An insightful argument for a linear relation between the entropy and the area
of a black hole was given by Bekenstein using only the energy-momentum
dispersion relation, the uncertainty principle, and some properties of
classical black holes. Recent analyses within String Theory and Loop Quantum
Gravity describe black-hole entropy in terms of a dominant contribution, which
indeed depends linearly on the area, and a leading log-area correction. We
argue that, by reversing the Bekenstein argument, the log-area correction can
provide insight on the energy-momentum dispersion relation and the uncertainty
principle of a quantum-gravity theory. As examples we consider the
energy-momentum dispersion relations that recently emerged in the Loop Quantum
Gravity literature and the Generalized Uncertainty Principle that is expected
to hold in String Theory.Comment: 7 pages, LaTex; this essay received an "honorable mention" in the
  2004 Essay Competition of the Gravity Research Foundation; submitted to IJMPD
  on 23 June 2004; published as Int.J.Mod.Phys.D13:2337-2343,200

ANDREA PROCACCINI

Bekenstein J. D.

GIOVANNI AMELINO-CAMELIA

Lifshitz E. M.

Majid S.

MICHELE ARZANO

English

arXiv

An insightful argument for a linear relation between the entropy and the area of a black hole was given by Bekenstein using only the energy-momentum dispersion relation, the uncertainty principle, and some properties of classical black holes. Recent analyses within String Theory and Loop Quantum Gravity describe black-hole entropy in terms of a dominant contribution, which indeed depends linearly on the area, and a leading log-area correction. We argue that, by reversing the Bekenstein argument, the log-area correction can provide insight on the energy-momentum dispersion relation and the uncertainty principle of a quantum-gravity theory. As examples, we, consider the energy-momentum dispersion relations that recently emerged in the Loop Quantum Gravity literature and the Generalized Uncertainty Principle that is expected to hold in String Theory

AMELINO-CAMELIA, Giovanni

M. Arzano

Andrea Procaccini

Archivio della ricerca- Università di Roma La Sapienza

A glimpse at the flat-spacetime limit of quantum gravity using the bekenstein argument in reverse

Crossref

A GLIMPSE AT THE FLAT-SPACETIME LIMIT OF QUANTUM GRAVITY USING THE BEKENSTEIN ARGUMENT IN REVERSE

An insightful argument for a linear relation between the entropy and the area of a black hole was given by Bekenstein using only the energy-momentum dispersion relation, the uncertainty principle, and some properties of classical black holes. Recent analyses within String Theory and Loop Quantum Gravity describe black-hole entropy in terms of a dominant contribution, which indeed depends linearly on the area, and a leading log-area correction. We argue that, by reversing the Bekenstein argument, the log-area correction can provide insight on the energy-momentum dispersion relation and the uncertainty principle of a quantum-gravity theory. As examples we consider the energy-momentum dispersion relations that recently emerged in the Loop Quantum Gravity literature and the Generalized Uncertainty Principle that is expected to hold in String Theory

AMELINO-CAMELIA, GIOVANNI

PROCACCINI, ANDREA

ARZANO, MICHELE

Carolina Digital Repository

arXiv:hep-th/0506182v1  21 Jun 2005A glimpse at the flat-spacetime limit of quantum gravityusing the Bekenstein argument in reverse1Giovanni AMELINO-CAMELIAa, Michele ARZANOb, Andrea PROCACCINIaaDip. Fisica Univ. Roma “La Sapienza” and Sez. Roma1 INFN, Piazzale Moro 2, Roma, Italyb Dept Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, USAABSTRACTAn insightful argument for a linear relation between the entropy and the area ofa black hole was given by Bekenstein using only the energy-momentum disper-sion relation, the uncertainty principle, and some properties of classical blackholes. Recent analyses within String Theory and Loop Quantum Gravity de-scribe black-hole entropy in terms of a dominant contribution, which indeeddepends linearly on the area, and a leading log-area correction. We argue that,by reversing the Bekenstein argument, the log-area correction can provide in-sight on the energy-momentum dispersion relation and the uncertainty principleof a quantum-gravity theory. As examples we consider the energy-momentumdispersion relations that recently emerged in the Loop Quantum Gravity lit-erature and the Generalized Uncertainty Principle that is expected to hold inString Theory.As first observed by Bekenstein [1], the fact that the entropy of a black hole should be proportionalto its (horizon-surface) area, up to corrections that can be neglected when the area A is much largerthan the square of the Planck length Lp, can be derived using very simple ingredients. In (classical)general-relativity one finds [2] that the minimum increase of area when the black hole absorbs aclassical particle of energy E and size s is ∆A ≃ 8πL2pEs (in “natural units” with h̄ = c = 1). Takinginto account the quantum properties of particles one can estimate the size of the particle as roughlygiven by its position uncertainty δx, and one can assume that a particle with position uncertainty δxshould at least [3] have energy E ∼ 1/δx. This leads to the conclusion[1, 4] that the minimum changein the black-hole area must be of order L2p, independently of the size of the area. Then using the factthat, also independently of the size of the area, this minimum increase of area should correspond tothe minimum (“one bit”) change of entropy, (∆S)min = ln 2, one easily obtains [1] the proportionalitybetween black-hole entropy and area.Over the last few years both in String Theory and in Loop Quantum Gravity some techniquesfor the analysis of entropy on the basis of quantum properties of black holes were developed. Theseresults[5, 6] now go even beyond the entropy-area-proportionality contribution: they establish that1This essay received an “honorable mention” in the 2004 Essay Competition of the Gravity Research Foundation –Ed.1the leading correction should be of log-area type, so that one expects (for A ≫ L2p) an entropy-arearelation for black holes of the typeS =A4L2p+ ρ lnAL2p+O(L2pA). (1)While all analyses agree on the coefficient of the term A/L2p, different quantum-gravity theories appearto give rise [5, 6, 7] to different predictions for the coefficient ρ of the leading ln(A/L2p) correction.Here we propose that the log-area leading correction might admit a simple description, just likethe argument presented by Bekenstein in Ref. [1] gives a simple description of the dominant linear-in-A contribution. Our simple description will also explain why the coefficient ρ of the log-area termtakes different values in different quantum-gravity theories. And we stress that the availability ofresults on the log-area correction might provide motivation for reversing the Bekenstein argument:the knowledge of the black-hole entropy-area law up to the leading log correction can be used toestablish the Planck-scale modifications of the ingredients of the Bekenstein analysis.We start by observing that, as stressed above, a key point for the Bekenstein analysis is theE ≥ 1/δx relation between the energy of a particle and the uncertainty in its position. One obtainsE ≥ 1/δx by combining appropriately [3] the Heisenberg uncertainty principle, the special-relativisticenergy-momentum dispersion relation, and the fact that a procedure for the measurement of the po-sition of a particle must not introduce an energy uncertainty for the particle which is greater than itsmass (in its rest frame), since otherwise the procedure is subject to the possibility of production ofadditional copies of the particle. But nearly all approaches to the quantum-gravity problem predicta Planck-length-dependent modification of E ≥ 1/δx, since various quantum-gravity results suggestthat the uncertainty principle and/or the energy-momentum dispersion relation should be modified.Modifications of the uncertainty principle have been discussed in quantum-gravity theories that pre-dict a new role for the Planck length in the localization of particles, as in the case of String Theory [8].Several scenarios for a modified energy-momentum dispersion relation [9] have been considered, par-ticularly in quantum pictures of spacetime which involve some level of spacetime discreteness [10] ornoncommutativity [11, 12, 13, 14]. In Loop Quantum Gravity, which indeed predicts a certain form ofspacetime granularity (discrete spectrum of areas and volumes), several studies[15, 16] found evidencefor a modification of the dispersion relation.In order to accommodate the differences between alternative quantum-gravity scenarios we considera rather general parametrization of the Planck-length modified E ≥ 1/δx relation:E ≥ 1δx− η1Lp(δx)2− η2L2p(δx)3+O(L3p(δx)4)(2)We will argue that a log-area leading correction to the entropy-area relation requires η1 = 0. For η1 6= 0the leading correction turns out to behave like the square-root of the area. A result in agreement with2(1) is obtained for η1 = 0, η2 ∼ ρ (and the sign convention in (2) turns out to ensure that η2 and ρare either both positive or both negative).As in the original Bekenstein argument [1], we take as starting point the general-relativity resultwhich establishes that the area of a black hole changes according to ∆A ≥ 8πEs when a classicalparticle of energy E and size s is absorbed. In order to describe the absorption of a quantum particleone must describe the size of the particle in terms of the uncertainty in its position [1, 4], s ∼ δx,and take into account a “calibration factor” [17, 18, 19] (ln 2)/2π that connects the ∆A ≥ 8πEsclassical-particle result with the quantum-particle estimate ∆A ≥ 4(ln 2)L2pEδx. Following the originalBekenstein argument [1] one then enforces the relation E ≥ 1/δx (and this leads to ∆A ≥ 4(ln 2)L2p),but we must take into account the Planck-length modification in (2), obtaining∆A ≥ 4(ln 2)[L2p −η1L3pδx−η2L4p(δx)2]≃ 4(ln 2)[L2p −η1L3pRS−η2L4p(RS)2]≃ 4(ln 2)[L2p −η12√πL3p√A−η24πL4pA]where we also used the fact that in falling in the black hole the particle acquires [18, 21, 22] positionuncertainty δx ∼ RS , where RS is the Schwarzschild radius (and of course A = 4πR2S).Next, following again Bekenstein[1], one assumes that the entropy depends only on the area ofthe black hole, and one uses the fact that according to information theory the minimum increase ofentropy should be ln 2, independently of the value of the area:dSdA≃ min(∆S)min(∆A)≃ ln 24(ln 2)L2p[1− η12√πLp√A− η24πL2pA] ≃(14L2p+η1√π2Lp√A+η2πA). (3)From this one easily obtains (up to an irrelevant constant contribution to entropy):S ≃ A4L2p+ η1√π√ALp+ η2π lnAL2p. (4)As anticipated the sought agreement with (1) requires η1 = 0 and η2 ∼ ρ (precisely η2 = ρ/π).From this we conjecture that the log-area contribution to the black-hole entropy might be deeplyconnected with a corresponding L2p/(δx)3 correction to the E ≥ 1/δx relation. And, as anticipated, ourdescription encourages the possibility that the coefficient of the log-area correction might be differentin different quantum-gravity theories. The dominant linear-in-A term must be “universal” since itwas obtained in Ref. [1] using only observations that do not involve any quantum-gravity effects (thePlanck length appears in the Bekenstein formula only as a result of the classical-gravity aspects of theanalysis, which of course involve the gravitational constant). Instead the coefficient of the log-arealeading correction depends directly on the coefficient of the L2p/(δx)3 correction to the E ≥ 1/δxrelation, and, since different quantum-gravity theories could lead to different modifications of theHeisenberg uncertainty principle and of the energy-momentum dispersion relation, this coefficientshould not be expected to take the same value in all quantum-gravity theories.3Amusingly this observation provides an opportunity to use the Bekenstein argument in reverse.In the 1970s there was no result on entropy from the analysis of quantum properties of black holesand analyses “a la Bekenstein”, using some simple ingredients, allowed to derive the linear term inthe entropy-area formula. Now that the analysis of quantum properties of black holes allows to derivealso the log-area correction we might be able to use this information to infer which Planck-scalemodifications should be introduced in the ingredients of the Bekenstein analysis.A first application of this “reversed Bekenstein argument” can be found in Loop Quantum Gravity.There is a large number of Loop-Quantum-Gravity studies (see, e.g., Refs. [15, 16] and referencestherein) reporting results in support of the possibility of a Planck-scale modified energy-momentumdispersion relation, although there is still no consensus on the precise form of this modification. Theideal framework for establishing Planck-scale modifications of the dispersion relation is the study of theself-energy of a particle in a low-energy effective theory on “quasi-Minkowski spacetime” (a quantumspacetime which most closely approximates Minkowski spacetime in a given quantum-gravity theory).But in Loop Quantum Gravity the well-known “classical-limit problem” [23], which of course includesa “Minkowski-limit problem”, has provided so far a rather serious obstacle for the analysis of Planck-scale modifications of dispersion relations. In particular, two key alternatives for the Planck-scalemodification of the energy-momentum dispersion relation are still being considered [15, 16]: one inwhich the function p(E) admits an expansion with leading Planck-scale correction of order LpE3,~p2 ≃ E2 −m2 + α1LpE3 , (5)and one in which the function p(E) admits an expansion with leading Planck-scale correction of orderL2pE4,~p2 ≃ E2 −m2 + α2L2pE4 . (6)We intend to show that, since also in Loop Quantum Gravity there is evidence [6] that the leadingcorrection to the entropy-area black-hole formula is logarithmic, our “reversed Bekenstein argument”favors the scenario (6). The scenario (5) would lead to a√A leading correction.On the basis of the analysis reported above it will be sufficient to study the implications of thesemodified dispersion relations for the relation between the energy of a particle and its position uncer-tainty, and show that (6) leads to a contribution of the type L2p/(δx)3, while from (5) one obtains aLp/(δx)2 term.In order to establish the implications for the relation between the energy of a particle and itsposition uncertainty, we can follow the familiar derivation [3] of the relation E ≥ 1/δx, substituting,where applicable, the standard special-relativistic dispersion relation with the Planck-scale modifieddispersion relation. Let us first consider the case of the dispersion relation (6). It is convenient tostart by focusing on the case of a particle of mass M at rest, whose position is being measured by aprocedure involving a collision with a photon of energy Eγ and momentum pγ . In order to measure the4particle position with precision δx one should use a photon with momentum uncertainty δpγ ≥ 1/δx.This δpγ ≥ 1/δx relation originates from Heisenberg’s uncertainty principle for which, as mentioned,modifications have not been suggested in the Loop-Quantum-Gravity literature (but they are expectedin other approaches to the quantum-gravity problem). Following the standard argument [3], one takesthis δpγ ≥ 1/δx relation and converts it into the relation δEγ ≥ 1/δx, using the special-relativisticdispersion relation, and then the relation δEγ ≥ 1/δx is converted into the relation M ≥ 1/δx becausethe measurement procedure requires2 M ≥ δEγ . If indeed Loop Quantum Gravity hosts a Planck-scale-modified dispersion relation of the form (6), it is easy to see that, following the same reasoning,one would obtain from δpγ ≥ 1/δx the requirementM ≥ 1δx(1− α23L2p2(δx)2). (7)These results strictly apply only to the measurement of the position of a particle at rest, but theycan be straightforwardly generalized [3] (simply using a boost) to the case of measurement of theposition of a particle of energy E. In the case of the standard dispersion relation (without Planck-scale modification) one obtains the familiar E ≥ 1/δx, as required for a purely linear dependence ofentropy on area. In the case of (6) one instead easily finds that E ≥ 1δx(1− α23L2p2(δx)2), which fulfillsthe requirements of our derivation of a leading-order correction of log-area form.The careful reader can easily adapt to the case of the dispersion relation (5) this simple analysiswhich we discussed for the dispersion relation (6). The end result is that from (5) it follows thatE ≥ 1δx(1 + α1Lpδx), and this in turn leads to a leading correction to the entropy-area formula thatgoes like the square-root of the area, rather than the predicted log-area leading correction. Thisconfirms that, using the reversed Bekenstein argument, the dispersion relation (6) should be preferredto the dispersion relation (5).In closing let us turn to String Theory. Also in String Theory there are robust indications of alogarithmic contribution to the black-hole entropy-area relation, but at present the mainstream String-Theory literature provides no evidence for a Planck-scale modification of the dispersion relation. In thecase of String Theory our proposed “reversed Bekenstein argument” leads us to consider the so-called“GUP” or “Generalized Uncertainty Principle”,δx ≥ 1δp+ λ2sδp , (8)which finds support in various String-Theory studies [8]. The scale λs in (8) is an effective stringlength (a characteristic length scale of the GUP which should be closely related to the string lengthand therefore the Planck length).The presence of a logarithmic term in the entropy-area relation can be viewed as a consequenceof the presence of the λ2sδp term in the GUP. In fact, as the careful reader can easily verify, from the2We are here again using the fact [3] that the measurement procedure must ensure that the relevant energy uncer-tainties are not large enough to possibly produce extra copies of the particle whose position one intends to measure.5GUP one obtains (following again straightforwardly the standard measurability line of analysis [3]) amodification of the relation E ≥ 1/δx. The modification is of the type E ≥ 1/δx+∆, with ∆ of orderλ2s/δx3, and originates from the fact that according to the GUP, (8), one obtains δpγ ≥ 1/δx+λ2s/δx3,instead of the δpγ ≥ 1/δx which follows from Heinsenberg’s uncertainty principle. As discussed abovethis type of relation between the energy and the position uncertainty of a particle, with a 1/δx3contribution in addition to the familiar 1/δx term, leads, through the Bekenstein argument, to alogarithmic contribution to the black-hole entropy-area relation.The examples of Loop Quantum Gravity and String Theory show that we might have at our disposala rather powerful tool for the analysis of quantum-gravity theories, assuming that it is legitimate tothink of a connection between the log-area contribution to black-hole entropy and a 1/(δx)3 correctionto the E ≥ 1/δx relation. Whereas in Loop Quantum Gravity the L2p/(δx)3 correction is motivated byresults on Planck-scale-modified dispersion relations of the type (6), in String Theory the analogouscorrection of order λ2s/δx3 is motivated by the GUP. In general, in any given quantum-gravity theory,the presence of a logarithmic term in the black-hole entropy-area relation may be a manifestation ofa modified dispersion relation and/or a modified uncertainty principle.AcknowledgmentsG. A.-C. gratefully acknowledges conversations with O. Dryer, D. Oriti, C. Rovelli and L. Smolin. Thework of M. A. was supported by a Fellowship from The Graduate School of The University of NorthCarolina. M. A. also thanks the Department of Physics of the University of Rome for hospitality.References[1] J. D. Bekenstein, Phys. Rev. D7 (1973) 2333.[2] D. Christodoulou, Phys. Rev. Lett. 25 (1970) 1596; D. Christodoulou and R. Ruffini, Phys. Rev. D4 (1971)3552.[3] E. M. Lifshitz, L. P. Pitaevskii and V. B. Berestetskii, “Landau-Lifshitz Course of Theoretical Physics,Volume 4: Quantum Electrodynamics” (Reed Educational and Professional Publishing, 1982).[4] S. Hod, Phys. Rev. Lett. 81 (1998) 4293.[5] A. Strominger and C. Vafa, Phys. Lett. B379 (1996) 99; S. 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A glimpse at the flat-spacetime limit of quantum gravity using the
  Bekenstein argument in reverse

A glimpse at the flat-spacetime limit of quantum gravity using the Bekenstein argument in reverse

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