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Black hole entropy in modified gravity models F. Briscesea∗ and E. Elizaldeb† a Dipartimento di Modelli e Metodi Matematici and GNFM Università degli Studi di Roma “La Sapienza”, Via A. Scarpa 16, I-00161, Roma, Italy b Institut de Ciències de l’Espai (CSIC) & IEEC-CSIC, Campus UAB Facultat de Ciències, Torre C5-Parell-2a pl, E-08193 Bellaterra (Barcelona) Spain Abstract An analysis of some modified gravity models, based on the study of pure Schwarzschild and of Schwarzschild-deSitterblack holes, and involvingtheuse of theNoethercharge method, is carried out. Corrections to the classical Einsteinian black hole entropy appear. It is shown explicitly how thecondition of positive entropycan beused in ordertoconstrain theviability of modified gravity 8 theories. 0 0 PACSnumbers: 04.50.+h,95.36.+x,98.80.-k,95.30.Tg 2 n I. INTRODUCTION. ported to pass all solar system tests; in addition, they a exhibitanumberofveryinterestingfeatures. In[13]pos- J sible Newton law corrections to such models have been 3 Increasinginterestis attractedby modified versionsof considered. 1 general relativity [1]. They have been proposed as seri- We present here an analysis of the models above ous alternatives to Einstein’s theory of gravitation, and ] could be used to describe more accurately the observed based on the study of pure Schwarzschild and also h Schwarzschild-de Sitter black holes (SBH, SdSBH), cal- t accelerated expansion of our universe [2, 3, 4, 5, 6]. In - culated through use of the Noether charge method. We addition,ithasbeenshown[7]thatitisactuallypossible p start with a discussion of two examples considered in e to reconstruct the explicit form of the postulated curva- [6, 14] and go over to study more recent ones [9, 11, 12]. h ture function f(R), from the universe expansion history. [ The directconfrontationofbasic quantities,asthe black Itisquitewellknownthatmodifiedversionsofgeneral hole entropy, with the well established, classical (Ein- 2 relativity are mathematically equivalent to scalar fields steinian) result can offer a further insight into the con- v models (see e.g. [1]), meaning that a solution in a modi- structionofageneralf(R)theory. Westartwithashort 2 fied gravitymodel canalwaysbe mapped into a solution review of modified f(R) gravity and the Noether charge 3 of the corresponding scalar field theory. In spite of this 4 method to compute the BH entropy and then calculate mathematical correspondence, physical equivalence does 0 the BH entropy for the models in Refs. [2, 7]. After notalwaysfollow. Infact,twocorrespondingsolutionsof . that, we extend our analysis to the models [9, 11, 12], 8 twoequivalenttheoriescanactuallyexhibitratherdiffer- 0 taking due care of the sign of the BH entropy and also ent physical behaviors. Furthermore, it is not necessary, 7 discussingthestabilityconditionsaswellastheexistence in order to justify modified gravity, to do it by always 0 of a Schwarzschild BH solution. We conclude by provid- : using this relation with scalar field theories. Because of ingabriefcomparisonofthedifferentmodelsconsidered. v the new situation, in the following we will disregardthis i X mathematical equivalence and do consider in our analysis modified gravity as an independent theory aiming r a directly at some measurable physical properties. What II. BLACK HOLE ENTROPY IN MODIFIED is more,inourtreatmentmodifiedgravitywill infactbe GRAVITY. viewed just as a different classical theory of gravitation. Although other models have been considered [8] with The action for f(R) gravity (see e.g. [1] for a review) the Gauss-Bonnet scalar in the action, here we shall re- is strict our attention to pure f(R) models. As often discussed, there are limitations on the function f(R) when J = 1 d4x√ g = 1 d4x√ g(R+f(R)), (1) trying to construct a theory which is in agreement with k2 − L k2 − Z Z the very precise solar system tests carried out so far, as with f(R) a generic function. As discussed in [2], in well as with all the known cosmological bounds [3]. Re- order to give rise to a realistic cosmology this function cently, different models of that kind have been studied needstofulfillsomelimitingconditions. However,forthe [2, 7, 9, 11, 12], the last three of them having been re- moment we can ignore them, because they do not affect the considerations which will follow. The equations of motion for this theory, in the presence of matter, are ∗†EE--mmaaiill:: eblriizsacledsee@@ideemcm.umab..uensiroma1.it (1+f′(R))Rµν − 21(R+f(R))gµfν′(+Rg)µ=ν(cid:3)kf2T′(R), (2) µ ν µν −▽ ▽ 2 whereT isthematterstressenergytensor. Contracting III. BLACK HOLE ENTROPY FOR TWO µν the indices in the last equation, we obtain the relation MODIFIED GRAVITY MODELS WITH NO R=0 SOLUTIONS. R(f′(R) 1) 2f(R)+3(cid:3)f′(R)=k2Tµ. (3) − − µ Since we areinterestedin the study ofthe Schwarzschild In order to illustrate the method with explicit exam- solution (in the case when there is a R = 0 solution) or plesofentropycalculation,weanalyzeheretwomodified either in Schwarzschild-De Sitter black holes, we restrict gravity models that appeared some time ago and which ourreasoningto the caseofmetric tensorswithconstant have been quite successful up to now. In those models scalar curvature in the vacuum. In that case, we simply no R = 0 solution occurs. The first one, introduced in have [15] [6], is given by R0(f′(R0) 1) 2f(R0)=0. (4) f(R)= a(R Λ1)−n+b(R Λ2)m, (10) − − − − − For completeness, let us recall that, in order to build with m,n,a,b > 0. The condition to obtain a SdSBH a realistic modified gravity, f(R) needs satisfy the two (namely, 2f(R0)=R0(f′(R0) 1)) leads to − conditions: R0 an(R0 Λ1)−n−1+bm(R0 Λ2)m−1 1 lim f(R)=const, lim f(R)=0, (5) − − − R→∞ R→0 h =2 a(R0 Λ1)−n+b(R0 Λ2)m , i(11) The first condition corresponds to the existence of an − − − effective cosmological constant at high curvature. The h i so that we have for the entropy second one allows for vacuum solutions, as for example MinkowskiorSchwarzschildspace-times. Then,although S = AH 1+na(R0 Λ1)−n−1+mb(R0 Λ2)m−1 . an effective cosmological constant exists, vacuum solu- 4G − − tionarepreservedanditislegitimatetostudy themalso h (1i2) in a large scale universe with nonzero R. Moreover, in Thus the SdSBH entropy is positive for all R0 >Λ1,Λ2. order to give rise to stable solutions [16], the following The second model, studied in [14], is defined by additional condition needs to be fulfilled: f(R)=αln(R/µ2)+βRm. (13) [1+f′(R)]/f′′(R)>R. (6) This modified gravity model does not admit vacuum so- We can now consider the Schwarzschild-De Sitter met- lutions,thuswecancalculatetheentropyfortheSdSBH. ric,asphericallysymmetricsolutionof(2)withconstant The Ricci scalar is such that it satisfies the relation curvature R0 (see [17]) 2αln(R0/µ2)+β(2 m)R0m α=0. (14) ds2 =a(r)dt2 dr2/a(r) r2dΩ, (7) − − − − The SdSBH entropy is given by where a(r)=1 2m/r R0r2/12and R0(f′(R0) 1) − − − − A 2fW(Re0)w=ill0u.se the Noether charge method, as discussed S = 4GH 1+α/R0+βmR0m−1 , (15) in [8, 18], in order to calculate the entropy for the and turns out to be (cid:0)positive for all values o(cid:1)f R0 >0. Schwarzschild-De Sitter BH. The entropy formula reads ∂ S =4π √ g L, (8) ZS2 − ∂R GIRVA.VIBTLYAMCKODHEOLLSETEHNATTRCOOPMYPINLYMWOIDTIHFITEDHE and the integration is made on the external horizon of SOLAR SYSTEM TESTS. events surface. In the case of constant curvature, for a generic modified theory, the result is We now analyze three recent models [9, 11, 12] which S =[1+f′(R0)]AH/4G, (9) havebeenproventocomplywiththesolar-systemaswell as with other cosmologicalparameter constraints. Their where AH is the area of the BH horizon. This enables respective authors have given a complete discussion of ustocalculatethe BHentropyforagenericf(R)theory. each model, taking care to provide a range for the free Wemustherestressthefactthattherequirementofpos- parameterscontainedinthef(R)function,andhavealso itive black hole entropy simply avoids the appearance of produced stable solutions. Here we just want to stress, ghost or tachyon fields in the corresponding scalar field with the help of these examples, how the correspond- theory. Then a negative entropy is simply a footprint of ing BH entropy calculation offers a further tool in order some instabilities in the Einstein frame. What is new in to confront each of those modified gravity theories with this picture is that we do not need to involve the (math- Einstein’sgeneralrelativity,giventhe fact thatthe pres- ematical) equivalence of these models in order to give a enceofsphericallysymmetricBHsolutionsisanecessary physically meaningful interpretation of such constrain. element of all local tests. 3 A. The Hu-Sawicki model. WecanthereforeconsidertheSdSBHsolutions,withcur- vature R0 given by In this model [9] (with n,c1,c2 >0) we have R0 2nλ(R0/C) 1+(R0/C)2 −n−1+1 R n f(R)=−m2c1c2 (cid:0)mRm22(cid:1)n+1. (16) n =−2λC (cid:2)1+(R0/C)2(cid:3)−n−1 ,o (22) h(cid:0) (cid:1) i In [9], m2 is chosen such that(cid:0), at(cid:1) cosmological scale, that satisfies also (21). In this case, the entropy is just R >> m2 at the present epoch, and f(R) satisfies the condition f′′(R) > 0 for R >> m2. This also ensure S = AH 1 2nλ(R0/C) 1+(R0/C)2 −n−1 . (23) that solutions with R >> m2 are stable. Moreover the 4G − requirementthatc1/c22 0atfixedc1/c2 givesacosmo- Therefore, nin this case a n(cid:2)on trivial cor(cid:3)rectioon of the → logical constant, in both cosmological and local tests of SdSBH entropy is found. We just stress the fact that gravity. In spite of this fact, since f(0) = 0, this theory the SBH solutions have classical entropy but are unsta- admits the Schwarzschild solution (i.e. R = 0). By the ble, and that the SdSBH ones have a modified entropy way we note that the stability condition for the vacuum which,underthelimitationsstatedin[11],isstrictlypos- solution is not satisfied unless n = 1 and 1 c1 > 0. itive. − Therefore, except of this case, vacuum solutions (than also SBH) are unstable. Note also that n = 1 corresponds to a Lagrangian L = R(1 c1)/k2 for small − C. The Appleby-Battye model. R, so it is associated with a correction to the gravita- tional coupling constant for small R, giving an effective G =G/(1+f′(0)) (see [10]). Here f(R) is given by [12] eff. The entropy formula gives for the SdS metric f(R)= R/2+log[cosh(aR) tanh(b)sinh(aR)]/2a − − (24) S(R0)= A4GH 1−nc1 c2(cid:0)mmRR0022(cid:1)nn−+11 2. (17) and f′(R)=[ 1+tanh(aR b)]/2. (25)   − −  (cid:16) (cid:0) (cid:1) (cid:17)  The entropy for the Schwarzschild solution is This model admits a SBH solution. The entropy for the SdSBH is simply S(0)=(1−c1)4AG, for n=1; S(0)= 4AG, for n>1, as in the Einstein theory; (18) S =[1+tanh(aR0−b)]AH/8G. (26) S(0)= , for 0<n<1,c1 >0. −∞ We can use the stability condition given in [12], aR0 − Then, inthe only stable case,with n=1 and1 c1>0, b>>1, to obtain − a correction to the classical Einstenian BH entropy is found. From (17) it also follows that, for SdS BH with S AH/4G. (27) ≃ R0 >>m2, the entropy is positive and corrections to its Einstenian value are of order (m2/R0)n+1. oFvoerrt,hbeeSinBgHf,′′t(hRe)st>ab0ilfiotyraclolnRd,itiinonthiissjmusotdbe<lt<he0v.aMcuourme- solutions are always stable and there are no substantial corrections to the classical result. B. The Starobinsky model. In this model f(R) is [11] D. Comparison of the behavior of f(R) for the different models. f(R)=λC 1+(R/C)2 −n 1 , (19) − n(cid:2) (cid:3) o It is interesting to put together all three models and from where explicitly compare the behavior of the function f(R), in f′(R)= 2nλ(R/C) 1+(R/C)2 −n−1. (20) particular, the stability of the Euclidean limit and the − asymptotic behavior at large curvature. To simplify the Note that in this case f′′(0(cid:2)) < 0 an thu(cid:3)s, although this comparison, we do not play with the values of the dif- modeladmitsaSBHsolution,itisunstabletogetherwith ferent parameters and set all coefficients equal to 1 and all its vacuum solutions. In [11] the author limits his the curvature powers equalto 2 or 4. For the case of the analysis to solutions that satisfy the following stability Hu-Sawicki model, with conditions x4 1+f′(R)>0, f′′(R)>0, 1+f′(R)>Rf′′(R) (21) fHS(x)=−1+x4, (28) 4 fHS fAB R(cid:144)m2 aR 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 -0.1 -0.2 -0.2 -0.4 -0.3 -0.6 -0.4 -0.8 -0.5 -1 -0.6 FIG. 1: Plot of the function f(R) for the Hu-Sawickimodel. FIG. 3: Plot of the function f(R) for the Appleby-Battye model. f V. BH ENTROPY IN A NEW MODEL THAT S R(cid:144)R UNIFIES INFLATION AND COSMIC 0 0.5 1 1.5 2 2.5 3 ACCELERATION. -0.2 Very recently, a modified gravitymodel has been pub- -0.4 lished [20], that unifies inflation and cosmic acceleration -0.6 under the same picture and also complies with the solar system tests. It is -0.8 R R2n x -1 f(R)=−f0Z0 exp"−α(x−R11)2n −f0Λi#dx, (31) where 0 < f < 1 and R1 is a constant given by f0R1 FIG. 2: Plot of the function f(R) for the Starobinsky model. 1e−α/x2dx = R , and R is the Ricci scalar at 0 now now present. The effective cosmologicalconstant in the early R universeissimply f( )=Λ andthe presentcosmo- i − −∞ being x R/m2, the corresponding plot is given in logicalconstantis2R0. Becauseofthefactthatf(0)=0, Fig. 1. Fo≡r the case of the Starobinsky model, with this model allows for SBH. To be general,we first calculate the SdSBH entropy, which is given by fHS(x)=−1+ (1+1x2)2, (29) S = A4GH (1−f0exp"−α(R0R12Rn1)2n −f0RΛ0i#), − (32) being x R/R0, we obtain Fig. 2. And for the case of where R0 is the SdSBH curvature that fulfills condition ≡ the Appleby-Battye model, with (4). Thus,blackholesarelessentropicthaninEinstein’s theory. For the SBH, we have x 1 A fAB(x)=−2 + 2log(coshx+sinhx), (30) S = 4GH 1−f0e−α . (33) Note that the stability con(cid:0)dition is he(cid:1)re f′′(R)>0, thus being x aR and b=1/2, Fig. 3. Inafir≡stcomparisonofthesedifferentmodels,wenote f0 >2n R12n . (34) that the one of Starobinsky, as remarked by the author Λi (R0 R1)2n+1 − himself [11], has unstable vacuum solutions. This seems To have stability for SBH, we need that true also for the Hu-Sawicki model, except for the case when0<n<1,thatleadstoanegativeandinfinite BH R1 f0 n< . (35) entropy. The Appleby-Battye model has the important 2 Λ i propertytopossessstablevacuumsolutionsforasuitable rangeofthefreeparameters. Italsoyieldsanunmodified This can be considered, together with the condition n> expression of the BH entropy. 10 12statedin[20],toavoidNewtonlawcorrectionsin − our solar system and on the earth surroundings. 5 VI. CONCLUSIONS. interpretationintheframeworkofmodifiedgravity,with- outneedingtopassthroughthe(mathematicallybutnot Comparison of BH entropy in modified gravity theo- physicallyequivalent)scalarfieldtheories. Wehopethat ries and in the usual Einstenian gravity case have been this quite simple considerations may be useful for future carried out. We have here analyzed different suitable analysis. models, recently considered in the literature, and have shownexplicitlyhowcorrectionstothe‘classical’BHen- WethankSergeiOdintsovforhelpfuldiscussions. This tropy can in fact appear. We have also argued that the investigation is partly based on work done by EE while condition of positive entropy can be used as an extra on leave at the Department of Physics and Astronomy, condition in order to constrain the viability of modified Dartmouth College, 6127 Wilder Laboratory, Hanover, gravity theories. 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