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PNUTP-04/A01 Brane World Confronts Holography ∗ Deog Ki Hong† Department of Physics, Pusan National University, Pusan 609-735, Korea Stephen D. H. Hsu‡ Department of Physics, University of Oregon, Eugene OR 97403-5203, U.S.A. HolographyprincipleimposesastringentconstraintonthescaleofquantumgravityM∗ inbrane- 4 world scenarios, where all matter is confined on the brane. The thermodynamic entropy of astro- 0 physical black holes and sub-horizon volumes during big bang nucleosynthesis exceed the relevant 0 bounds unless M∗ > 10(4−6) TeV, so a hierarchy relative to the weak scale is unavoidable. We 2 discuss theimplications for extra dimensions as well as holography. n a I. INTRODUCTION J 0 1 The idea that our universe might have extra dimen- sionsbeyondfourspace-timedimensionsdateslongback 1 to early last century. In 1921 Kaluza [1] and later Gravity v Klein [2] introduced extra dimensions to unify all forces 0 in nature. The metric for higher dimensions was postu- 6 lated to be 0 1 g ϕA A ϕA 40 gˆµˆνˆ = (cid:18) µνϕ−Aν µ ν ϕ µ (cid:19) , (1) Hidden SM 0 / wheregµν istheordinary4dmetricandAµ isinterpreted h as the photon field. After the dimensional reduction to FIG. 1: Onlygravity propagates in bulk. t - M4 S1, the general coordinate transformationinduces p × the U(1) gauge transformation in M4. e h String theory proposed as the theory of everything is scale, is given by : consistent only in 10 dimensions, where 6 extra dimen- v Xi sioSnosoanreacfotemrpHacotriafivead-WtoibtteeanC[5a]lapboi-iYnateudmoauntifotlhda[t3,o4u]r. VwM∗D−2 =MP2 , Vw ≡Z dD−4x −g(D−4). (3) p r world may be confined on a brane embedded in eleven ByadjustingV , thevolumeoftheextradimensions,we a dimensional spacetime, a low-scale gravity or a brane- w may take M = O(1) TeV to solve the hierarchy prob- ∗ world scenario [6, 7] is proposed as a solution to the lem. Since this low-scale gravity will be different from gauge hierarchy problem. Letting gravity propagate in the usual gravity at short scales, Newton’s law deviates extra dimensions while all standard model particles are at sub millimeter. The model is soonto be tested. How- confined on a brane, the scale of gravity can be made ever,inthistalkwewillarguethattheholographybound arbitrary. Inthe braneworldthe Einsteinactionhasthe requires the fundamental Planck scale can not be too form small [8]. To reproduce the successful nucleosynthesis in BigBangcosmologyandtoaccountforthesupernovaex- S =Z d4x M∗2R dD−4x M∗D−4 √−g . (2) plosion, M∗ > 104−6 TeV and the brane-world solution (cid:0) (cid:1) thushasalittlehierarchyproblem,evenifitisoperating. The relation between the fundamental Planck scale M ∗ and the apparent one M , the four-dimensional Planck P II. WHAT IS HOLOGRAPHY BOUND Bekenstein [9, 10] conjectured that for a system of en- ∗TalkdeliveredbyDKHfortheInternationalconferenceonFlavour ergyM ina radiusR, itsentropyisboundedfromabove physics(ICFP-II)inKIAS,Seoul,Korea,Oct. 6-11,2003. †Electronicaddress: [email protected] 2πMR ‡Electronicaddress: [email protected] S < ~ . (4) 2 For weak gravity, the size of the system is much larger thanitsSchwarzschildradius,R (=2GM)<R. There- s fore we get 2πMR A S < < , A=4πR2. (5) ~ 4G~ The entropy of a system is less than one quarter of its area in the unit of Planck area, l2 =G~/c3. p The Bekenstein bound for entropy lead ’t Hooft [11] and Susskind [12] to formulate the holographyprinciple, which states that the entropy in a spatial volume V en- closed by a surface area A cannot exceed A/4 in Planck FIG. 2: Closed universe and a collapsing star units. Consider a system in a box. In quantum field theory, a state in a box can have arbitrarily large energy. However, if its compton length is smaller than its time, violates the Bekenstein bound (See Fig. 2). For Schwarzschild radius, an observer outside the box can a closed universe, near the big crunch, the area of the not access such a state. Therefore, the energy of states closed universe can be made arbitrarily small while its in the box is limited to outside observers. Such states entropy never decreases. Similarly in the evolution of a do not contribute to entropy of the system measured by collapsing star the spatial area of the collapsing star be- outside observers. For outside observers, the number of comes arbitrarily small, violating the Bekenstein bound. accessible states of the system is much less than that of Since the spacelike volume does not have an intrin- the states allowed by a local quantum field theory. sic meaning in general relativity, one may introduce an InD dimensions,theSchwarzschildradiusR ofasys- s intrinsic entropy bound [14]. In 1999 Bousso [15, 16] in- tem with energy E is determined roughly by the condi- troduced a covariant entropy bound, which states that tion that the gravitational potential energy is of order the entropy on any light-sheet of a surface B will not one at R = R . If R is smaller than the radius of the s s exceed the area of B: extra dimensions, A(B) E SL . (10) Φ R (M2−DE)1/(D−3). (6) ≤ 4 ∼ MD−2RD−3 −→ s ∼ ∗ ∗ The null geodesics extended from the surface will merge Therefore, the energy of a system of size R must have at a focal point in the future direction [17] (See Fig. 3), a upper bound not to collapse into a black hole. If defining a null hypersurfaceL. As time evolves,all mat- E isthemaximumenergyofthesystem,thenE < max max a−DRD−1,where a−1 isthe ultraviolet(UV) cutoff. Not to collapse into a black hole, the size of the system has to be bigger than its Schwarzschild radius (M2−DE )1/(D−3)<(M2−Da−DRD−1)1/(D−3)<R.(7) ∗ max ∗ WefindthattheUVcutoffisrelatedtotheinfraredcutoff of the system, a>M−1 (RM )2/D. (8) ∗ ∗ A The entropy of the system [13] is given as (k = 1), B neglecting all quantum numbers except positions, FIG. 3: Entropyon a nullhypersurface, L. S =ln2(R/a)D−1 <(RM )D−3+2/Dln2. (9) ∗ ter inside B of spatial volume V will pass through the ForD =4,thisboundgivesS <CA3/4,whichissmaller null hypersurface, L. By the second law of thermody- than the area A. namics, the entropy in the null hypersurface is not less than the entropy inside B. Therefore, the covariant entropy bound gives the Bekenstein bound, III. COVARIANT ENTROPY BOUND A S <S < . (11) V L 4 The Bekenstein bound for entropy applies to a static systemonly. Forinstanceonecaneasilyseethataclosed Furthermoreitdoesremainvalidinthecaseofdynamical universeoracollapsingstar,wherethesystemevolvesin systems like collapsing universe or collapsing stars. 3 IV. BLACK HOLE THERMODYNAMICS (1)The holographic boundis violated duringthebigbang. Consider a spacelike region V of extent r on the 3- h Original Bekenstein conjecture on entropy bound was brane,andcomparetheapparent(3+1)entropywiththe motivated by black hole thermodynamics, where the en- holographic bound applied to a hypersurface B which is tropyofblackholeisfoundtobeproportionaltothearea the boundary of V. Let V have the same shape as the of the black hole horizon. brane, with thickness of order M∗−1, so that its surface Here,wereviewthederivationoftheblackholeentropy area is of order rh2 in units of M∗. (It is possible that by’tHooft[18],whichisbasedontwodistinctproperties the brane is thicker than M∗−1, forcing us to use a larger of black holes. The first property is that black holes hypersurface with more entropy density, however it is radiateasblackbodieswithacertaintemperature,called hard to imagine that the brane thickness is parametri- Hawking temperature, cally larger than the fundamental length scale.) Impose that this region saturate the holographic bound, so r h 1 satisfies [19] (See Fig. 4) T = , (12) H 8πM T3r3 M2r2 , (18) whereM isthemassoftheblackhole. Thesecondprop- h ∼ ∗ h erty is that black holes have an event horizon. If one or drops an object with energy ∆E into a black hole with 2 ambassosrpEti.on(∆crEoss≪sec1ti≪onEis tihnenPlanck units.) Then, the rh ∼T−1(cid:18)MT∗(cid:19) . (19) σ =πR2, R 2E. (13) ≃ From the Hawking’s result, the emission probability is W πR2ρ e−βH∆E, (14) T−4 ∆E ≃ S T−3/2 bewryhgeyare∆HρaE∆m.EiNltiooswntih,aeinfdwaecentssinuitgpypionofsHestitalhbteeesrptfroosrpcaeascspee.asrTtaihrceelend,wesictrhibeend- (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) σ = E+∆E T E,∆E 2ρ(E+∆E) |h | | i| T−3 W = E,∆E T E+∆E 2ρ(E)ρ . (15) |h | | i| ∆E 3 Brane T T T T ByPCT invariance,thematrixelementshavetobesame * d BBN and we get FIG. 4: Entropy in thehorizon σ e∆E/TH ρ(E+∆E) = = (16) Now consider a cosmological horizon volume of size W ρ ρ(E)ρ ∆E ∆E d M /T2 (assuming radiation domination). The H P ∼ Therefore, we find the density of states of the black hole ratio of r to d is h H ρ(E)=exp(4πE2) and the black hole entropy becomes d T M T H P . (20) A r ∼ M M ∼ 10−4eV S =lnρ(E)=4πE2+S or S = +S . (17) h ∗ ∗ 0 0 4 For the matter-dominant epoch, the horizon distance is whIenretShe0 insetxhte sseucbtlieoand,inwgetewrmill. apply the holography tghiveenonassetdHtem∼p(cid:0)eMraPtu/rTed2o(cid:1)f(Tmda/tTte)r3/d2,omwhinearetioTnd.≃T1h0eerVatiios bound and the black hole entropy bound to the brane then becomes world scenarios,both ADD [6] and RS models [7]. d M T3 1/2 T 3/2 H P . (21) r ∼ M (cid:18)M2T (cid:19) ∼(cid:18)10−2 eV(cid:19) h ∗ ∗ d V. HOLOGRAPHY BOUNDS ON BRANE WORLD SCENARIO Wefindthatforanytemperaturehigherthan10−2eVthe causalhorizoncontainsmoredegreesoffreedomthanare Consider an ADD/RS world in which the standard allowedaccording to the HB applied to the fundamental modeldegreesoffreedomareconfinedtoa3-branewhile theory. the gravitational degrees of freedom propagate in D di- Our understanding of thermodynamics and statistical mensions. The large effective volume V of the bulk al- physics is based on counting states. If the HB is cor- w lows the apparent Planck scale M to be much larger rect, the early universe in the brane worlds under con- P than the true dynamical scale of gravityM TeV. sideration will likely not obey the usual laws of thermal ∗ ∼ 4 physics at temperatures >10−2eV. This makes our understanding of nucleosynthesis and the microwave back- ground problematic. In order that our thermodynamic description of nucleosynthesis (at T 10 MeV) not be invalidated by Gravity holography, we find ∼that M > 104 TeV. (This bound ∗ is reduced slightly from (20) when prefactors in the ex- pressions for the entropy density and horizon size are BH included.) Hidden (2)The holographic bound is violated by supernova cores. Consider the supernova of a star of mass M > 8M , SM ⊙ whichispoweredbythecollapseofanironcoreandleads to neutron star or black hole formation. In this process FIG. 5: Black holes in a braneworld the entropy of the collapsed neutron star is of order one per nucleon, so the total entropy is roughly 1057. The radius of the core is a few to ten kilometers, so that its us assume,motivated by holography,thatthe entropyof area(1012 cm2)inM∗ units isonly1046,whereagainwe a pancake black hole continues to be of order its surface takeafiducialvolumeofthicknessjustgreaterthanthat area in units of M . The surface area of a large hole is ∗ of the brane. (As in the cosmologicalcase the degrees of dominated by the r3lD−5 component, so the black hole freedom we are counting are all confined to the brane.) entropybound arising fromthe Susskind constructionin Unless M∗ > 106 TeV there is a conflict between the RS worlds is of the form usual thermodynamic description of supernova collapse and the holographic entropy bound. S <(rM )3 . (22) ∗ (3) Black hole entropy bound vs. covariant bound That is, the upper bound on the entropy growswith the Susskind[12]imaginesaprocessinwhichathermody- apparent 3-volume of the region. In this case the black namicsystemisconvertedintoablackholebycollapsing hole bound is clearly weaker than the covariant bound, a spherical shell around it. Using the GSL, one obtains because the surface B used in the application of the lat- a bound on the entropy of the system: S A/4, ter is much smaller than the area of the pancake hole. matter ≤ where A is the area of the black hole formed. This is Interestingly, (22) is the same result one would have ob- a weaker conjecture than the covariant bound, and has tained naively from D = 3+1 quantum field theory in considerable theoretical support [9, 10, 12, 16]. In the the absence of gravity, with ultraviolet cutoff M∗! applicationoftheCBwearefreetochoosethehypersur- faceB,aslongasitslightsheetintersectsallofthematter whose entropy we wish to bound, whereas in the black hole bound the area which appears is that of the black VI. DISCUSSION hole which is formed. The black hole entropy bound is sensitive to the dynamics of horizon formation. Our results can be interpreted in two ways, depend- In TeV gravity scenarios, the black hole size on the 3- ing on how one views holography and related entropy brane is controlled by the apparent Planck scale M = bounds. P 1019 GeV.Theextentofthehorizonintheperpendicular Itseemslikelythatholographyisadeepresultofquan- directionsoffthebranedependsonthemodel,unlessthe tum gravity,relatinggeometryandinformationina new hole is very small. way[16]. Ifso,itprovidesimportantconstraintsonextra In ADD worlds, the horizon of an astrophysical black dimensional models. Our analysis shows that the ordi- hole likely extends to the boundary of the compact ex- nary thermodynamic treatment of nucleosynthesis and tra dimensions. As discussed in [20], large black holes supernovae are in conflict with the covariant bound. In have geometry S2 TD−4, and the horizon includes all otherwords,braneworldsobeyingholographydonotre- × of the extra volume V . Due to this additional extra- producethe observedbig bangthermalevolutionorstel- w dimensional volume, the resulting entropy density is the lar collapse. Exactly what replaces the usual behavior same as in 3+1 dimensions and there is no obvious vio- is unclear - presumably it is highly non-local - but the lation of any bounds. number of degrees of freedom is drastically less than in In RS scenarios, however, black holes are confined to the thermodynamic description. thebraneandhaveapancake-likegeometry[21,22]. (See An alternative point of view is to regardbrane worlds Fig.5.) Theblackholesizeinthedirectiontransverseto as a challenge to holography. If such worlds exist they the brane grows only logarithmically with the mass M. have the potential to violate the entropic bounds by ar- Thusfar,noonehascomputedtheHawkingtemperature bitrarily large factors. However, it must be noted that or entropy of a pancake black hole. In fact, exact solu- the basic dynamical assumptions underlying the scenar- tionsdescribingthisobjectshaveyettobeobtained. Let ios(thatthe3-braneandbulkgeometryariseasaground 5 state of quantum gravity) have never been justified. All 070-C00022. The work of S.H. was supported in part violationsdiscussedhererequireahierarchybetweenM under DOE contract DE-FG06-85ER40224 and by the P and M , or equivalently that the extra-dimensional vol- NSF through through the USA-Korea Cooperative Sci- ∗ ume factor V = dD−4x g exceed its “natu- ence Program,9982164. w (D−4) − ral” size M−(D−R4). p ∗ ∼ Finally, we note that the brane, or whatever confines matter to 3 spatial dimensions, is absolutely necessary for these entropy violations. Without the brane, matter initiallyinaregionwithsmallextentintheextra(D 4) − dimensions will inevitably spread out due to the uncer- tainty principle. For ordinary matter in classical general relativity, in the absence of branes, Wald and collabora- tors[23]haveproventhecovariantentropyboundsubject to some technical assumptions. Acknowledgements TheworkofD.K.H.issupportedbyKRFPBRG2002- [1] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin [12] L. Susskind, J. Math. Phys. 36 (1995) 6377 (Math. 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