Entropy Corrections for a Charged Black Hole of String Theory
aa r X i v : . [ g r- q c ] M a r Entropy Corrections for a Charged Black Holeof String Theory
Alexis LarrañagaOctober 30, 2018
National University of Colombia, National Astronomical Observatory.Bogota, [email protected]
Abstract
We study the entropy of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger(GMGHS) charged black hole, originated from the effective action thatemerges in the low-energy of string theory, beyond semiclassical approxi-mations. Applying the properties of exact differentials for three variablesto the first law thermodynamics we derive the quantum corrections tothe entropy of the black hole. The leading (logarithmic) and non leadingcorrections to the area law are obtained.
PACS: 04.70.Dy, 04.70.Bw, 11.25.-wKeyWords: quantum aspects of black holes, thermodynamics, strings andbranes
When studying black hole evaporation by Hawking radiation using the quantumtunneling approach , a semiclassical treatment is used to study changes in ther-modynamical quantities. The quantum corrections to the Hawking temperatureand the Bekenstein- Hawking area law have been studied for the Schwarzschild,Kerr and Kerr-Newman black holes [1, 2] as well as BTZ black holes [3].It has been realized that the low-energy effective field theory describingstring theory contains black hole solutions which can have properties whichare qualitatively different from those that appear in ordinary Einstein gravity.Here we will analyse the quantum corrections to the entropy of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) charged black hole [4], whichis an exact classical solution of the low-energy effective heterotic string theorywith a finite amount of charge. To obtain the quantum corrections we use the1riterion for exactness of differential of black hole entropy from the first law ofthermodynamics for two parameters. We find that the leading correction termis logarithmic, while the other terms involve ascending powers of inverse of thearea.In the quantum tunneling approach, when a particle with positive energycrosses the horizon and tunnels out, it escapes to infinity and appear as Hawk-ing radiation. Meanwhile, when a particle with negative energy tunnels inwardsit is absorbed by the black hole and as a result the mass of the black hole de-creases. Therefore, the essence of the quantum tunneling argument for Hawkingradiation is the calculation of the imaginary part of the action. If we considerthe action I ( r, t ) and make an expansion in powers of ~ we obtain I ( r, t ) = I ( r, t ) + ~ I ( r, t ) + ~ I ( r, t ) + ... (1) = I ( r, t ) + X i ~ i I i ( r, t ) , (2)where I gives the semiclassical value and the terms from O ( ~ ) onwards aretreated as quantum corrections. The work of Banerjee and Majhi [6] shownthat the correction terms I i are proportional to the semiclassical contribution I . Since I has the dimension of ~ , the proportionality constants should havethe dimension of inverse of ~ . In natural units ( G = c = k B = 1) , the Planckconstant is of the order of square of the Planck Mass. Therfore, from dimensionalanalysis the proportionality constants have the dimension of M − i , where M isthe mass of black hole, and the series expansion becomes I ( r, t ) = I ( r, t ) + X i β i ~ i M i I ( r, t ) (3) I ( r, t ) X i β i ~ i M i ! , (4)where β i ’s are dimensionless constant parameters. If the black hole has othermacroscopic parameters such as angular momentum or electric charge, one canexpress this expansion in terms of those parameters, as done in [1] and [3]. Inthis work, the dimensional analysis suggest that the constants of proportionalityfor charged rotating black holes have the dimensions of (cid:0) r H − Q e − φ (cid:1) i , sowe will use an expansion in terms of the horizon radius and the electric chargeas I ( r, t ) = I ( r, t ) X i β i ~ i ( r H − Q e − φ ) i ! . (5)Using this expansion we will calculate the quantum corrections to the entropyof the GMGHS black hole. 2 Entropy as an Exact Differential
In order to perform the quantum corrections to the entropy of the black hole wewill follow the analysis of [1, 3]. The first law of thermodynamics for chargedblack holes is dM = T dS + Φ dQ, (6)where the parameters M and Q are the mass and charge of the black hole, while T, S and Φ are the temperature, entropy and electrostatic potential, respectively.This equation can be re-written as dS ( M, Q ) = 1
T dM − Φ T dQ, (7)from which one can infer that in order for dS to be an exact differential, thethermodynamical quantities must satisfy ∂∂Q (cid:18) T (cid:19) = ∂∂M (cid:18) − Φ T (cid:19) . (8)If dS is an exact differential, we can write the entropy S ( M, J, Q ) in theintegral form S ( M, Q ) = ˆ T dM − ˆ Φ T dQ − ˆ (cid:18) ∂∂Q (cid:18) ˆ T dM (cid:19)(cid:19) dQ. (9)
The low energy effective action of the heterotic string theory in four dimensionsis given by A = ˆ d x √− ge − φ (cid:18) − R + 112 H µνρ H µνρ − G µν ∂ µ φ∂ ν φ + 18 F µν F µν (cid:19) , (10)where R is the Ricci scalar, G µν is the metric that arises naturally in the σ model, F µν = ∂ µ A ν − ∂ ν A µ (11)is the Maxwell field associated with a U (1) subgroup of E × E , φ is the dilatonfield and H µνρ = ∂ µ B νρ + ∂ ν B ρµ + ∂ ρ B µν − [Ω ( A )] µνρ , (12)where B µν is the antisymmetric tensor gauge field and3 Ω ( A )] µνρ = 14 ( A µ F νρ + A ν F ρµ + A ρ F µν ) (13)is the gauge Chern-Simons term. Considering H µνρ = 0 and working in theconformal Einstein frame, the action becomes A = ˆ d x √− g (cid:16) − R + 2 ( ∇ φ ) + e − φ F (cid:17) , (14)where the Einstein frame metric g µν is related to G µν through the dilaton, g µν = e − φ G µν . (15)The charged black hole solution, known as the Gibbons-Maeda- Garfinkle-Horowitz-Strominger (GMGHS) solution, is given by [4, 5] ds = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r (cid:18) − Q e − φ M r (cid:19) d Ω (16)where e − φ = e − φ (cid:18) − Q e − φ M r (cid:19) (17) F = Q sin θdθ ∧ dϕ (18)and φ is the asymptotic constant value of φ at r → ∞ . Note that this metricbecome Schwarzschild´s solution if Q = 0 . The GMGHS solution has a sphericalevent horizon at r H = 2 M (19)and its area is given by A = ˆ √ g θθ g ϕϕ dθϕ = 4 π (cid:0) r H − Q e − φ (cid:1) . (20)Equation (20) tell us that the area of the horizon goes to zero if r H = 2 Q e − φ , (21)i.e. the GMGHS solution becomes a naked singularity if M ≤ Q e − φ . (22)The Hawking temperature is T H = κ ~ π = ~ πM , (23)4hich is independent of charge. Finally, the electric potential computed onthe horizon of the black hole is Φ = Qr H e − φ . (24)One can easily check that thermodynamical quantities for the GMGHS blackhole satisfy condition (8), making dS an exact differential. Thus, the integralform of the entropy (9) gives S ( M, Q ) = ˆ T H dM − ˆ Φ T H dQ (25) S ( M, Q ) = 4 π ~ (cid:20) M − e − φ Q (cid:21) (26)that corresponds to the standard black hole entropy S ( M, Q ) = A ~ = π (cid:0) r H − Q e − φ (cid:1) ~ . (27) When considering the expansion for the action (5), it affects the Hawking tem-perature by introducing some correction terms [1, 3, 6]. Therefore the temper-ature is now given by T = T H X i β i ~ i ( r H − Q e − φ ) i ! − , (28)where T H is the standard Hawking temperature and the terms with β i arequantum corrections to the temperature. It is not difficult to verify that theconditions to make dS an exact differential are satisfied when considering thisnew form for the temperature. Therefore, the entropy with correction terms isgiven by S ( M, Q ) = ˆ T dM = ˆ T H X i β i ~ i ( r H − Q e − φ ) i ! dM (29)or S ( M, Q ) = ˆ T H dM + ˆ β T H ~ ( r H − Q e − φ ) dM + ˆ β T H ~ ( r H − Q e − φ ) dM + ... (30)This equation can be written as 5 ( M, Q ) = S + S + S + ...., (31)where S is the standard entropy given by equation (27) and S , S , ... are quan-tum corrections. The first of these terms is S = β ~ ˆ T H ( r H − Q e − φ ) dM. (32)Solving the integral, we obtain S = πβ ln (cid:12)(cid:12) r H − Q e − φ (cid:12)(cid:12) . (33)The following terms can be written, in general, as S j = β j ~ j ˆ T H ( r H − Q e − φ ) j dM (34)By calculating the integral, we obtain S j = πβ j ~ j − − j (cid:0) r H − Q e − φ (cid:1) − j (35)for j > . Therefore, the entropy with quantum corrections is given by S ( M, Q ) = π (cid:0) r H − Q e − φ (cid:1) ~ + πβ ln (cid:12)(cid:12) r H − Q e − φ (cid:12)(cid:12) + X j> πβ j ~ j − − j (cid:0) r H − Q e − φ (cid:1) − j . (36)Using equation (20), and doing a re-definition of the β i , we can write theentropy in terms of the area of the horizon as S ( M, Q ) = A ~ + πβ ln | A | + X j> πβ j ~ j − − j (cid:18) A π (cid:19) − j . (37)The first term in this expansion is the usual semiclassical entropy while thesecond term is the logarithmic correction found earlier for some geometries andusing different methods [7, 8] . The value of the coefficients β i can be evalatedusing other approaches, such as the entanglement entropy calculation. Finallynote that the third term in the expansion is an inverse of area term similar tothe one obtained by S. K. Modak [7] for the rotating BTZ black hole, for thecharged BTZ black hole [8] and also in the general cases studied in [1] and [3].6 Conclusion
As is well known, the Hawking evaporation process can be understood as a con-sequence of quantum tunneling in which some particles cross the event horizon.The positive energy particles tunnel out of the event horizon, whereas, the neg-ative energy particles tunnel in, resulting in black hole evaporation. Using thisanalysis we have studied the quantum corrections to the entropy of the GMGHSblack hole of heterotic string theory. With the help of the conditions for exact-ness of differential of entropy we obtain a power series for entropy. The firstterm is the semiclassical value, while the leading correction term is logarithmicas has been found using other methods[7, 8]. The other terms involve ascendingpowers of the inverse of the area. This analysis shows that the quantum correc-tions to entropy obtained in the literature [1, 3], also hold for the black hole ofstring theory studied here.
Acknowledgements . This work was supported by the Universidad Nacionalde Colombia. Project Code 2010100.
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