On the Davies-Unruh effect in a wide range of temperatures Carlos. E. Navia Instituto de F´ısica, Universidade Federal Fluminense, 24210-346, Nitero´i, RJ, Brazil (Dated: January 13, 2017) The Debye model of the specific heat of solid at low temperatures is incorporate in the Entropic GravityTheory(EGT).Ratherofasmoothsurface,theholographicscreenisconsideredasanoscil- latingelasticmembrane,withacontinuousrangeoffrequencies,thatcutsoffatamaximum(Debye) temperature, T . We show that at low temperatures T <T , the conservation of the equivalence D D principle in EGT requires a modification of the Davies-Unruh effect. While the maintenance of Davies-Unruh effect requires a violation of the equivalence principle. These two possibilities are equivalents, because both can emulate the same quantity of dark matter. However, in both cases, thecentralmechanismistheDavies-Unruheffect,thisseemstoindicatethatthemodificationofthe 7 Davies-Unruheffectemulatesdarkmatterwhichinturncanbeseeasaviolationoftheequivalence 1 principle. This scenario is promising to explain why MOND theory works at very low temper- 0 atures (accelerations) regime, i. e., the galaxies sector. We also show that in the intermediate 2 region,fortemperaturesslightlylowerorslightlyhigherthanDebyetemperature,EGTpredictsthe n mass-temperature relation of hot X-ray galaxy clusters. a J 2 I. INTRODUCTION is no evidence of a new physics beyond standard model, 1 in the Large Hadron Collider (LHC) data [13]. There is ] The Davies-Unruh effect (DHE) [1, 2], essentially pre- no evidence for SUSY (super-symmetric particles) [14], A dict that in an accelerated frame of reference; a vacuum where the “neutralino” is a kind of natural candidate for a dark matter particle. G statemayseenasathermalbathofphotonswithablack boddy spectrum at a temperature T, the main point of In the 80s, Milgrom [15, 16], proposed a modification . h the DUE is that this temperature is proportional to the in the Newtonian law of gravity as solution to the miss- p acceleration of the frame. ing mass problem, without dark matter. Rather than a - scientific theory MOND is considered only as an empir- o The connection between thermodynamic and gravity r began in the 70s with Bekenstein [3] and Hawking [4], ical model by most of the scientific community, because st researching the nature of black holes. In 1995 Jacobson predicts a violation of the equivalence principle. How- a ever, MOND has been very well successful to describe [5]shownathermodynamicdescriptionofgravityobtain- [ thegalaxiesdynamics[17–20]. Indeed,MONDpredicted ing the Einstein’s equations. 1 According to Padmanabhan [6], the association be- the Tully-Fisher relation. v The connection between entropic gravity and MOND tween gravity and entropy leads in a natural way to de- 2 is not new, there are some literature on this topic such scribes gravity as an emergent phenomenon, and a for- 4 as reported in [21, 22]. In this Letter we gives empha- 4 malism of gravity as a entropic force is derived by Ver- sis to the formalism of Debye model [23] of the specific 3 linde [7] in 2010. The dependence of information on sur- heatofsolidatlowtemperatures,incorporatedtotheen- 0 facearea,ratherthanvolume(Holographicprinciple)[8], . it is one of the key of black hole thermodynamic theory, tropic gravity. In section II, the basis of EGT within the 1 Debye formalism is presented and we defined the Debye 0 as well as in EGT. temperature, T , in EGT. Section III, is devoted to a 7 On the other hand, the Tully-Fisher relation [9] is an D analysis of the inertia at low temperatures (T <T ). In 1 empirical result, very well established for spiral galax- D v: ies. ThisrelationishardtobeobtainedfromNewtonian the intermediate region, i.e., temperatures close to TD, EGTseemstoindicatethatisthegalaxyclustersregion, i gravity, at least if only the visible mass of the galaxy is X this topic is discussed in section IV, and the section V is considered. The output for this impasse was postulating r the presence of a galactic halo of dark matter. Nowa- devoted for our conclusions. a days the empirical roots of the missing mass problem goes from the flat rotation curves of galaxies, cluster of galaxies, gravitational lensing, large scale structure, and II. GRAVITY AT LOW TEMPERATURES it is needed to describe the spectrum of the cosmic mi- crowave background radiation (CMB). In 1912, Debye [23] developed a theory to explain the The nature of dark matter is unknown. But the most heat capacity of solid as low temperatures. He assumed widely accepted hypothesis is that dark matter is com- that the vibration of the atoms of the lattice of a solid, posed of weakly interacting massive particles (WIMPs) followsacontinuousrangeoffrequencies,suchasanelas- that interact only through gravity and the weak force tic structure, that cuts off at a maximum frequency, ω . D [10]. However,sofar,thereisnodirectevidenceofwimps Inthistheoryeachsolidhasaspecifictemperature,called orotherdarkparticlessuchastheaxions,andonlyupper asDebyetemperature,T =(cid:126)ω /k . TheDebyemodel D D B limits were reported [11, 12]. In addition, so far, there correctlypredictsthelowtemperaturedependenceofthe 2 heatcapacityofsolidandcoincideswiththeapproaching the Dulong-Petit law at high temperatures. a (a /a ) InEGTanholographicscreenistheclosedareawhere 1 0 is stored the information of the surrounding matter en- 1 ) 0 ) = = caTlnohdseeidhsobclyoongtrshaiedpeshcriercedesnac.sreTtehnheseficrneofeiondrcmoimdaetsdioenwgriitsehecsotdohifefiteNhdeeiwnsytNostnbeimiatns. D (a /a) 10 0.1 aMOND ( aNewton ( equipotential surfaces. 1.0 Acentralargumentoftheholographicprincipleiscon- 1 > a > 0 siderthateachbitofinformationonthescreencarriesan 0.01 0.01 0.1 1 10 100 energy 1/2k T and the number of bits N on the screen a /a B 0 surface is proportional to the area of the screen, and ex- pressed as N = (c3/G(cid:126))A, where G is Newton gravita- FIG.1: Debyefirstfunction,asafunctionoftheacceleration and parametrized as a power function. tional constant. Taking into account the equipartition 0.5 of energy principle, the specific potential energy on the screen can be written as means low temperature regime T/T (cid:28) 1 and Eq.5 re- D 1 (cid:18)T (cid:19) producetheTullyFisherrelationM ∝v4 observedinthe U = 2NkBTD1 TD . (1) galaxiesdynamicandplottedaslogM =4v+log(1/Ga0) [17]. The slope, 4, fall precisely with that observed 0.0 Following the analogy with Debye model, we have sub- in galaxies, whereas the normalization require a = 0 10 20 30 40 50 60 70 0 stituted k T, for k T D (T /T). The main difference 10−10ms−2 [17]. The acceleration a is the Milgrom ac- B B 1 D 0 withtheDebyetheoryisinthatthethirdDebyefunction, celerationparameter[15]andinthislimitEGTcoincides D (x) was replaced by the first Debye function, D (x), with the MOND theory. In this limit the Debye first 3 1 because the information bits located on the screen have function(Eq.2)canberelatedasD (x)=π2/xwithx= 1 only a vibrational state along of the gradient of Newton T /T =2πck T /((cid:126)a)andleadtoa =12ck T /(π(cid:126)). D B D 0 B D potential and it is assumes that the vibrations follows a Finallytheintermediateregion,0<α<1,EGTseems continuous range of frequencies. D is defined as to indicate that is the galaxy cluster region, see section 1 IV. (cid:18)T (cid:19) (cid:18) T (cid:19)(cid:90) TD/T x D D = dx, (2) 1 T T ex−1 D 0 III. INERTIA AT LOW TEMPERATURES The shape of Debye function reflects the Bose-Einstein statistic formula, used in its derivation. The starting point for development of the general the- IF M represents the all mass enclosed by the screen ory of relativity was the equivalence principle, it is also surface, the specific potential energy can be written as valid in the Newtonian gravity. There are strong evi- U = Mc2, and considering that the entropy variation, dences indicating that the equivalence principle holds in ∆S of the screen, happens when a particle of mass m is all experiments at Earth[24]. at a distance ∆X (close to the Compton wave length). Westartingtheanalysis,takenintoaccounttheBeken- The Bekestain entropy variation can be expressed as stein entropy variation expressed in the Eq. 3, and the mc entropic force concept F =T∆S/∆x to obtain ∆S =2πk ∆X. (3) B (cid:126) F =m a=2π[k T D (T /T)]m c/(cid:126). (6) i B 1 D g theserelations,allowstoobtaintheentropicforcedefined as F = T∆S. The more simple case is for a spherical According to the Debye framework, the quantity, kBT ∆x screen of radius R, (A=4π R2), and the acceleration of wassubstitutedbykBT D1(TD/T). InEq.6,Fintheleft the mass m is side represent the force of inertia, that can be expressed as F = m a, where m is the inertia mass, while the (cid:18) (cid:19) i i T GM aD D = . (4) mass, m, in the right side of the equation, represent the 1 T R2 mass of the particle, at a distance equal to Compton wavelength of the holographic screen, when the entropic Considering that T/T ∝a/a (see below), the Debye D 0 force emerges, then it is linked with the gravitational functioncanbeparametrizedbyapowerfunctionoftype mass, m . Here we have two possibility: (a/a )α, as shows in Fig. 1 and Eq.4 becomes g 0 (a) If the equivalence principle holds for all tempera- (cid:18) a (cid:19)α GM ture regions, we have mi =mg and Eq. 6 becomes a = . (5) a0 R2 2πc a= k T D (T /T). (7) (cid:126) B 1 D The two asymptotically cases are: (a) α = 0 and means high temperature regime T/T (cid:29) 1 and the Eq.5 coin- ThisequationwecalledasgeneralizedDUE,athightem- D cides with the Newtonian gravity theory, and (b) α = 1 peratures, T /T (cid:28)1, D (x)(cid:39)1, and coincides with the D 1 3 10-8 Thefactthatm (cid:54)=m formotionsinlowtemperatures kkPPTT D1(TD/T) regimen,canemuliatedagrkmatter. Ifwecalledasmd the 10-9 dark matter mass, it can be obtained as md = mg −mi and taking into account Eq.8, we obtain -1 a (ms)10-10 md = 1 −1; (10) m D (T /T) 10-11 i 1 D This expression coincides with Eq. 8, and means that 10-112E-3 0.01 0.1 1 10 100 we have a duality. However, in both cases, the central T /T D mechanism is the DUE. This suggested that the modi- FIG. 2: Comparison between the conventional DUE, dotted fication of DUE is more fundamental in EGT and the black line and the generalized DUE effect, solid red curve. emulated dark matter is seen as an apparent violation of the equivalence principle. 1 = Newton α = 0 αND < 1α IV. GALAXY CLUSTERS 101 MO 0 < It is well known also that the MOND theory has its mi m / g “Achilles’ heel”. The galaxy cluster, seems to indicate that still is necessary a residual mass. In most cases the 100 MOND critics largely use this to reject MOND. Indeed, the residual mass required by MOND was supply with an exotic neutrino, the “sterile neutrino”, considered as promisingcandidatestohotdarkmatter[27,28]. Sofar, 10-1 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 there is no direct evidence of these neutrinos [29, 30]. a ( m s-2 ) On the other hand, in the central part of clusters the observed acceleration is usually slightly larger than a 0 [31]. This clearly shows the limitations of MOND in FIG. 3: The ratio between the gravitational and inertial cluster analysis. However, there is not this limitation in masses,asafunctionoftheacceleration. Thefigureextended EGTandmeansthatthetemperatureoftheholographic to solar-system scales (each planet is labelled). screen that surrounding clusters enclosed by the screen has a temperature slightly larger than the Debye tem- conventional DUE. Fig. 2 shows a comparison between perature. This means that the clusters analysis requires the generalized and conventional DUE. The discrepancy 0<α<1. for T/T (cid:28) 1 can emulate dark matter. Considering The relationship between various galaxy cluster mass D thatm∝1/aandm ∝1/a fordarkmatterandcalling estimators and X-ray gas temperature agree to within d d as a and a the accelerations according to the modified 40% [32]. Virial theorem mass estimates based on clus- m and conventional DUE, we have 1/a =1/a −1/a and ter galaxy velocity dispersions seem to be accurately re- d m multiplying this last expression by a, we have lated to the X-ray temperature as M ∝Tδ with δ =3/2 [32]. This results are consistent with that predicted by m 1 d = −1. (8) simulations [33]. However, when is combined several in- m D1(TD/T) dependentobservation,i.e.,awiderangeoftemperatures ofgalaxyclusters,seemstoindicateasteeperδ (cid:46)2index For high temperatures T/T (cid:29) 1, D (T /T) = 1 and D 1 D [18]. Even so, in all cases, the greatest discrepancy is in m =0. d normalization, with differences around 40% to 50%. (b)Iftheequivalenceprincipleisviolatedm (cid:54)=m and i g An analysis of galaxy cluster on the basis of entropic keeping the DUE without modification Eq.6 becomes gravity is presented in [22]. However, in this section we mi =D (T /T). (9) presentanalternativestraightforwardanalysisongalaxy m 1 D clusters,onthebasisofEGTwithintheDebyeformalism. g Under the assumption of spherical symmetry and fol- Taking in account Eq. 5 D (T /T)=(a/a )α, the ratio 1 D 0 lowing the Eq. 5, the asymptotic (r → ∞), allow us m /m is plotted in Fig. 3 as function of acceleration. i g calculate the mass of cluster as Following the Fig.3, we can see that the ratio m /m g i for low accelerations increases as the acceleration de- r2 a M(r →∞)= a( )α. (11) creases, the extreme case (strong violation) coincides G a 0 with the MOND theory prediction. While, for high ac- celerations (temperatures), a/a (cid:29)1, and it includes all Theaccelerationinthegravitationalpotentialoftheclus- 0 solar system, the equivalence principle remains valid. ter is related by a=C2dlnρ /dr, with C2 =k T /νm . s x s b x p 4 The red dots line in Fig. 4, represent the prediction M (M)bSun 1213141510 10 10 10 δfαorfo==Emαq0M..+916O12Na,=nDfdo1rn.a9αo06r.∼Vm=T.a10lh0i.ze9−Ca61ntO0ioamoNrnnmsdC−baLey1lxi.UzapaSrtfeIiaosOcsnteNodfroSaras10t−hMαis ∝c=asT0e.Xδ4d0iwff,eiftorhsr EGT within the Debye formalism can emulate dark matter of two equivalent ways, The first case requires 100 101 T (keV) modification of the DUE in order to maintain the equiv- x alenceprincipleandthesecondonerequiresmodification FIG.4: ThemassX-raytemperaturerelationforgalaxyclus- of equivalence principle in order to maintain the DUE. ters(graytriangles[34,35])andgroupsofgalaxies(greentri- In both cases the DUE is the fundamental mechanism. angles[27]). ThedashedlineindicatestheexpectedinΛCDM This means that the modification of the DUE emulate [33]. The solid line indicates the prediction of MOND [18], dark matter which in turn emulates a violation of the (α=1),andthereddotlineisthepredictionofEGTwithin equivalence principle. This scheme is promising, because the Debye formalism to α=0.96. recently results [36], on the SPARC database of galax- ies [37], seems indicate a challenge to the dark matter The density distribution ρ is well described by the hypothesis. x “β−model(cid:48)(cid:48), whose asymptotic expression is ρ ∼r−3β, TheEGTwithintheDebyeformalismisalsopromising x where β has a typical value of ∼ 2/3 [26]. Under these in the analysis of galaxy clusters. However, there is also conditions the Eq. 11 becomes aseconddarkentitytoconsider,thedarkenergy,respon- sibleoftheacceleratingexpansionofUniverse. Thedark r1−α energy in EGT is discussed in [22, 38]. Even so, there is M(r →∞)= C2(1+α)(−3β)1+α. (12) Gaα s more a complication, the increase in a factor of 10 of the 0 numberofsupernovaeIA,inrelationtothefirstanalyses, For α = 1 we have the MOND prediction to the seems to indicate that the accelerated expansion signal mass-Xray temperature relation of clusters, expressed as is only marginal [39]. M ∝T2. This relation is represented by the black solid We believe that as more information is gathered, we X line in Fig. 4 [18], where the mass function of an X-ray will have more conditions to test the EGT within the fluxofseveralsamplesofgalaxyclustersareplotted. We Debye formalism, in other complex systems. can see that data is closer with MOND’s predicted slope thanthatprevisionedbystandardΛCDMandexpressed as M ∝ T3/2 [33] and represented by the dashed black Acknowledgments X line in Fig. 4, and it is better than MOND only at high X-ray temperatures. 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