Horizon Areas and Logarithmic Correction to the Charged Accelerating Black Hole Entropy

It has been shown by explicit and exact calculation that the geometric product formula i.e. horizon area (or entropy) product formula of outer horizon (${\cal H}^{+}$) and inner horizon (${\cal H}^{-}$) for charged accelerating black hole (BH) should \emph{ neither be mass-independent nor it be quantized}. This implies that the horizon area (or entropy ) product is mass-independent conjecture has been~\emph{broken down} for charged accelerating BH. This also further implies that the mass-independent feature of the area product of ${\cal H}^{\pm}$ is \emph{not} a generic feature at all. We also compute that the \emph{Cosmic-Censorship-Inequality} for this BH. Indeed it is violated for this BH. Moreover, we compute the specific heat for this BH to determine the local thermodynamic stability of this BH. Under certain criterion, the BH shows the second order phase transition. Furthermore, we compute logarithmic corrections to the entropy for the said BH due to small statistical fluctuations around the thermal equilibrium.


Introduction
BH is a universal thermal object [1,2]. Their thermodynamic property is also universal. New thermodynamic product relations [particularly area (or entropy) product relations] of event horizon (EH) area and Cauchy horizon (CH) area for several class of BHs have been found also are universal. This relation is called universal because the product of area of multihorizons particularly two physical horizons namely H ± is mass-independent. When a thermodynamic product relation satisfied this criterion then it is said to be universal. This is the only motivation behind this work and this is in fact an interesting topic in recent years in the scientific community particularly in the general relativity (GR) community [3] and in the string theory community [4,5].
For example, in case of Kerr-Newman (KN) BH [3] which is an electrovacuum solution of Einstein's equations, it has been shown that the product of inner horizon (IH) area and outer horizon (OH) area should read where A − and A + are area of CH and EHs respectively. This relation indicates that the universal product depends only quantized angular momentum and quantized charges respectively. When the charge parameter Q = 0, one obtains the area product for Kerr BH: and it indicates that the universal product depends only quantized angular momentum parameter. Again, when the angular momentum parameter J = 0, one obtains the area product formula for spherically symmetric charged BH: and it implies that the universal product depends only quantized charge parameter. In the above three cases, it is indeed true that the area (or entropy) product formula is mass-independent thus it is universal in this sense. Motivated by the Visser's work [6] (see also [6,7,8,9,10,11]), we have tried to extend our analysis here for charged accelerating anti-de Sitter (AdS) BH. By explicit and exact calculation, we show that the area (or entropy) product formula of OH and IH for charged accelerating BH should not be mass-independent and it should also not to be quantized. Thus we conclude that the theorem of Ansorg-Hennig [3] "The area (or entropy) product formula is independent of mass" is not universal for charged accelerating BH. Moreover, we study other thermodynamic properties particularly the local thermodynamic stability by computing the specific heat. Under appropriate condition the BH posessess second order phase transition.
One aspect is that the leading-order logarithmic corrections to BH entropy due to quantum fluctuations around the thermal equilibrium BH temperature for charged AdS BH has not been studied previously, we compute here the logarithmic corrections to Bekenstein-Hawking entropy for the said BH and it appears to be a generic feature of the BH. It should be noted that we have assumed that the BH is a thermodynamic system which is in equilibrium at Bekenstein-Hawking temperature.
The another strong motivation comes from the work of Cvetič et al. [4], where the authors suggested that if any how the cosmological parameter is quantized then the area (or entropy) product relations for rotating BH in D = 4 and D > 4 should provide a "looking glass for probing the microscopics of general BHs". Thus it is quite interesting to investigate the area (or entropy) product formula after incorporating the cosmological constant.
The interesting property of the charged AdS BH is that it has an accelerating horizon and the OH posessess conical singularity [12]. In spite of the non-asymptotic structure, the first law of BH thermodynamics and Smarr formula are satisfied for this accelerating spacetime [12] (See also [15]). But the cosmological horizon don't have accelerating horizon. This type of BH is said to be slowly accelerating BH. The other novel properties of this accelerating BH is that it is described by the C metric [16,17,19,20]. Again the C metric has some peculiar features in the sense that it accelerates by pulling with a 'cosmic string' which described by a 'conical deficit in the spacetime' which connects the OH of the BH to infinity [12].
Actually the idea of vacuum C metric was first given by Levi-Civita in 1918 [21]. Then it was rediscovered by Newman and Tamburino in 1961 [22], also by Robinson and Trautman [24] in same years, and by Ehlers and Kundt [23] in 1963 but there were no explanation has given. First Robinson and Trautman discovered the interesting features of this metric that is it emits gravitational radiation. Later Podolsky et al. [25] studied the gravitational and electromagnetic radiation emitted by the uniformly accelerated charged BH in AdS spacetime.
In the next section, we have given the basic characteristics of the charged accelerating BH and we have also derived the area functional relation in terms of BH mass, charge, acceleration, cosmological constant. We have also derived the cosmic censorship inequality for this BH and finally we have computed the specific heat which determines the local thermodynamic stability. In the third section, we have discussed the logarithmic correction to BH entropy for this class of BHs. Finally in the last section, we have given the conclusion.

Thermodynamic properties of Charged Accelerating BH:
The metric of the charged accelerating BH [20,12] is described by with the gauge potential F = dB and B = − q r dt, and where G(θ) = 1 + 2mχcosθ + q 2 χ 2 cos 2 θ. (6) and the conformal factor is Ω = 1 + χrcosθ. It determines the conformal infinity of the AdS BH. The quantities m and q are BH mass and BH charge respectively, χ > 0 determines the acceleration of the BH and − Λ 3 = 1 ℓ 2 where ℓ is the radius of the AdS BH. It should be noted that when χ < 1 ℓ , a single BH is present with single horizon [18] whereas when χ > 1 ℓ , two BHs of opposite charge are separated by accelerating horizon [19,26] and when χ = 1 ℓ is a special case and it has been explicitly described in [27]. To obtain the angular coordinates as usual form on S 2 we have set the restriction mχ < 1 2 . Now we discuss the angular part of the metric and the properties of G(θ) at the north pole (θ = 0) and south pole (θ = π). In general, K = G(θ) but at north pole fixed with K = K N = 1 + 2mχ + q 2 χ 2 and at south pole K S = 1 − 2mχ + q 2 χ 2 that indicates the choices at the north pole and at the south pole are mutually incompatible. Thus the metric cannot be made regular at both the poles at the same time. Therefore the convention is to make a choice that makes it regular at one of the polesthis leads to a 'conical deficit' at the other pole which is δ = 8πmχ 1+2mχ+q 2 χ 2 and which corresponds to a 'cosmic string' [12] with tension µ = δ 8π = mχ 1+2mχ+q 2 χ 2 . Thus the C metric is described by the five physical parameters: the mass m, the charge q, the negative cosmological constant −Λ = 3 ℓ 2 , the acceleration χ and the 'tension of the cosmic string' on each axis which is represented by the periodicity of the angular coordinate. More discussion regarding the thermodynamic properties ( particularly first law of BH thermodynamics [15], Smarr mass formula, thermodynamic volume, Gibbs free energy and Reverse isoperimetric inequality) of the C metric could be found in [12]. Now we evaluate the radii of BH horizons by imposing the condition F(r i ) = 0 i.e.
Apply the Vieta's theorem, we find . .
Eliminating the mass parameter, one obtains a single mass-independent relation as In terms of two BH physical horizons area A i = the mass independent functional relationship is given by where, Eq. (12) is indeed mass independent but the difficulties arise when we write the expression in terms of area of the BH physical horizons [Eq. (13)] because there is a factor K where a mass term m is present. Therefore it is indeed true that the mass-independent relation has been violated for charged accelerating BH. Thus the "Ansorg-Hennig [3] area theorem" conjecture breaks down for charged accelerating BH. This is an another example we have provided in the literature that the area product of OH and IH is not always universal. The BH entropy [12] is given by and the electric potential on the horizon should read Now the BH temperature is given by We know the famous Cosmic-Censorship-Inequality [which requires cosmic-censorship hypothesis [28] (See [30,31,32,33,34])] for Schwarzschild BH is given by This inequality for accelerating BH becomes This idea had first given in 1973 by Penrose [28] which is an important topic in GR which relates ADM mass (i.e. total mass of the spacetime) and the area of the even horizon. It is called Cosmic-Censorship-Inequality or Cosmic-Censorship-Bound [29].
This inequality implies that it provides the lower bound on the mass (or energy) for any timesymmetric initial data which satisfied the famous Einstein equations with negative cosmological constant, and which also satisfied the dominant energy condition which possesses no naked singularities.
Local thermodynamic stability and phase transitions [13,14] particularly second order phase transition are important phenomena in BH thermodynamics and it can be determined by computing the specific heat which is calculated to be for accelerating BH: Now we analyze the above expression of specific heat for different parameter space. Case I: When the specific heat is positive i. e. C i > 0, which indicates that the BH is thermodynamically stable. Case II: When the specific heat is negative i. e. C i < 0, which implies that the BH is thermodynamically unstable. Case III: When the specific heat C i diverges that signals a second order phase transition for such BHs could be observed from the Fig. 2. In Fig. 2, we have plotted specific heat with the horizon radius. From the figure one can observed that the phase transition occurs at the positive value of the EH radius.

Logarithmic Corrections to Entropy for charged accelerating BH:
In this section, we should derive the logarithmic corrections to BH entropy for charged accelerating BH by assuming that the BH considered as a thermodynamic system which is in equilibrium at Bekenstein-Hawking temperature, and due to the effects of statistical thermal fluctuations around the equilibrium.
To derive this correction, we should follow the classical work of Das et al. [37]. Where the authors first studied the general logarithmic corrections to BH entropy for higher dimensional AdS spacetime and BTZ BH. Since the BHs have been considered as macroscopic object compared to the  Figure 1: The inequality for specific heat of the Case-I is plotted in Fig. 1(a), Case-II is plotted in Fig. 1(b) and Fig. 1(c). Along the abcissa we have chosen the value of horizon radius r + and along the ordinate we have chosen the value of q. We have set ℓ = χ = 1. Planck scale length and this indicates that the logarithmic terms are much smaller compared to the Bekenstein-Hawking terms and therefore it should be treated as corrections. Now we can define the canonical partition function [36] as where T i = 1 β i is the temperature of H i . We have chosen the value of Boltzman constant k B to be unity.
The density of states can be defined as an inverse Laplace transformation of the partition function: where a is a real positive constant and is the entropy of the system near its equilibrium. Near equilibrium of the inverse Hawking temperature β i = β 0,i , we can expand the entropy function as where, S 0,i := S i (β 0,i ) and S ′′ 0,i = ∂ 2 S i ∂β 2 i at β i = β 0,i . Putting the Eq. (28) in Eq. (24), one obtains Let us choose β i − β 0,i = iy i and setting a = β 0,i , y i is a real variable and evaluating a contour integral we have The logarithm of ρ i (E) gives the corrected entropy of the thermodynamic system: Next our task is to compute S ′′ 0,i , for this we must choose any specific form of function S i (β i ) which is modular invariant partition function and which is also admitted an extremum at some specific value β i,0 of β i followed by underlying conformal field theory (CFT) [35,37]. Therefore the exact entropy function followed by CFT is of the form: where c, d are constants. It can be rewritten as more general form which admits saddle point as where m, n, c, d > 0. The special case we have considered here when m = n = 1 and it is due to the CFT. After some algebraic computation (See for more details [37,38]) one can find the value of S ′′ 0,i = T 2 i S 0,i , then we get the leading order corrections to BH entropy as Putting the values of S 0,i = and T i , one can obtain the logarithmic correction to BH entropy for charged accelerating BH: This corrected entropy formula indicates that it is a function of horizon radius, mass parameter, charge parameter, acceleration and cosmological constant.
It should be noted that the product is explicitly depends on the value of mass parameter and etc. Thus the logarithmic corrected entropy formula is not mass-independent and it does not quantized.
We have plotted the logarithmic corrected entropy and without logarithmic corrected entropy in Fig. 3. It follows from the graph [Fig. (a)] when there is no acceleration i. e. χ = 0 and there is no logarithmic correction, the entropy is increasing when horizon radius increases. When there is an acceleration the value of entropy diverges at a certain horizon radius this indicates the phase transition occurs at r + = 1. On the other hand, when we have taken the logarithmic correction of entropy in the presence of acceleration then the divergence disappears [ Fig. (b)] and all the phenomena occurs at 0.5 < r + < 1. Three situations are also qualitatively different. This is an another interesting feature of charged accelerating BH.

Conclusion:
We have studied the thermodynamic properties of slowly accelerating BH consists of five parameters, the mass, the charge, the acceleration, the cosmological constant and cosmic string tension. We derived the geometric product formula i.e. area product formula of OH and IH for charged accelerating BH. We showed that this area product formula should not be mass-independent nor does it quantized. This suggests that the mass-indepedent conjecture of Ansorg-Hennig breaks down for charged accelerating BH. This is an another example we have added in the literature that the area product of two physical horizons is not always mass-independent. We also derived the famous Cosmic Censorship Inequality for this slowly accelerating BH. The physical significance of this inequality determined the lower bound of mass or energy for a time-symmetric initial data which fulfilled the dominant energy condition. Moreover, we evaluated the criterion under which the BH showed the second order phase transition. Finally, we computed the logarithmic correction to entropy due to statistical quantum thermal fluctuations near the BH equilibrium temperature for this charged accelerating BH. The implication of these logarithmic correction may useful to understanding the Suskind's Holographic hypothesis [39].