Corrected entropy of high dimensional black holes
aa r X i v : . [ h e p - t h ] O c t Corrected entropy of high dimensional black holes
Tao Zhu, ∗ Ji-Rong Ren, † and Ming-Fan Li ‡ Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China (Dated: September 9, 2018)Using the corrected expression of Hawking temperature derived from the tunneling formalismbeyond semiclassical approximation developed by
Banerjee and
Majhi [12], we calculate the correctedentropy of a high dimensional Schwarzschild black hole and a 5-dimensional Gauss-Bonnet (GB)black hole. It is shown that the corrected entropy for this two kinds of black hole are in agreementwith the corrected entropy formula (2) that derived from tunneling method for a ( n +1)-dimensionalFriedmann-Robertson-Walker (FRW) universe[19]. This feature strongly suggests deep universalityof the corrected entropy formula (2), which may not depend on the dimensions of spacetime andgravity theories. In addition, the leading order correction of corrected entropy formula alwaysappears as the logarithmic of the semiclassical entropy, rather than the logarithmic of the areaof black hole horizon, this might imply that the logarithmic of the semiclassical entropy is moreappropriate for quantum correction than the logarithmic of the area. PACS numbers: 04.70.DyKeywords: high dimensional black hole, corrected entropy, tunneling
One of the intriguing properties of a black hole isthat it carries entropy[1]. Understanding this entropyis an enormous challenge in modern physics. Any de-velopments in this direction might lead to important in-sights into the structure of quantum gravity which in-cludes in particular the notion of “holography” and theemerging notion of “quantum spacetime”. There aremany approaches to calculate the entropy of a blackhole. With the semiclassical approximation, the blackhole entropy obeys the celebrated Bekenstein-Hawkingarea law. When full quantum effect is raised, the arealaw should undergo corrections, and these correctionscan be obtained from field theory methods[2], quan-tum geometry techniques[3], general statistical mechani-cal arguments[4], Cardy formula[5], etc[6]. All these ap-proaches show that the corrected entropy formula takesthe form S c = S + α ln S + · · · , (1)where α is a dimensionless constant and S denotes theuncorrected semiclassical entropy of black hole. It is wellknown that the corrected entropy formula of eq.(1) isuniversal. This implies that eq.(1) could be valid for allblack holes.Hawking radiation from the horizon of a black hole[7]also provides an approach to calculate thermodynamicentities like temperature and entropy of a black hole.Many different derivations of Hawking radiation exist inthe literature. Among these a simple and physically in-tuitive picture is provided by the tunneling mechanism.It has two variants namely null geodesic method[8] andHamilton-Jacobi method[9]. Recently, the connection be-tween the anomaly approach and tunneling mechanism ∗ [email protected] † [email protected] ‡ [email protected] is discussed[10] and the Hawking black body spectrum isobtained from tunneling machanism[11]. However, mostof these derivations are confined to the semiclassical ap-proximation. Recently, a general formalism of tunnelingbeyond semiclassical approximation has been developedby Banerjee and
Majhi [12]. This formalism has beenused to investigate the quantum corrections to the semi-classical entropy for various black holes[13, 14, 15, 16, 17].More interestingly, the corrected entropy formulas ofblack holes calculated from this formalism all take theform same as eq.(1).In our recent work[18, 19], this formalism has been ex-tended from black holes to Friedmann-Robertson-Walker(FRW) universe. We have shown that the corrected en-tropy of apparent horizon for a FRW universe takes theform[19] S c = S + α ln S + X i =2 α i S i − + const . (2)It is obvious that the first and the second terms havethe same form as eq.(1). As pointed out in ref.[19], thiscorrected entropy formula has three important features: • Eq.(2) not only holds in Einstein gravity, but alsois valid for Gauss-Bonnet gravity, Lovelock gravity, f ( R ) gravity and scalar-tensor gravity. This featuremight imply that the corrected entropy formula ofeq.(2) is independent of gravity theories. • Eq.(2) is derived from the tunneling method inan arbitrary dimensions ( n + 1)-dimensional FRWspacetime. This might imply that it is independentof dimensions of the spacetime. • The corrected entropy formulas of different blackholes calculated from the tunneling method all takethe form same as eq.(2), such as those of the BTZblack hole, Kerr-Newmann black hole, etc.These features strongly suggest deep universality of thecorrected entropy formula of eq.(2).Up to now, the given examples of black holes are allconfined to low dimensional spacetimes. Since the FRWuniverse is different from black holes and eq.(2) is de-rived for a FRW universe, one may ask that whether thecorrected entropy formula of eq.(2) is still valid for highdimensional black holes in Einstein gravity or in othergravity theories. In order to answer this question, weinvestigate the corrected entropy formula of high dimen-sional black holes in tunneling perspective. We explicitlycompute the corrected expressions for the temperatureand the entropy for an ( n + 1)-dimensional Schwarzschildblack hole and a 5-dimensional Gauss-Bonnet black hole.It is shown that the corrected entropy formula of eq.(2)is also valid for this two kinds of black hole, and thereforethe universality of eq.(2) is more plausible.Consider an ( n + 1)-dimensional static, sphericallysymmetric spacetime of the form ds = − f ( r ) dt + dr g ( r ) + r d Ω n − , (3)where d Ω n − denotes the line element of an ( n − r = r H is given by f ( r H ) = g ( r H ) = 0. In this space-time, a massless scalar particle obeys the Klein-Gordonequation − ~ √− g ∂ µ ( g µν √− g∂ ν ) φ = 0 . (4)Note that, in the tunneling framework, the tunneling par-ticle is considered as a spherical shell. For this the tra-jectory of the tunneling process is radial and thereforeonly the ( r, t ) sector of the metric (3) is important. Inthis case, tunneling of a particle from a black hole canbe considered as a two-dimensional quantum process in( r, t ) plane. For a two dimensional theory, the standardWKB ansatz for the wave function φ can be expressed as φ ( r, t ) = exp (cid:20) i ~ I ( r, t ) (cid:21) , (5)where I ( r, t ) is one particle action which will be expandedin powers of ~ as I ( r, t ) = I ( r, t ) + X i ~ i I i ( r, t ) . (6)Here I ( r, t ) is the semiclassical action and the otherterms are treat as quantum corrections. Since only the( r, t ) sector is relevant and the other dimensions can notaffect the tunneling process, the treatment and the re-sult for low dimensional black holes are same as thesehere. As shown in ref.[12, 14], I i ( r, t ) are proportional to I ( r, t ), thus we have I ( r, t ) = X i γ i ~ i ! I ( r, t ) . (7) With this expression of action, the corrected Hawkingtemperature can be expressed as[12, 14] T c = T H X i γ i ~ i ! − , (8)where T H = ~ Im Z r dr p f ( r ) g ( r ) ! − (9)is the standard semiclassical Hawking temperature of theblack hole.With the corrected Hawking temperature (8) we nowproceed with the calculation of the corrected entropy ofblack holes. We first consider an ( n + 1)-dimensionalSchwarzschild black hole whose metric has the form[20] ds = − (cid:16) − mr n − (cid:17) dt + (cid:16) − mr n − (cid:17) − dr + r d Ω n − . (10)The ADM mass of the black hole is given by M = ( n − n − m π . Its semiclassical Hawking temperature canbe obtained from eq.(9) as T H = ( n − ~ πr H . (11)The mass is related to the horizon radius as M = ( n − n − π r n − H , (12)where r H = m / ( n − is the location of the horizon. Inthe semiclassical approximation, the entropy of the hori-zon obeys the Bekenstein-Hawking area law S BH = A ~ , (13)where A = Ω n − r n − H is the area of the horizon. Withsemiclassical temperature (11) and entropy (13), the firstlaw of thermodynamics holds on the horizon T H dS BH = dM. (14)From this expression the Bekenstein-Hawking entropycan be computed as S BH = Z dMT H . (15)In the Hawking temperature expression (8), there areun-determined coefficients γ i . Obviously, γ i should havethe dimension ~ − i . Now, we will perform the follow-ing dimensional analysis to express these γ i in terms ofdimensionless constants by invoking some basic macro-scopic parameters of high dimensional black hole. Inthe ( n + 1)-dimensional spacetime, one sets the unitsas G n +1 = c = k B = 1, where G n +1 is the ( n + 1)-dimensional gravitation constant. In this setting, thePlanck constant ~ is of the order of l n − p , where l p is thePlanck length. Therefore, according to the dimensionalanalysis, the proportionality constants γ i have the dimen-sion of l i (1 − n ) p [14, 17]. For Schwarzschild black hole, theonly macroscopic parameter is the radius of horizon r H .Therefore, one can express the proportionality constants γ i in terms of black hole parameters as γ i = α i r − i ( n − H , (16)where α i are dimensionless constants. Now the correctedHawking temperature can be written as T c = T H X i α i ~ i ( r n − H ) i ! − = T H X i ˜ α i ( S BH ) i ! − , (17)where ˜ α i = ( Ω n − ) i α i are also dimensionless constants.Note that for Schwarzschild black hole S BH is proportionalto the area of horizon, thus it only dependent on the ra-dius of horizon r H . Things will be a bit different forGauss-Bonnet black hole while the entropy is not pro-portional to the area of horizon.Replace the semiclassical Hawking temperature T H with the corrected Hawking temperature (17), one candetermine the corrected entropy by the integral S c = Z dMT c = Z dMT H X i ˜ α i S i BH ! . (18)Integrating the above expression, we obtain the correctedBekenstein-Hawking entropy of an ( n + 1)-dimensionalSchwarzschild black hole as S c = S BH + ˜ α ln S BH + X i =2 ˜ α i − i S i − BH + const . (19)Interestingly the leading order correction is logarithmicin S BH , which is consistent with eqs.(1) and (2).Now we turn to the Gauss-Bonnet black hole, whichis the black hole solution in Gauss-Bonnet gravity. The(4 + 1) dimensional static, spherically symmetric blackhole solution in this theory is of the form ds = − f ( r ) dt + f ( r ) − dr + r d Ω , (20)where the metric function is f ( r ) = 1 + r α " − (cid:18) αmr (cid:19) / . (21)Here m is related to the ADM mass M by the relationship M = π m . The event horizon is located at r H whichsatisfies r H + α − m = 0 . (22) For the horizon to exist at all, one must have r H +2 α > r H as M = 3Ω π ( r H + α ) . (23)Substituting the metric (20) into eq.(9), we obtain theHawking temperature for this black hole T H = ~ π r H r H + 2 α . (24)For black holes in Einstein gravity, the entropy of thehorizon is proportional to its area. Gauss-Bonnet grav-ity is the natural generalization of Einstein gravity by in-cluding higher derivative correction term, i.e., the Gauss-Bonnet term to the original Einstein-Hilbert action. Inthis gravity theory, the semiclassical Bekenstein-Hawkingentropy-area relationship that the entropy of the horizonis proportional to its area, does not hold anymore. Therelationship is now[21] S GB = A ~ (cid:18) αr H (cid:19) . (25)But the first law of thermodynamics still holds on thehorizon of a Gauss-Bonnet black hole T H dS GB = dM. (26)For Schwarzschild black hole, one can express the cor-rected temperature (8) in terms of S BH as (17). Thisis always correct because S BH is only dependent on r H ,which is the only independent macroscopic parameter ofthe Schwarzschild black hole. Unlike Schwarzschild blackhole that has only one independent macroscopic param-eter r H , the Gauss-Bonnet black hole have two indepen-dent parameters r H and α . Thus eq.(16) in which the un-determined coefficients γ i have been expressed in termsof r H might be not appropriate here. In this case, themost general form of the proportionality constants γ i canbe expressed in terms of r H and α as γ i = ( c i r H + d i r H α + e i α / ) − i , (27)where un-determined coefficients c i , d i , and e i are dimen-sionless constants. Thus the corrected temperature nowfor Gauss-Bonnet black hole is T c = T H X i α i ~ i ( c i r H + d i r H α + e i α / ) i ! − i . (28)In order to determine c i , d i , and e i , one should treatthe independent parameter α as a variable. Due to thedifference in α , the semi-classical first law of thermody-namics of Gauss-Bonnet black hole should be modifiedas T H dS = dM + 3Ω π r H − αr H + 2 α dα, (29)where the added term is just the “work term” inducedby the differentiation of α . Now with the corrected tem-perature (28), the first law of thermodynamics is dS c = 3Ω ~ T H T c (cid:2) ( r H + 2 α ) dr H + 2 r H dα (cid:3) . (30)From the principle of the ordinary first law of thermo-dynamics one interprets entropy as a state function. Inrefs.[13, 14, 19], this property of entropy has been used toinvestigate the first law of thermodynamics and entropyfor black holes. The entropy must be a state functionmeans that dS c has to be an exact differential. As aresult the following integrability condition must hold: ∂∂α (cid:20) T H T c ( r H + 2 α ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) r H = ∂∂r H (cid:20) r H T H T c (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) α . (31)From this condition one can easily determine c i , d i , and e i as d i = 6 c i , e i = 0 . (32)Substituting above results into (28), it is easy to showthat the corrected temperature for Gauss-Bonnet blackhole can be expressed as T c = T H X i ˜ α i ( S GB ) i ! − . (33)This expression have two important features. First, itinvolves both the parameters r H and α to express theun-determined coefficients γ i . Second, it ensures thatthe corresponding corrected entropy is a state functionwhen we treat the parameter α as a variable.With the corrected Hawking temperature (33), the cor-rected expression of the entropy for this black hole canbe determined as S c = S GB + ˜ α ln S GB + X i =2 ˜ α i − i S i − GB + const . (34)It is clear that this corrected entropy formula is consistentwith eq.(2).Thus we have derived the corrected entropy formulafor a high dimensional Schwarzschild black hole and a 5 dimensional Gauss-Bonnet black hole. By using the cor-rected expression of Hawking temperature derived fromtunneling formalism beyond semiclassical approximationand applying the semiclassical first law of thermodynam-ics for this two black holes, the corrected expressions ofthe entropy of the horizon are determined. All the highorder quantum corrections to the entropy are computed.It is shown that these corrected expressions of entropyare both in agreement with eq.(2). It seems that, thecorrected entropy formula of black holes in tunneling per-spective does not depend on the dimensions of spacetimeand gravity theories. This supports the universality ofthe corrected entropy formula (2) by tunneling method.There is another significant point for the corrected en-tropy, is that it involves the term of the logarithmic of S as the leading order correction. From the Table I, itis easy to see that for a large number of black holes, theleading order corrections in the corrected entropy appearsas the logarithmic of the semiclassical entropy S . Thisfeature agrees with the universal quantum corrected en-tropy expression (1). In Einstein gravity, the correctedentropy expression is usually written as another form S c = A ~ + α ln A + · · · . (35)Of course, because the entropy of the horizon is propor-tional to its area in Einstein gravity, this form is con-sistent with eq.(1). But for other non-Einstein gravitytheories, the semiclassical Bekenstein-Hawking entropy-area relationship that the entropy of the horizon is pro-portional to its area, does not hold anymore. Then, theexpression (35) is not valid in these cases. So one canconclude that, in the expression of quantum correctedentropy of the horizon, the logarithmic of the semiclassi-cal entropy is more appropriate than the logarithmic inthe area of horizon. Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (No.10275030) and the CuiyingProgramme of Lanzhou University (225000-582404). [1] J.D. Bekenstein, Phys. Rev.
D 7 (1973) 2333.[2] D.V. Fursaev, Phys. Rev.
D 51 (1995) R5352; R.B. Mannand S.N. Solodukhin, Nucl. Phys.
B 523 (1998) 293;D.N. Page, New J. Phys. (2005) 203.[3] R.K. Kaul and P. Majumdar, Phys. Rev. Lett. (2000)5255; S. Kloster, J. Brannlund, A. DeBenedicts, Class.Quantum Grav. (2008) 065008.[4] S. Das, P. Majumdar, R.K. Bhaduri, Class. Quant. Grav. (2002) 2355; S.S. More, Class. Quant. Grav. (2005)4129; S. Mukherjee and S.S. Pal, JHEP (2002) 026; Y.S. Myung, Class. Quant. Grav. (2009) 065007.[5] S. Carlip, Class. Quantum Grav. (2000) 4175; M.R.Setare, Eur. Phys. J. C 33 (2004) 555.[6] A.J.M. Medved and E.C. Vagenas, Phys. Rev.
D 70 (2004) 124021; R. Banerjee and B.R. Majhi, Phys. Lett.
B 662 (2008) 62; R. Banerjee, B.R. Majhi, and S.Samanta, Phys. Rev.
D 77 (2008) 124035; J.E. Lidsey, S.Nojiri, S.D. Odintsov, and S. Ogushi, Phys. Lett.
B 544 (2002) 337; S. Nojiri, S.D. Odintsov, and S. Ogushi, Int.J. Mod. Phys.
A 18 (2003) 3395; S. Hod, Class. Quant.
Does ln S exist?3 D BTZ black hole[13] Yes4 D Schwarzchild black hole[12, 14] Yes4 D Schwarzschild-AdS black hole[12] Yes4 D Schwarzschild-AdS black hole in Rainrow gravity[15] Yes4 D Reissner-Nordstrom black hole[14] Yes4 D Kerr black hole[14] Yes4 D Kerr-Newmann black hole[14] Yes5 D Gauss-Bonnet black hole Yes( n + 1) D Schwarzchild black hole Yes( n + 1) D FRW universe for various gravity theories[19] YesTABLE I: The leading order correction in the corrected entropy appear as the logarithmic of the semiclassical entropy S forvarious black holes.Grav. 21 (2004) L97.[7] S.W. Hawking, Nature (1974) 30; S.W. Hawking,Commun. Math. Phys. (1975) 199.[8] M.K. Parikh, F. Wilczek, Phys. Rev. Lett. (2000)5042.[9] K. Srinivasan and T. Padmanabhan, Phys. Rev. D 60 (1999) 024007.[10] R. Banerjee and B.R. Majhi, Phys. Rev.
D 79 (2009)064024.[11] R. Banerjee and B.R. Majhi, Phys. Lett.
B 675 (2009)243.[12] R. Banerjee and B.R. Majhi, JHEP 06 (2008) 095; R.Banerjee and B.R. Majhi, Phys. Lett.
B 674 (2009) 218.[13] S.K. Modak, Phys. Lett.
B 671 (2009) 167.[14] R. Banerjee and S.K. Modak, JHEP (2009) 063.[15] Y.Q. Yuan and X.X. Zeng, Int. J. Theor. Phys 48 (2009)1937.[16] B.R. Majhi, Phys. Rev. D 79 (2009) 044005; H.M. Sia-haan and Triyanta,
Hawking Radiation from a VaidyaBlack Hole: A Semi-Classical Approach and Beyond ,arXiv:0811.1132; R. Banerjee, B.R. Majhi, and D.Roy, arXiv:0901.0466; B.R. Majhi and S. Samanta, arXiv:0901.2258; K.X. Jiang, T. Feng, and D.T. Peng,Int. J. Theor. Phys 48 (2009) 2112; Y.Q. Yuan, X.X.Zeng, Z.J. Zhou, and L.P. Jin, Gen. Relat. Grav.,10.1007/s10714-009-0806-x.[17] R. Banerjee and S.K. Modak,
Quantum Tunneling,Blackbody Spectrum and Non-Logarithmic Entropy Cor-rection for Lovelock Black Holes , arXiv:0908.2346[hep-th].[18] T. Zhu, J.R. Ren, and D. Singleton, arXiv:0902.2542.[19] T. Zhu, J.R. Ren, and M.F. Li, JCAP 08 (2009) 010[arXiv:0905.1838[hep-th]]; T. Zhu and J.R. Ren, Euro.Phys. J.
C 62 (2009) 413 [arXiv:0811.4074[hep-th]].[20] R.C. Myers and M.J. Perry, Ann. Phys. 172 (1986) 304.[21] R.C. Myers and J.Z. Simon, Phys. Rev.
D 38 (1988)2434; R.G. Cai, Phys. Rev.
D 65 (2002) 084014; R.G.Cai and Q. Guo, Phys. Rev.
D 69 (2004) 104025; M.Cvetic, S. Nojiri, and S.D. Odintsov, Nucl. Phys.
B 628 (2002) 295; S. Nojiri and S.D. Odintsov, Phys. Lett.
B521 (2001) 87; G. Kofinas and R. Olea, Phys. Rev.