aa r X i v : . [ phy s i c s . g e n - ph ] D ec Modified Planck units Yu.L.Bolotin , , V.V.Yanovsky
January 5, 2017 A.I.Akhiezer Institute for Theoretical Physics, National Science Cen-ter Kharkov Institute of Physics and Technology, NAS Ukraine, Akademich-eskaya Str. 1, 61108 Kharkov, Ukraine Institute for Single Crystals, NAS Ukraine, Nauky Ave. 60, Kharkov31001, Ukraine V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv,61022, Ukraine
Abstract
Planck units are natural physical scales of mass, length and time,built with the help of the fundamental constants ~ , c, G . The func-tional role of the constants used for the construction of Planck unitsis different. If the first two of them represent the limits of the actionand the speed of light and underlie quantum mechanics and specialrelativity, the Newton’s constant G ”only” fixes the absolute value ofthe gravitational forces. It seems natural to make a set of fundamentalconstants more consistent and more effective if used to build Planckunits only limit values. To this end, in addition to the limit values ~ and c we introduce an additional limit value - a maximum powerin nature. On the basis of these values, a modification of the Planckunit system is proposed. The proposed modification leaves unchangedthe numerical values of Planck units, however, opens up exciting newpossibilities for interpreting the known results and for obtaining newones. Introduction
Introduction of the Planck system of units [1] prior to creation of quantummechanics as well as of special and general relativity theory, was a significantevent in physics. The system of ”natural measurement units”, as Planckcalled it, put forward the questions which were significantly ahead of the stateof physics at that time. To answer these questions, there had to be achieveda considerable progress in the development of physics. Now, more than acentury later, we are only groping for approaches to the raised problems, andthe final understanding seems to be achieved solely after creation of quantumtheory of gravitation. The systems of units based on different fundamentalconstants not only reflect the history of the development of physics, but alsopermit to estimate the prospects of this development. Thereat, the Planckunits still give an opportunity to look into the future.Various problems bound up with the Planck system of units are consideredin a great number of papers (see e.g. [2, 3, 4, 5, 6]) in the field of elementaryparticle physics and cosmology. Among such works one should mention thosedevoted to attempts to build alternative sets of fundamental scales whichdiffer both in the choice of constants and in their number [7, 8, 9, 10].In the present paper, a new modification of the Planck system of unitsis proposed. While considering the fundamental constants which belong tothis system of units it should be emphasized that the functional role of theconstants ~ , c and G is different. The constants ~ and c are limit valuesand underlie quantum mechanics ( ~ ) and special relativity ( c ), whereas theNewton constant G ”merely” fixes the interaction strength. It seems naturalto make the set of fundamental constants more consistent and more effectivefor subsequent analysis of physics on the Planck scale, if to replace the grav-itation constant G by a new limit value associated with general relativity. Inthis case the Planck units will be expressed only in terms of the limit valueswhich underlie quantum mechanics ( ~ ), Special Relativity ( c ) and GeneralRelativity (maximum force or maximum power). At such transition the nu-merical values of the Planck units remain unchanged, however there appearnew interesting opportunities for interpretation of well-known results and forobtaining new ones.In this paper we discuss advantages of the proposed modification and givea number of examples to illustrate its effectiveness2 Modification of Planck system of units
Dimensional analysis is a powerful method which makes it possible to obtainresults (both qualitative and quantitative) on the base of general knowledgeof the phenomenon under consideration Dimensional analysis (along withsymmetry considerations) is especially significant for construction of initialapproaches to description of those systems for which any theory is absentat present. As is well-known, many astonishing results have been achievedowing to dimensional considerations which at first sight seem to be quitesimple. There are even such results are still not obtained by other, morerigorous way. A classic illustration of such a situation is quantum gravity.The latter has practically become a synonym of Planck-scale physics whichdescription to a considerable extent reduces to endless shuffle of fundamentalconstants. However, dimensional analysis is not all-powerful, and the resultsobtained with its help should be interpreted carefully.Especially significant role in clarification/understanding of the founda-tions of the future theory Planck-scale processes belongs to the same nameunits. The Planck units are fundamental physical scales of mass, length andtime built by means of the fundamental constants ~ , c, G : m P l = q ~ cG ≃ . × − kg,l P l = q ~ Gc ≃ . × − m,t P l = q ~ Gc ≃ . × − secWhen presenting new universal constants in his report for Prussian Academyof Sciences in Berlin (1899), Planck noted: ”These units preserve their nat-ural value till the law of gravity as well as both laws of thermodynamicshold true, and till the speed of light propagation in the vacuum remains in-variable. Therefore, though measured by a variety of intellectuals and usingdifferent methods, they will always have the same value”.It should be noted that the choice of ~ , c, G in the capacity of initial con-stants necessary for construction of fundamental scales, is not unique. As aninitial material, there may be used various combinations of these constants.One can give many examples of the cases when the use of one or anothercombination of fundamental constants essentially simplifies the theory. One besides ~ , c, G , Planck considered the Boltzmann constant to be a universal one, too
3f typical examples is the fine-structure constant α = e / ~ c . It is difficult toimagine quantum electrodynamics in which this combination is not used.Our choice of the starting constants is based on the following considera-tions. It should be reminded that the statement concerning the existence oflimit values may be a foundation of physical axiomatics. As is well-known,quantum mechanics can be constructed proceeding from the existence of theminimum quantum of action ~ , whereas Special Relativity implies the exis-tence of the maximum speed c . Not long ago it has become clear that ananalogous approach may be also realized in General Relativity if to postulatethe existence of a maximum force [11]. The proof based on the Jacobsonsargumentation [12] shows that the Einstein equations can be derived fromthermodynamic identities on the Rindler horizon. Gibbons [13] postulatedthe existence of limit (maximum) force in the principle: I suggest that classi-cal General Relativity in four spacetime dimensions incorporates a Principalof Maximal Tension and give arguments to show that the value of the maxi-mal tension is F max = c G ≈ . × N (1)The limit does not depend on the nature of forces and holds for gravita-tional, electromagnetic, nuclear and any other forces. Naturally, this value isindependent of the choice of reference system (inertial or non-inertial), andmay be interpreted for our World as an additional (on a par with ~ and c )fundamental limit constant.Alternatively, it is possible to use as a basic principle equivalent statementas a basic principle: there is a maximum power, P max = η = c G ≈ . × W (2)Both values are the components of the 4-vector F λ = dp λ dt . The limit poweradmits a trivial physical interpretation. Let us consider the power releasedin the process of ”annihilation” of a black hole with the mass M . Theminimum duration of this process is the time t = 2 R g /c = 4 M G/c , where R g = 2 M G/c is the gravitational radius. The power released in such aprocess is expressed as P = M c M G/c = c G = η (3)4he multiplier 1 / A ≈ B corresponds to therelation lg A ≈ log B .The mechanism of the occurrence of the limit force (power) is extremelysimple [13]. Let us consider two bodies with the masses M and M , thedistance between which being R . According to the Newtons theory, the forceacting between them are expressed as F = G M M R = (cid:18) GM c R (cid:19) (cid:18) GM c R (cid:19) c G (4)Since M M ≤
14 ( M + M ) , then F ≤ (cid:20) ( M + M ) Gc R (cid:21) c G (5)Approaching of the bodies is limited by the condition R > R g (cid:0) R g = MGc (cid:1) which prevents the formation of a black hole with the mass M + M . There-fore, F ≤ c G (6)Surfaces on which implemented the maximum force (maximum pulse flow),or the maximum power (maximum energy flow) are horizons. Any attemptto exceed the force limit leads to the appearance of horizon. In its turn, thelatter does not permit to exceed the limit.Here is another example to clarify the mechanism of the occurrence ofthe maximum force. In the Newtonian mechanics F = dp/dt , therefore F max = (∆ p ) max (∆ t ) min ≈ mct P l = mc l P l (7)At first sight one may expect that unlimited growth of the mass will give riseto an arbitrarily great force. However, this is not so, and the limitation isbound up with the appearance of horizon at the increase of the mass on afixed scale of length ( l P l ). Indeed, when omitting the numerical multipliers5(1) we find the mass with the gravitational radius equal to the Plancklength: m ≈ l P l c G = r ~ Gc c G = r ~ cG = m P l (8)Consequently, the maximum mass which can be used in (7) for preventingthe appearance of the horizon is the Planck mass, so F max ≈ m P l c l P l = c G (9)It is surprising that the result (9) can be obtained in the form of the combi-nation of the Planck units with the dimension of force: F P l = m P l l P l t P l = r ~ cG r ~ Gc c ~ G = c G (10)Note that for the Planck mass the gravitational radius coincides with theCompton wavelength. It should be emphasized that all our statements con-cern solely D = N + 1 = 4. It is only in the space D = 4 that the Planckforce is independent of ~ : F P l ( D ) = M P l ( D ) L P l ( D ) T P l ( D ) = G − D − D ~ D − c D +4 D − (11)More strictly the expression for the limit force (limit power) can be obtainedin the frames of GR [11].The force on a test mass m at a radial distance d from a Schwarzschildblack hole is F = G M md q − GMc d (12)At first sight it seems that this relation contains a singularity and makes itpossible to achieve an arbitrarily large value of force. However, if to discardnon-physical point-like masses, it turns out that the limit value of force exists.Indeed, any mass cannot reduce its size down to the gravitational radius.Otherwise it will be transformed into a black hole, and it will be unobservable.The minimum distance is the sum of the gravitational radii of these masses d ≥ d min = mGc + MGc = G ( m + M ) c . Having substituted the minimum distanceinto (12) we obtain F = c G M m ( M + m ) q − MM + m ≤ c G (13)6ow consider the use of the limit power for construction of modified Planckunits. To this end we replace the gravitational constant by the limit power, G = c η . In other words, by using the set ( ~ , c, η ) instead of ( ~ , c, G ) we getthe modified system of Planck units (which consists only of the limit values): m P l = r ~ ηc ≃ . × − kg,l P l = s ~ η c ≃ . × − m,t P l = p ~ /η ≃ . × − sec (14)It is interesting to note that the Planck time is defined only by two of the in-troduced fundamental constants. That is, by changing c and G concordantlyin such a way that η preserves its value one can preserve the value of thePlanck time, herewith changing the values of the Planck mass and length.Naturally, such a possibility, though less conspicuous, also existed for theoriginal the Planck choice of fundamental constants.It should be emphasized again that the necessary condition for the exis-tence of a event horizon is finiteness of the realized power and of the speedof light. Thereat, as we have already noted, the magnitude of the limit valueis less significant than the fact of its existence. It is easily seen thatlim c →∞ R g = lim c →∞ mGc = 0;lim P max →∞ R g = lim c →∞ mc P max = 0 (15)In other words, at η → ∞ or c → ∞ the concept of gravitational radius and,consequently, the event horizon, becomes meaningless.While considering the maximum force (maximum power) as a fundamen-tal constant it is natural to use it instead of the gravitational constant. Forinstance, Newton’s law of universal gravitation acquires the form F = G mMR = c mM F max R = 1 F max mc · M c R Here the relation between the value of gravitational interaction and the max-imum force becomes more transparent: a gigantic maximum force gives rise7o a weak gravitational interaction. Naturally, if to choose the gravitationalconstant G in the capacity of initial fundamental constant, the converse state-ment is true, too.The choice of the maximum power as a new fundamental constant leads tothe Planck scales which preserve their previous numerical values. However,such a changeover opens up interesting opportunities for interpretation ofestimations made using the modified Planck units, as well as for the obtainingof new results. Below we will give a number of examples. If a space undergoes quantum fluctuations, the latter must manifest them-selves as uncertainties in different kinds of measurements [14, 15, 16]. Amongthem, measurement of lengths is significant. Let δl is an uncertainty withwhich the length l can be measured. To solve this problem, Wigner [17, 18]proposed a gedanken experiment. Let us place a clock and a mirror at differ-ent ends of the segment to be measured. By measuring the time of registra-tion of a reflected signal one can measure the length of the segment. However,quantum fluctuations will generate the uncertainty δl for the distance to bemeasured. Wigner showed that δl ≥ ~ lmc (16)Here m is the clock mass. As first sight it seems that the influence of quantumfluctuations will be eliminated if the clock mass tends to infinity. However,the growth of the mass is strictly limited. It is obvious that the watch size d islimited by the condition of the experiment ( d ≤ δl ). On the other hand, thesize of the clock must exceed its Schwarzschild radius d > Gm/c to preventtransformation of the clock into a black hole, since otherwise the indicationsof the clock will be inaccessible for observers. From here it follows that δl ≥ Gmc (17)By combining (14) and (15) we obtain δl ≥ (cid:0) ll P l (cid:1) / = l P l (cid:18) ll P l (cid:19) / , l P l ≡ s ~ η c (18)8imilar relations can be obtained for measuring time intervals [19, 20],( δt ) ≥ ~ tmc , δt ≥ Gmc (19)where t is the measured time interval. By combining these two expressionswe find δt ≥ (cid:0) tt P l (cid:1) / (20)Relation (20) connects the minimum uncertainty during measurement of timewith the measured time interval. The absolute value of uncertainty δt ∼ t / rises, whereas its relative value δt/t ∝ t − / diminishes. Now rewrite relations(18) and (20) in the form δl ≥ (cid:18) lc ~ η (cid:19) / c ; δt ≥ (cid:18) t ~ η (cid:19) / (21)which clearly shows that the existence condition for the minimum uncer-tainty during measurement of distance (time) is equivalent to the existencecondition for the limit power which, in its turn, is dictated by the existenceof the horizon.To avoid confusion, we emphasize that in the first and in the second caseit is not about the accuracy of the particular design of ”ruler” or hours, andthe universal limitations on the accuracy of the measurement of length andtime, which are based on fundamental physical laws. According to the traditional viewpoint, a dominating share of the degrees offreedom for our world belongs to the fields that fill the space. But graduallyit has become clear that such an estimation impedes construction of thequantum field theory. To avoid the problem of divergence at short distancessince our world should be described on a three-dimensional discrete gratingwith a period on the order of the fundamental Planck length. Recentlyanother, more radical theory the so-called holographic principle [21, 22] hasbecome popular. Its title is bound up with optical hologram which has the9orm of two-dimensional image of a three-dimensional object. This principleis based on two key statements:1. All the information contained in a space domain can be recorded(presented) on the boundary of this region, which is called the holographicscreen;2. A theory on the boundary of the considered region of space mustcontain not more than one degree of freedom per Planck area, or in otherwords total number of degrees of freedom N satisfies the inequality N ≤ Al P l = Ac G ~ (22)It means that the information density on the holographic screen is limitedby the value l − P l = 10 bit/m .The relation between the number of the degrees of freedom and the vol-ume of the ”memory” of the system is based on the fundamental statementthat information always needs a material carrier: without matter informationdoes not exist. In the context of this statement relation (22) gives an answerto an exceptionally intricate question concerning the maximum density ofinformation record on a material carrier. Following Susskind [22], assumethat we have a physical object with the surface area A and the entropy S .Let this object collapse into a black hole. Naturally, the aria of the horizon ofthe formed black hole is less than A . However, according to thermodynamicsof black holes, its entropy cannot decrease herewith. Accordingly, we obtainthe holographic restriction S ≤ A l p (23)Here equality is achieved for the objects capable to independently collapseinto a black hole. The relation between the entropy and information makesit possible to formulate the limit density of information record I ≤ A l p inthe form of holographic limit. In other words, information capacity rises inproportion to the area, but not to the volume. However, such an approachcontradicts our intuitive ideas of the relation between information capacityand the increase of the volume. Below we will show how to calm down ourintuition.As an elementary information carrier, it is convenient to choose the degreeof freedom of the system, irrespective of the concrete structure of the matter.Such a choice will make it possible to establish a relation between the changes10n information and those in entropy expressed directly in terms of the degreesof freedom of the system.Philosophy of the holographic principle brings forth holographic dynamicsin which all (!) the forces known in nature are replaced by the so-calledentropic forces. The latter are generated by the changes of the information I on a holographic screen. Taking into account that ∆ I = − ∆ S assumethat the entropic forces result from the changes in the entropy due to thedisplacement of matter. In other words, the entropic forces originate inthe universal tendency of any macroscopic system to increase the entropy.Holographic dynamics may be constructed in terms of changes in the entropy,and does not depend on details of microscopic theory. For instance, there isno fundamental field associated with the entropic force.Consider a physical system of macroscopic size (e.g. Universe) differentparts of which contain certain information. Let us imagine that we have toconcentrate some part of the available information in a finite volume duringa maximally short period of time. What fundamental limitations will arisewhile solving this problem?Concentration of information in a certain volume is inevitably bound upwith concentration of material carriers in the same volume. The existingphysical limitations for energy concentration lead to direct prohibition of theprocesses of excessive information concentration. Most concisely these twotypes of limitations are formulated in the form of the principle of maximumforce and the holographic principle. Therefore, it seems natural to investigatepossible relations between these principles.Earlier we have shown (see (18)) that the existence condition for the min-imum uncertainty at measurements of distance is equivalent to the existencecondition for the limit power which, in its turn, is dictated by the existence ofhorizon. Now reproduce this condition using the holographic principle [23].Let a volume l be partitioned into minimal constituents permissible fromthe viewpoint of physical laws (e.g. cubes). It seems natural to impart onedegree of freedom to each elementary volume (by analogy with the dimen-sionless cell of phase volume of quantum system dp dq/ (2 π ~ ) N ). If the mini-mum uncertainty at measurements of the distance l is δl , then the elementaryvolume component has the volume ( δl ) , and the number of the degrees offreedom for the system is ( l/δl ) . According to the holographic principle,( l/δl ) ≤ l l P l (24)11hat immediately sends us back to (18).It is significant to note that while deriving [24] (24) we have found theexpression for the minimum uncertainty δl using the holographic principle.As we have earlier seen, the existence of this fundamental characteristic ofspace is the direct consequence of the principle of the maximum force, andits value can be obtained without taking into account the holographic prin-ciple. Therefore, the statement to the effect that the holographic principleis a consequence of quantum fluctuations of space-time [24] is of the samereliability.Equivalence of the holographic and the maximum force principles maybe also proved quantitatively. As we have earlier seen, the limit densityof information on a holographic screen is bounded by the value ρ inf , max = l Pl = 10 bit/m . Using the modified Planck length l P l = q ~ c η we find that η ≈ . × W . The value of the limit power obtained with the help ofthe holographic principle coincides with η = c G postulated by the principleof maximum force. The limit values ~ , c, η control (restrict, bound) the rates of any physicalprocesses, in particular, the rate of information transmission. Its significanceis beyond the scope of purely technological applications. To a great extent,the level of development of human society is defined by the rate of informationtransmission and processing.One of the initial estimations of the limit rate of information processingis the so-called Bremermann’s limit [25] M c / ~ = ( M/kg ) 10 bit per second( M is the mass of processor ). For M = 1 kg the time of execution of oneoperation ∆ t B ≈ − sec. is much less than the Planck time, that givesrise to doubts about the estimation adequacy. A weak point of the saidestimation is disregard of the processor size and, consequently, the absenceof limitation of the rate of transmission of signal inside a computing device.If to take into account [26] both the uncertainty principle ∆ E ∆ t ≥ ~ andthe finite rate of signal propagation ∆ t ≥ L/c , then ∆ t > max [ ~ /M c , L/c ].Consequently, if the condition L < ~ /M c is fulfilled we can reproduce theBremermann’s limit ∆ t B = ~ /M c . However, it is obvious that in this casewe will also be confronted with the fact that taking into account gravitation12akes independent choice of the size L and the mass M . impossible. Theprocessor size is limited by the condition L > r g = MGc that prevents theformation of a black hole (horizon). In view of this restriction the minimumtime required for execution of one operation∆ t min = (cid:0) G ~ /c (cid:1) / ∼ − sec (25)is the Planck time; the limit rate of any computing device ν , ν = t − P l = (cid:0) c /G ~ (cid:1) / ∼ bit/ sec (26)In terms of limit power, interpretation of the result (26) is extremely trans-parent: ν = t − P l = (cid:0) c /G ~ (cid:1) / = (cid:16) η ~ (cid:17) / (27)The rate of information processing by an arbitrary computing device isbounded by the limit concentration of energy inside the device. The limitpower η is the quantitative measure of this limitation. At η → ∞ (limitationis absent) a computing device could work at an arbitrarily high rate. For description of physical processes with essentially different characteristicscales (of length, time or energy) the traditional point of view dictates touse different approaches. In fact, this means that macro- and micro-scalesare tacitly assumed to be independent. At present the situation has cardi-nally changed. It has turned out that even macro-objects possess quantumfeatures. A classical example of such a symbiosis is investigation of blackholes. Finding of relations between different scales facilitates solution ofsome fundamental problems that have remained unsolved in the frameworkof traditional approaches.The new approach [27] has been called UV/IR (ultraviolet- infrared) re-lation. The hypothesis is based on the following arguments.In any efficient quantum field theory defined in a spatial domain with thecharacteristic size l and using UV cutoff Λ, the entropy S ∝ Λ l . Assumeadditionally that in the framework of the considered theory there are fulfilledthe thermodynamic laws of black holes. In particular, this means that in such13 theory the entropy S of any object must be less than the entropy of a blackhole S BH of the same size: S ≤ S BH ≈ (cid:18) ll P l (cid:19) (28)From here it follows that l Λ ≤ (cid:18) ll P l (cid:19) (29)It is natural to identify the inverse UV-scale with the minimum uncertainty oflength measurement δl = Λ − . In this case (29) is immediately transformedinto δl = l / P l l / .From the viewpoint of physics of limit values, the relation between smalland large scales can be obtained from a natural condition: the total energyconfined within a domain of the linear size l must not exceed the energy ofa black hole of the same size, i.e. l ρ Λ ≤ M BH ∼ lm P l (30)Here ρ Λ is the energy in the volume l . Violation of this inequality will leadto the formation of a black hole with the event horizon preventing further riseof the energy density. Thus, the relation δl = l / P l l / or its equivalent (29) for δl = Λ − can be considered the relation between the infrared l and ultraviolet δl scales in effective quantum field theory in which the thermodynamic lawsof black holes are fulfilled.It is interesting to note that the UV/IR relation has broken down [see(28)] seemingly unquestionable statement to the effect that the obtaining ofinformation about the structure of a spatial object of the size ∆ x requiresthe energy E ∆ x ≈ ~ c ∆ x (31)Relation (31) dictates an evident strategy: investigation of smaller spatialscales necessitates construction of more and more powerful accelerators. Sucha strategy does not take into account gravitational effects and is not doubton spatial scales essentially exceeding the Planck ones. What will happen ifwe achieve energies of the order of the Planck E P = m P l c and higher? Ac-celerators of such energies will turn out to be pointless. It will be impossible14o analyze the result of collision of high-energy particles, since the collisionproducts will be hidden by horizon of the radius R S = 2 Gc E ∆ x c = 2 Gc E ∆ x ≈ E ∆ x F max (32)Due to finiteness of F max a giant ”Super Plancketron” collider [28] will notpermit to obtain information about spatial scales which are less than thePlanck scale, no matter how high the energy of this accelerator may be. The existence condition for the traditional space-time in the presence ofvacuum polarization (virtual processes of production and annihilation of pairscaused by quantum fluctuations) leads to limitation of proper accelerationrelatively to the vacuum, or, in other words, to the occurrence of the maximalacceleration [29, 30, 31, 32].The proper acceleration of the particle a in curved space-time is the scalardefined by the relation a = − c g µν Dv µ ds Dv ν ds (33)where g µν is the metric tensor, v µ ≡ dx µ /ds , the dimensionless four-velocityof the particle, D/ds is the covariant derivative with respect to the lineelement on the world line of the particle, Dv µ ds ≡ dv µ ds + Γ µαβ v α v β (34)Here Γ µαβ are the affine connections (the Christoffel symbols) of space-timewith the metric g µν , ds = g µν dx µ dx ν , the linear element of this space-time.From the energy-time uncertainty principle it follows that the lifetime ofthe virtual pair particle-antiparticle (with the particle mass m ) generated dueto vacuum fluctuations, is ≈ ~ / mc , whereas the distance covered duringthis time is ≈ ~ / mc (the Compton wavelength of the particles). If a virtualparticle acquires the energy equal to its rest mass, it will be transformedinto a real particle. When considering the rest system of a particle which is,generally speaking, non-inertial, we find that it undergoes the inertial force F in = | ma | , where a is the proper particle acceleration. The work executed15y the inertial force during the particle lifetime A = ma × ~ mc . If A = mc ,then there arises acceleration a = 2 mc ~ (35)At this acceleration, particles of the mass m will be copiously producedfrom the vacuum. The growth of acceleration will lead to the rise of themass of the produced particles. What critical consequences may arise atunlimited growth of acceleration? If the value of acceleration is high enough,the produced particles can be transformed into black holes. This will occurin the case when the Compton wavelength of a particle (particle ”size”) ~ /mc is less than its Schwarzschild radius 2 Gm/c , ~ /mc < Gmc (36)From here it follows that the threshold for black hole formation is a mass ofthe order of the Planck mass ( ~ η ) / /c . By substituting m = m P l into (1)we find a ≈ r η ~ c (37)(as before, we omit the multipliers of the order of unity). At such an ac-celeration, production of black holes with the Planck mass due to vacuumpolarization will result in breakdown of the traditional knowledge of thestructure of space-time, and the acceleration concept itself will lose its con-ventional sense. Therefore, the value a should be considered the maximalproper acceleration relatively to the vacuum. Note that the presence of themaximum acceleration leads to the formation of a horizon even in SR. Infact, from the viewpoint of SR, the length l of an object moving with theacceleration a is limited by the relation l ≤ c a . On the other hand, it cannotbe less than l ≥ l P l = q ~ η c . When using this inequality for accelerationone obtains a ≤ c p η ~ . As is seen, the maximal acceleration corresponds tothe fundamental acceleration in the Planck system of units, and is a simplecombination of the three limit values ~ , c, η . The necessary condition for itsexistence is finiteness of all the three limit values : at c → ∞ , ~ → η → ∞ the maximum acceleration is absent.The presence of the maximum proper acceleration a (33) automaticallyleads to the existence of the minimum radius of curvature R min of the particle16orld lines. The radius of curvature of the world line is R = c /a (since thecentripetal acceleration during motion along the circle of the radius R is a = v /R ). Therefore, the minimum radius of curvature has the form R min = c a ≈ (cid:18) ~ Gc (cid:19) / = c (cid:18) ~ η (cid:19) / (38)Again, we clearly see the key role of the horizon which produces the limitpower and, as a consequence, the maximal proper acceleration and the min-imum radius of curvature of the world line. Achievement of required accuracy in any quantum measurement inevitablyimposes certain limitations on characteristics of the device designed to per-form it. All possible methods to measure the time always involve observationof some periodical physical process. As an example (following [33]), considera quantum clock based on observation of radioactive disintegration describedby the following equation dNdt = − λN (39)where N ( t ) is the current number of radioactive particles in the sample.Average number of the decayed particles during the time interval ∆ t ≪ λ − is ∆ N = λN ∆ t . It enables us to measure the time intervals calculatingnumber of the decaying particles∆ t = ∆ NλN (40)The relative error of such a method of time measurement ε = ( λN ∆ t ) − / =1 / √ ∆ N ≤
1. At first glance, it seems that increasing size of the quantumclock (the number N ), we would gain unlimited improvement in accuracy ofthe time interval measurement. However, such a process is limited by thefollowing condition: the rise of the clock mass must not lead to transforma-tion of the clock into a black hole (i.e. to occurrence of horizon). Let usanalyze the quantitative limitations which may be caused by this condition.17y using the uncertainty principle ∆ E ∆ t ≥ ~ / t ≥ ~ ε c M (41)where M = N m p (with m p corresponding to the mass of one particle) isthe clock mass. If the clock radius R (the clock is assumed to be spherical)becomes less than the gravitational radius R g , it will be impossible to usethe clock for time measurements. The condition R > R g is transformed into1 M > Gc R (42)When substituting (42) into (41) we obtain∆ tR > ε Gc ~ (43)Treating R as uncertainty ∆ r in position of the physical object (the clock),which is the basis for the time measurement process, and taking into accountthat ε ≤ t ∆ r > Gc ~ (44)The obtained inequality limits the possibility to determine the time and spacecoordinates of events to an arbitrary precision.Let us analyze expression (44) using the notion of the limit force [34]. Forthis purpose present it in the form∆ t ∆ r > F max ~ (45)At the fixed Planck constant ~ , it is only the limit force F max defines thelimitation imposed on the quantum clock size. If such a force is absent inthe theory, i.e. F max = ∞ , then R g →
0, and limitation for the quantumclock size is absent too. The main cause of the discussed limitation is therequirement
R > R g equivalent to the condition preventing the formation ofhorizon. Therefore, the occurrence of the force F max in relation (45) whichcan be achieved only at the horizon seems absolutely natural.The structure of relation (45) does not contain any information concern-ing the process which has been the base for construction of the clock. This18uggests the idea that this relation may be obtained from general considera-tions. To prove this statement let us use the uncertainty relation∆ x min ∆ p max ≥ ~ F max = ∆ p max ∆ t min we immediately obtain that the minimum size ∆ x min ofthe clock necessary for measurement of the time intervals ∆ t min obeys thelimitation ∆ x min ∆ t min ≥ ~ F max = ~ cη (47)in complete correspondence with (45). This is just the relation that describesthe structure of space-time foam! A simple form of relation (47) points tothe fact that the limit values ~ , c, η have a fundamental character.Certainly, the earlier obtained restrictions for the limits of measurabilityof distance and time (21) are in accord with relations (47). In fact, multipli-cation of the uncertainties (21) gives δl · δt ≥ (cid:18) l ~ cη (cid:19) / c (cid:18) t ~ η (cid:19) / = ( l · t ) / (cid:18) ~ cη (cid:19) / (48)Suppose that we are to measure the minimum scales of length and time, i.e. l = δl and t = δt . In such a case (48) will reproduce relation (47). We propose to modify the system of Planck units by replacing the gravita-tional constant G by the limit power η ≡ c /G . Such a replacement makesit possible to use the system of fundamental scales of mass, length and timebased only on the limit values ~ , c, η . The latter play a special role in theattempts to describe Nature proceeding from the first principles. References [1] M.Planck, ¨Uber irreversible Strahlungsvorg¨ange. 5 Mitteilung // S.-B.Preu ββ