Generalized Bekenstein Hawking system: logarithmic correction

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1 ur. Phys. J. C (014) 74:876 DOI /epjc/s Regulr Article - Theoreticl Physics Generlized Bekenstein Hwking system: logrithmic correction Subenoy Chkrborty Deprtment of Mthemtics, Jdvpur University, Kolkt, Indi Received: 19 November 013 / Accepted: 8 April 014 / Published online: 3 June 014 The Author(s) 014. This rticle is published with open ccess t Springerlink.com Abstrct The present work is generliztion of the recent work [rxiv ] on the modified Hwking temperture on the event horizon. Here the Hwking temperture is generlized by multiplying the modified Hwking temperture by vrible prmeter α representing the rtio of the growth rte of the pprent horizon to tht of event horizon. It is found tht both the first nd the generlized second lw of thermodynmics re vlid on the event horizon for ny fluid distribution. Subsequently, the Bekenstein entropy is modified on the event horizon nd the thermodynmicl lws re exmined. Finlly, n interprettion of the prmeters involved is presented. 1 Introduction In blck hole physics semi-clssicl description shows tht blck hole behves s blck body emitting therml rdition with temperture (known s the Hwking temperture) nd entropy (known s the Bekenstein entropy) proportionl to the surfce grvity t the horizon nd re of the horizon [1,], respectively. Further, this Hwking temperture nd the Bekenstein entropy re relted to the mss of the blck hole through the first lw of thermodynmics [3]. Due to this reltionship between the physicl prmeters (nmely, entropy nd temperture) nd the geometry of the horizon, there is nturl specultion bout the reltionship between blck hole thermodynmics nd the instein field equtions. A first step in this direction ws put forwrd by Jcobson [4], who derived the instein field equtions from the first lw of thermodynmics: δq = T ds for ll loclly Rindler cusl horizons with δq nd T s the energy flux nd Unruh temperture mesured by n ccelerted observer just inside the horizon. Subsequently, Pdmnbhn [5,6] on the other side ws ble to derive the first lw of thermodynmics on e-mil: schkrborty.mth@gmil.com; schkrborty@mth.jdvu.c.in the horizon strting from the instein equtions for generl sttic sphericlly symmetric spce-time. This ide of equivlence between instein field equtions nd the thermodynmicl lws hs been extended in the context of cosmology. Usully, the universe bounded by the pprent horizon is ssumed to be thermodynmicl system with Hwking temperture nd the entropy s T A = 1, π S A = π (1) G where is the rdius of the pprent horizon. It ws shown tht the first lw of thermodynmics on the pprent horizon nd the Friedmnn equtions re equivlent [7]. Subsequently, this equivlent ide ws extended to higher dimensionl spce-time, nmely grvity theory with Guss Bonnet term nd Lovelock grvity theory [7 10]. It is presumed tht such n inherent reltionship between the thermodynmics t the pprent horizon nd the instein field equtions my led to some clue on the properties of drk energy. Although the cosmologicl event horizon does not exist in the usul stndrd big bng cosmology, in the perspective of the recent observtions [11 16], the universe is in n ccelerting phse dominted by drk energy (ω d < 1/3) nd the event horizon distinct from the pprent horizon. By defining the entropy nd temperture on the event horizon similr to those for the pprent horizon (given bove) Wng et l. [17] showed tht both the first nd the second lw of thermodynmics brek down on the cosmologicl event horizon. They justified it rguing tht the first lw is pplicble to nerby sttes of locl thermodynmic equilibrium, while the event horizon reflects the globl fetures of spcetime. As result, the thermodynmicl prmeters on the non-equilibrium configurtion of the event horizon my not be s simple s on the pprent horizon. Further, they speculted tht the region bounded by the pprent horizon my be

2 876 Pge of 7 ur. Phys. J. C (014) 74:876 tken s the Bekenstein system, i.e., the Bekenstein entropy or mss bound, S < R, nd the entropy or re bound, S < A/4, re stisfied in this region. Now due to universlity of the Bekenstein bounds nd s ll grvittionlly stble specil regions with wek self-grvity should stisfy the bove Bekenstein bounds, the corresponding thermodynmicl system is termed Bekenstein system. Further, due to the rdius of the event horizon being lrger thn the pprent horizon, Wng et l. [17] termed the universe bounded by the event horizon non-bekenstein system. In the recent pst there were published series of works [18 3] investigting the vlidity of the generlized second lw of thermodynmics of the universe bounded by the event horizon for instein grvity [18,19] nd in other grvity theories [18 1] nd for different fluid systems [18,19,,3] (including drk energy [19,,3]). In these works the vlidity of the first lw of thermodynmics on the event horizon ws ssumed nd it ws possible to show the vlidity of the generlized second lw of thermodynmics with some resonble restrictions. However, the vlidity of the first lw of thermodynmics on the event horizon ws still followed by question mrk. Very recently, the uthor [4] ws ble to show tht the first lw of thermodynmics is stisfied on the event horizon with modified Hwking temperture for two specific exmples of single D fluids. The present work is further extension of it. Here, by generlizing the Hwking temperture, or modifying the Bekenstein entropy, it is possible to show tht both the first nd the generlized second lw of thermodynmics (GSLT) re lwys stisfied on the event horizon. The pper is orgnized s follows: Sect. dels with bsic equtions relted to erlier works. The thermodynmicl lws with generlized Hwking temperture nd modified Bekenstein entropy re studied, respectively, in Sect. 3 nd in Sect. 4. The interprettion of the prmeters involved in generlized Hwking temperture nd modified Bekenstein entropy hs been nlyzed in Sect. 5. Finlly, summry of the work nd possible conclusions re presented in Sect. 6. Bsic equtions nd erlier works The homogeneous nd isotropic FRW model of the universe cn loclly be expressed by the metric ds = h ij (x i )dx i dx j + R d () where i, j cn tke vlues 0 nd 1, nd the two dimensionl metric tensor h ij, known s the norml metric, is given by h ij = dig( 1, /1 kr ) (3) with x i being the ssocited co-ordintes (x 0 = t, x 1 = r). R = r is the re rdius nd is considered s sclr field in the norml D spce. Another relevnt sclr quntity on this norml spce is χ(x) = h ij (x) i R j R = 1 (H + k/ )R (4) where k = 0, ±1 stnds for flt, closed or open model of the universe. The Friedmnn equtions re H + k/ = 8 Gρ (5) 3 nd Ḣ k/ = 4 G(ρ + p) (6) where the energy density ρ nd the thermodynmic pressure p of the mtter distribution obey the conservtion reltion ρ + 3H(ρ + p) = 0 (7) Usully, the pprent horizon is defined t the vnishing of the sclr i.e. χ(x) = 0, which gives = 1 H + k/. (8) Now the surfce grvity on the pprent horizon is defined s χ κ A= 1 R R= = 1. (9) So the usul Hwking temperture on the pprent horizon is given by (s in (1)) T A = κ A = 1. (10) It hs been shown by Wng et l. [17] nd others [7 9,5,6] tht universe bounded by the pprent horizon (with prmeters given by (1)) is thermodynmicl system stisfying both the first nd the second lw of thermodynmics, not only in instein grvity but lso in ny other grvity theory nd lso for bryonic s well s for exotic mtter. On the other hnd, the difficulty strts from the very definition of the event horizon. The infinite integrl in the definition R = t (11) converges only if t α with α>1 i.e. the event horizon does not exist in the decelerting phse, it hs only relevnce in the present ccelerting er. In the literture, the Hwking temperture on the event horizon is usully tken similr to the pprent horizon (replcing by R )s(see(10)) T = 1. (1) R This choice is lso supported from the mesurement of the temperture by freely flling detector in de Sitter bckground (where the two horizons coincide) using quntum

3 ur. Phys. J. C (014) 74:876 Pge 3 of field theory [7]. But unfortuntely, with this choice of temperture nd the entropy in the form of Bekenstein, i.e., T = 1, S = R R G, (13) the universe bounded by the event horizon is not relistic thermodynmicl system s both thermodynmicl lws fil to hold there [17]. Recently, the surfce grvity on the event horizon hs been defined similr to tht on the pprent horizon (see (9)) by [4] κ = 1 χ R R=R = R R, (14) A nd s result the modified Hwking temperture on the event horizon becomes T m = κ = R R, (15) A which for the flt FRW model (i.e. k = 0) becomes T = H R. (16) As the two horizons re relted by the inequlity < R, (17) we lwys hve T A < T. (18) Using this modified Hwking temperture the uthor [4] hs been ble to show the vlidity of the first lw of thermodynmics on the event horizon for two specific single fluid D models. 3 Generlized Hwking temperture nd thermodynmicl lws In this section, to proceed serching for generl prescription, we strt with generliztion of the modified Hwking temperture in the form T g = αr R (19) A where the dimensionless prmeter α is to be determined so tht α = 1 on the pprent horizon. The mount of energy flux cross horizon within the time intervl is [7,8] d h = 4 R h T bk k b, (0) with k null vector. So for the event horizon we get d = 4 R 3 H(ρ + p). (1) Now using the instein field q. (6) nd the definition of the pprent horizon (i.e. (8)), the bove expression for the energy flux simplifies to ( ) 3 R R A d =. () G From the Bekenstein entropy re reltion (see (13)) we hve ( ) R R T ds = α. (3) G Hence for the vlidity of the first lw of thermodynmics i.e. d = dq = T ds, (4) we hve R α = A /. (5) /R Thus reciprocl of α gives the reltive growth rte of the rdius of the event horizon to the pprent horizon. For the generlized second lw of thermodynmics, we strt with the Gibbs lw [17,9] to find the entropy vrition of the bounded fluid distribution: T f ds f = d + pdv, (6) where T f nd S f re the temperture nd entropy of the given fluid distribution, respectively, V = 4 R 3 /3 nd = ρv. The bove eqution explicitly tkes the form T f ds f = 4 R (ρ + p)( HR ). (7) Also using the first lw (i.e. (4)) we hve from (1) T ds = 4 R 3 H(ρ + p). (8) Now for equilibrium distribution, we ssume T f = T g, i.e., the inside mtter hs the sme temperture s the bounding surfce nd we obtin T ds T = 4 R (ρ + p), (9) with S T = S + S f, the totl entropy of the universl system. Agin using the instein field q. (6), the conservtion reltion (7), nd (8) we hve on simplifiction T g ds T = ( R ) R GH. (30) Now using the generlized Hwking temperture (19)the time vrition of the totl entropy becomes ds T = GH R, (31) which is positive definite for n expnding universe nd hence the generlized second lw of thermodynmics lwys holds on the event horizon.

4 876 Pge 4 of 7 ur. Phys. J. C (014) 74:876 4 Modified Bekenstein entropy nd thermodynmicl lws In the previous section we hve generlized the Hwking temperture, keeping the Bekenstein entropy re reltion unchnged nd we re ble to show the vlidity of both the first lw of thermodynmics nd GSLT on the event horizon, irrespective of ny fluid distribution nd we my term this universe bounded by the event horizon generlized Bekenstein system. However, it is possible to hve two other modifictions of the entropy nd temperture on the event horizon s follows. () S (m) (b) S (m) = βs (B), T = T (m), nd = δs (B), T = 1 δ T (m). We shll now exmine the vlidity of the thermodynmicl lws for these choices. () S (m) Here S (m) = βs (B) nd T = T (m). nd S (B) re, respectively, the modified entropy nd the usul Bekenstein entropy on the event horizon, T (m) is the modified Hwking temperture on the event horizon (given by (15)or(16)) nd β is prmeter hving vlue unity on the pprent horizon. Then s before from the vlidity of the Clusius reltion β cn be determined s β = R R d. (3) Thus for this choice of β the bove modified entropy nd modified Hwking temperture stisfy the first lw of thermodynmics on the event horizon. Now we shll exmine the vlidity of the generlized second lw of thermodynmics (GSLT) on the event horizon for this choice of entropy nd temperture on the horizon. Proceeding s before (ssuming tht the temperture of the inside fluid is the sme s the modified Hwking temperture for thermodynmicl equilibrium), we hve ds T = GH ( R ) A. (33) Thus vlidity of GSLT depends on the evolution of the two horizons (pprent nd event) if both horizons increse or decrese simultneously the GSLT is lwys stisfied. However, s long s the wek energy condition (WC) is stisfied A > 0 nd > 0ifR > nd GSLT is stisfied. But if WC is violted then A < 0 nd GSLT will be stisfied only if < 0 i.e. R <, which my be possible only in the phntom er. Hence for this choice of entropy nd temperture on the event horizon GSLT is lwys stisfied s long s WC is stisfied nd when WC is violted then GSLT will be vlid if R <. Further, it should be noted tht if we choose the temperture on the event horizon s the generlized Hwking temperture (i.e. T g ) then β turns out to be unity i.e. we get bck the previous generlized Bekenstein system (in Sect. 3). (b) S (m) = δs (B), T = 1 δ T (m). As before the prmeter δ should be unity on the pprent horizon to mtch with the Bekenstein system. Agin for the vlidity of the first lw of thermodynmics (i.e. the Clusius reltion) δ turns out to be R A /R nd s result the entropy on the event horizon becomes constnt (equl to tht t the pprent horizon). So this choice of entropy temperture is not of much physicl interest. 5 Interprettion of the prmeters α nd β (I) α-prmeter. In this section we shll try to find some implictions of the fctor α. Note tht α cn be termed the rtio of the expnsion rte of the two horizons. If we compre the expnsion rte of the expnding mtter with tht for both horizons, we hve H = 1 ( ) 3ω + 1, H = H (34) R R where the mtter in the universe is chosen s brotropic fluid with eqution of stte p = ωρ. As n event horizon exists only for the ccelerting phse we hve ω< 1 3. Hence both the horizons expnd slower thn comoving. So the expnsion rte of both the horizons coincide (i.e. α = 1) when HR = (3ω + 1). (35) Before proceeding, we present comprtive chrcteriztion of the two horizons in Tble 1. We shll now try to estimte the prmeter α for some known fluid systems. () Perfect fluid with constnt eqution of stte ω(< 1 3 ). For flt FRW model, the cosmologicl solution is (t) = 0 t [ 3(ω+1) ], i.e. H(t) = (36) 3(ω + 1)t nd 3(1 + ω) R (t) = t. (37) (1 + 3ω) Hence HR = (38) (1 + 3ω)

5 ur. Phys. J. C (014) 74:876 Pge 5 of Tble 1 A comprtive study of the horizons Horizon Loction or definition Cusl chrcter Velocity Accelertion Apprent horizon vent horizon = 1 H + κ R = t Time like if 1 <ω< 1 3,Nullif ω = 1or 1 3, Spce like if ω< 1orω > HR A (ρ + p) = 3 (1 + p ρ ) 9H (1 + p ρ )( p ρ ṗ ρ ) (t ) Null HR 1 H(1 + qhr ) i.e. (33) is identiclly stisfied for ll ω(< 1 3 ). So for perfect fluid with constnt eqution of stte (< 1 3 ) we lwys hve α = 1 nd hence the expnsion rtes of the two horizons re identicl throughout the evolution. Thus the universe bounded by the event horizon with modified/generlized Hwking temperture (given by (15)/(19)) is Bekenstein system nd it supports the results in [4]. (b) Intercting hologrphic drk energy fluid. We shll now study n intercting hologrphic drk energy (HD) model tht consists of drk mtter in the form of dust (of energy density ρ m ) nd HD in the form of perfect fluid: p d = ω d ρ d. The interction between them is chosen s 3b H(ρ m + ρ d ) with b the coupling constnt. If R is tken s the I.R. cutoff, then the rdius of the event horizon (R ) nd the eqution of stte prmeter ω d re given by [30] R = nd c d H ω d = 1 3 d 3c (39) b d (40) where d = ρ d /(3H ) is the density prmeter for the drk energy nd the dimensionless prmeter c crries the uncertinties of the theory nd is ssumed to be constnt. In this cse (33) is modified s HR = (41) (1 + 3ω t ) with p d ω t = (ρ m + ρ d ) = ω d d. (4) We shll now exmine whether for this model reltion (39) is stisfied or not. Using (37), (38), nd (40)in(39) we obtin cubic eqution in x(= d ), x 3 + cx x (1 b )c = 0. (43) This cubic eqution hs positive root (x p )ifb < 1(the other two roots re either both negtive or pir of complex conjugtes). In the Tble we present the vlue of x for different choices of b nd c within the observtionl bounds: Tble Vlue of x for different vlues of c nd b from (43) c b d x The tble shows tht for intercting D fluids it is possible to hve identicl expnsion rtes (i.e. α = 1) for the two horizons within the observtionl limit of d nd c. (II) β-prmeter. To interpret the prmeter β we consider therml fluctutions of the pprent horizon so tht the re chnges by n infinitesiml mount, i.e., A (m) = A + ɛ; then the entropy nd temperture of the modified pprent horizon cn be written s (from the choice ()) S (m) = βs B, T = T (m). Now the modified rdius of the pprent horizon is relted to the originl rdius s (in the first pproximtion) R (m) = R + ɛ, R ɛ = ɛ 4, (44) nd β pproximtes β = 1 ɛ R + ɛ R lnr. (45) Hence we hve S (m) nd = S (B) + ɛ G lnr (46) T = 1 + ɛ R R 3. (47) Thus there is logrithmic correction to the Bekenstein entropy, nd the Hwking temperture is corrected by term proportionl to R 3 due to this therml fluctution. However, if we consider the infinitesiml chnge in the rdius of the pprent horizon due to the therml fluctution, i.e. R (m) = R + ɛ, then the modified entropy nd temperture on the horizon become

6 876 Pge 6 of 7 ur. Phys. J. C (014) 74:876 S (m) = S (B) + 4 ɛ R G, T = T H + ɛ R, (48) i.e., the correction to the Bekenstein entropy is proportionl to the rdius of the horizon, nd tht of the temperture is proportionl to the inverse squre of the rdius. 6 Summry nd concluding remrks In this work we hve studied thermodynmicl lws on the event horizon for the following three choices of entropy nd temperture on the event horizon: (1) S = S (B), T = T (g), () S = βs (B), T = T (m) (3) S = δs (B),, T = 1 δ T (m) where S (B) nd T (m) re, respectively, the usul Bekenstein entropy nd modified Hwking temperture (given in (15)or (16)) nd the prmeters α, β, nd δ re evluted using the Clusius reltion. It is found tht the prmeters tke vlue unity on the pprent horizon so tht ll the three choices reduce to the Bekenstein Hwking system on the pprent horizon. However, for the third choice the entropy on the event horizon turns out to be constnt (equl to tht on the pprent horizon) nd hence it is not of much physicl interest. So we hve not discussed it further. On the other hnd, both thermodynmicl lws hold on the event horizon unconditionlly for ny fluid distribution for the first choice, while for the second choice of entropy nd temperture on the event horizon we must hve R < in the phntom er for the vlidity of the GSLT. Hence we cll universe bounded by the event horizon (for the bove two choices of entropy nd temperture) generlized Bekenstein Hwking thermodynmicl system. Also some interprettions of the prmeters α nd β hve been presented in Sect. 5. Lstly, if in the present model of the universl thermodynmics there is therml equilibrium with CMB photons then the temperture of the horizon must coincide with the CMB temperture (.73K ) tody [31,3]. Now restoring the dimension, the temperture of the event horizon (see (19)) cn be written (in Kelvin) s [3] T (g) = (1 + q)h 3 R 3 ( ) 1 hc (49) HR 1 π R k B where q = (1+Ḣ/H ) is the usul decelertion prmeter; h = erg s, c = cm/s, nd k B = erg/k re, respectively, the Plnck constnt, the speed of light, nd the Boltzmnn constnt. In prticulr, if we choose the cosmic fluid s hologrphic drk energy, then from (39) we get HR = c/ d nd we hve T (g) (1 + q)c3 = d (c 0.3 K. (50) d ) π R Now using the observed vlues of c,ω d, q, nd choosing R ppropritely, it is lwys possible to mtch T (g) with the temperture of the CMB photons. Finlly, the conclusions re point-wise presented below. I. The universe bounded by the event horizon (generlized Bekenstein Hwking system) is relistic thermodynmicl system where both thermodynmicl lws hold for ny mtter system within it. II. In deriving the thermodynmicl lws we hve used the second Friedmnn q. (6) nd the energy conservtion reltion (7). On the other hnd, ssuming the first lw of thermodynmics it is possible to derive the instein field equtions. So we my conclude tht the first lw of thermodynmics nd the instein field equtions re equivlent (i.e. one cn be derived from the other) on the event horizon irrespective of ny fluid distribution. III. The generlized Bekenstein Hwking system i.e. universe bounded by the event horizon supports the recent observtions, i.e., the results of the present work re comptible (qulittively) with the present observed. IV. If due to some therml fluctutions the pprent horizon is modified so tht its re chnges infinitesimlly then up to the first order of pproximtion the Bekenstein entropy is corrected by logrithmic term nd the correction to the Hwking temperture is proportionl to the inverse cube of the rdius of the pprent horizon. V. The horizon temperture cn be in therml equilibrium with CMB photons by n pproprite choice of the prmeters involved. For future work one my consider the following issues. (i) Do we hve vlidity of the thermodynmicl lws on the event horizon for other grvity theories? (ii) Is this generlized Hwking temperture or the modified Bekenstein vlid for other horizons (if it exists) of the universl thermodynmicl system? (iii) Further, wht re the physicl nd geometricl implictions of the prmeters α nd β? (iv) Is the present generlized Bekenstein Hwking system, i.e., S = S (B), T = T (g) or S = βs (B), T = T (m) or some other modified version on the event horizon physiclly more relistic? Acknowledgments The work is done during visit to IUCAA under its ssociteship progrmme. The uthor is thnkful to IUCAA for wrm hospitlity nd fcilities t its Librry. The uthor is thnkful to UGC- DRS progrmme in the Dept. of Mthemtics, Jdvpur University.

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