inside its own Schwarzschild hoop. In the first case, it is immediate to recognize that L 0 R s is the β-independent, trivial solution. In order to be seen as a black object the maximal length at rest must be smaller than the Schwarzschild radius R s. On the other hand, if one assumes that L 0 > R s and tries to determine β from (25), one finds β 2 = 1 + α 4R s L 0 > 1 (26) which is unphysical. Only in a super-luminal inertial frame would a macroscopic object be seen as a black hole. In other words, as one might expect, even in the presence of quantum corrections there is no transformation of coordinates between inertial frames such that a classical object with linear size L 0 > R s appears to be a black hole. Let us consider now a quantum particle, rather than a classical object. λ C 1 β 2 + α Θ H ( β ) 4λ R C 1 β 2 s 1 β 2 + α Θ H ( β ) 4R s 1 β 2 (27) Once again, the terminal boost is unphysical, i.e. β > 1, and the only acceptable solution is λ C = R s m = 1 α = M P (28) Thus, there does not exist any inertial frame where an isolated elementary particle with (invariant) mass m < M P looks like a black hole. However, this 13

14 conclusion does not prevent the production of a black hole in the final state of a two-body high energy scattering experiment [35]. This different case will be discussed in the next section. 3.3 High energy collisions and black hole production We are now ready to extend (20) to the case of a two-body system of colliding partons in the framework of higher dimensional quantum gravity. In this case, the gravitational coupling constant is G with dimensions (in natural units) [ G ] = l d 1, much below the Planck energy, and d is number of space-like dimensions ( 3 ). If the two partons have four-momenta p 1 and p 2, it is useful to introduce the Mandelstam variable s = ( p 1 + p 2 ) 2. In terms of s we can define the effective Schwarzschild radius of the system as r H ( s ) = ( 2 ) 1/(d 2) ( ) s G s L d 1 1/(d 2) (29) where L is the higher dimensional minimal length. The hoop-conjecture states that whenever the two partons collide with an impact parameter b r H ( s ), then a micro-black hole is produced. In our approach we can rephrase this statement as follows: the two-parton system will collapse into a black hole if the debroglie wavelength (20) is smaller, or equal, to the Schwarzschild radius (22) 1 + L2 s 4 ( ) s s L d 1 1/(d 2) L ( ) s L d 1 1/(d 2) (30) where we have switched to natural units, h = c = 1 and no step-function is needed as the two particles are by definition in a relative state of motion. Solving for s we find the threshold invariant energy for the creation of a micro black hole. This is a necessary, but not sufficient condition for this event to occur. As it can be expected, the production channel opens up once the quantum gravity energy scale is reached s 1 L = M (31) Equation (31) tells us that in a high energy scattering experiment we can probe distances down to L but not beyond. The would be trans-planckian region is shielded by the creation of a black hole with linear dimension increasing with s. This argument is the essence of a recent proposal by Dvali end co-workers [15,16] to explain how quantum gravity can self-regularize in 14

15 the deep ultraviolet region [17]. 7 Thus, the trans-planckian regime is actually inaccessible, and the deep UV region is dominated by large, classical field configurations. This mechanism has been dubbed classicalization. [18,19]. There is a second important consequence of the relation (30) regarding the final stage of black hole evaporation. Micro black holes are known to be semiclassically unstable because of Hawking radiation. However, the standard description of thermal decay breaks down when the black hole approaches the full quantum gravity regime. Even worse, no semi-classical model can foresee the end-point of the process which remains open to largely unsubstantiated speculations. Equation (30) on the other hand, shows not only the transition from ordinary particle to black hole, but the inverse process as well. Start from the black hole region and decrease the invariant mass of the object. Effective models of quantum gravity-improved black holes suggest that for M >> M the semi-classical model is correct and the particle emission is to a good degree of approximation a grey-body thermal radiation. However, as M M and the size of the black hole becomes comparable with L, the mass of the object reveals a discrete spectrum and the decay process goes on through the emission of few quanta while jumping quantum mechanically towards the ground state. In this late stage of decay the black hole behaves like a hadronic resonance, or an unstable nucleus, rather than a hot body. Thus, it is not surprising that after crossing the critical point M = M one is left with an ordinary elementary particle [36]. 4 Conclusions To our mind, an important conceptual result that emerges from our survey of Planckian physics is this: Once the Planck units are precisely defined, that is, including the numerical factors, one may assign a limiting, universal Planckian value to every other physical quantity. From here arises, in the first instance, the intriguing notion of universal boundedness of physics in the sense that all physical quantities may be bounded from above, or below, or at least have an extremal value expressible in terms of Planck units. We have illustrated this notion in the case of the Planckian value of length, mass, charge and force and have uncovered the existence of a new black hole constant, namely, the linear energy density, or maximal tension, and identified it with the limiting Planckian value of the unified static electro-gravitational force. Whether or not this value is also the super-unification point of all fundamental forces 7 The scenario of UV self-complete quantum gravity is especially attractive when realized in the more general framework of T ev quantum gravity. In this case, the Planck scale is lowered down to the T ev scale and opens the exciting possibility to detect quantum gravity signals at LHC. 15

16 of nature remains an open question. Be that as it may, we have argued that the existence of these Planckian universal constants demands a revision of some well-established notions in classical, as well as quantum physics. One such revision may be summed up as singularity-free black holes because the existence of an irreducible length and a maximal linear energy density within every black hole implies the existence of a hard, incompressible core that prevents the gravitational collapse into a singular point. Remarkably, this idea, which lies at the heart of General Relativity, has some rippling effects for Special Relativity as well, the link being provided by the so called hoop conjecture. More specifically, turning to ultra-relativistic high-energy physics, by which we mean elementary particle physics at the Planck scale of energy, ( possibly the T ev quantum gravity length scale ) we have proposed an empirical, but consistent framework to reconcile the notions of minimal length and maximal tension with the Lorentz-Fitzgerald length-contraction rule expected from Special Relativity. In agreement with the hoop conjecture, the presence of an ultimate length barrier has been related, once again, to the presence of a black hole barrier that arises as soon as the invariant mass of a particle is equal to the Planck mass, the greatest that an elementary particle can attain before collapsing into a Planckian black hole. We have noted that this transition from particle to black hole cannot be induced by boosting an isolated particle from one inertial frame to another, but is likely to happen in a high-energy scattering experiment. In any case, this argument leads us to confront, once again, the Planck-Gibbons limiting force F s, Eq.(15). With hindsight, the glaring absence of h from that expression seems to support the classicalization idea discussed in the previous section. On the other hand, the appearance of G N in (15) points to the pivotal role that gravity plays in the unification of fundamental forces: It seems clear, now, that this central aspect of Planckian physics, namely the existence of a classical superforce, can be traced back to the very definition of Planck units discussed in the Introduction, especially the definition (4) and the fundamental equality (2). That equality, on the one hand, sets the mass/energy scale of Planckian physics where one would expect the onset of Quantum Gravity; on the other hand, it tells us, effectively and concisely, that at that same energy scale there is a trade-off between a quantum length (Compton) and a classical one (Schwarzschild.) This trade-off, in turn, signals the end of quantum, special relativistic effects and the onset of a gravitational phase through the formation of singularity-free black holes characterized by an extended, incompressible core the size of the corresponding Schwarzschild radius. This is the domain of the superforce: it represents the ultimate linear energy density of all Schwarzschild black holes. This identity, while obvious on dimensional grounds, may seem surprising on physical grounds. In actual fact, the physical explanation rests on the duality between deep UV and far IR domains in quantum gravity. In this perspective, it seems natural to identify ρ with the energy density of a relativistic string and, therefore, we identify the super-force Eq.(11) with the universal string 16

17 tension ρ hc 2πα T s. (32) Therefore, there are two equivalent ways of writing ρ : classical, macroscopic form given by Gibbons Maximal Tension (11); quantum, microscopic form which is the String Tension (32) The two definitions are linked through: i) the existence of a universal linear energy density for black holes exposing their stringy nature. [38]; ii) the classicalization mechanism of quantum gravity that identifies trans- Planckian black holes with classical, macroscopic objects. On this basis, it seems to be a unique property of gravity to bridge the gap between micro and macro worlds. [37] Finally, in these concluding remarks about Planckian physics, we cannot fail to take stock of the fact that the most promising candidate for a self-consistent fundamental theory of quantum gravitational phenomena is Super-String Theory. The string length there, α, may be identified with the Planck Length L P cm as both play the same fundamental role in a quantum theory of gravity. On the other hand, there are significant aspects of Planckian physics addressed in this article that, to our knowledge, lie completely outside the purview of String Theory. Such is the case for the new black hole universal constant, or the remarkably simple expression of the unified electrogravitational force. Similarly, to our knowledge String Theory says nothing about the role of black holes in the ultra-relativistic limit of the Lorentz- Fitzgerald length-contraction rule and how to extend it into the Quantum Gravity regime. In sum, if String Theory is truly a fundamental theory of Quantum Gravity, and we believe it is, then it is for string theorists to consider and perhaps incorporate into the theory, or derive from it, if at all possible, the more phenomenological aspects that emerge from our survey of Planckian physics. References [1] J. A. Wheeler, Ann. Phys. (NY) 2, 604 (1957); J. A. Wheeler and K. Ford, Geons, black holes, and quantum foam: A life in physics, New York, USA: Norton (1998) 380 p [2] A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52, 1108 (1995) 17

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