Effects of thermal fluctuations on the Kerr【-逻*辑*与-】ndash;Newman【-逻*辑*与-】ndash;NUT【-逻*辑*与-】ndash;AdS

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Effects of thermal fluctuations on the Kerr【-逻辑与-】ndash;Newman【-逻辑与-】ndash;NUT【-逻辑与-】ndash;AdS

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M Sharif^,^∗, Qanitah Ama-Tul-Mughani^,Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan

First author contact: ^∗Author to whom any correspondence should be addressed.
Received:2021-01-29Revised:2021-05-7Accepted:2021-05-8Online:2021-06-25

Abstract
This paper is devoted to studying the impact of thermal fluctuations on thermodynamics of rotating as well as charged anti-de Sitter black holes with the Newman–Unti–Tamburino (NUT) parameter. To this end, we derive the analytic expression of thermodynamic variables, namely the Hawking temperature, volume, angular velocity, and entropy within the limits of extended phase space. These variables meet the first law of thermodynamics as well as the Smarr relation in the presence of new NUT charge. To analyze the effects of quantum fluctuations, we derive the exact expression of corrected entropy, which yields modification in other thermodynamical equations of state. The local stability and phase transition of the considered black hole are also examined through specific heat. It is found that the NUT parameter increases the stability of small black holes, while the logarithmic corrections induce instability in the system.
Keywords： black hole;thermal fluctuations;thermodynamics;NUT parameter

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M Sharif, Qanitah Ama-Tul-Mughani. Effects of thermal fluctuations on the Kerr–Newman–NUT–AdS black hole. Communications in Theoretical Physics, 2021, 73(8): 085402- doi:10.1088/1572-9494/abff1b

1. Introduction

Black hole (BH) thermodynamics, with finite temperature and entropy, not only discusses its classical aspect but also provides an elementary insight into quantum gravity. In thermodynamic systems, intrinsic entropy is assumed to play an essential role in the study of their physical features and is correlated to horizon area. It is intended that BHs must have maximum entropy to avoid the infringement of the second law of thermodynamics. This scenario will reduce the entropy of the Universe; consequently, the equilibrium phase between thermal radiations and BH physics cannot be obtained. The connection between the BH area with the maximum entropy has provoked the holographic principle [1], which only remains valid for large-scale structures and gets violated near the Planck scale due to the quantum corrections in the area–entropy relation. These correction terms do not perturb BHs that have a larger horizon radius but have certain implications on small BHs whose sizes reduce due to Hawking radiation [2]. As a pioneer, Das et al [3], developed the algorithm of corrected entropy and applied it to Schwarzschild, Reissner–Nordstrom (RN) and Bañados–Teitelboim–Zanelli (BTZ) BHs.

The effect of thermal fluctuations on numerous BHs has been studied in the literature [4]. Pourhassan et al [5] discussed the impact of logarithmic corrections on several quantities such as entropy and volume in the background of a modified Hayward BH. Using a similar approach in higher-dimensional charged BHs, Pourhassan et al [6] studied the influence of corrected entropy on thermodynamics quantities. They also investigated the validity of the first law of thermodynamics. Haldar and Biswas [7] graphically analyzed the behavior of enthalpy, Helmholtz and Gibbs free energies for Lovelock anti-de Sitter (AdS) BHs and concluded that the thermodynamic quantities follow a decreasing trend against logarithmic corrections. The same authors [8] explored thermodynamic characteristics of regular BHs by incorporating thermal fluctuation effects near the equilibrium phase. Nadeem-ul-Islam et al [9] discussed the effects of quantum corrections on BTZ BHs and found that small BHs show unstable behavior due to logarithmic corrections. Ganai et al [10] discussed thermodynamic potentials of a charged rotating BTZ BH in the presence of small statistical perturbations. Upadhyay [11] discussed the effects of thermal fluctuations on the stability of charged rotating AdS BHs and showed that thermodynamic potentials satisfy the first law of BH thermodynamics. He found that for small BHs, the specific heat takes negative values, which suggests that small BHs are thermodynamically in an unstable phase. However, the specific heat is found to be always positive for larger BHs, which means that these BHs are in a stable phase.

The Newman–Unti-Tamburino (NUT) metric [12] is one of the most interesting solutions of general relativity. This metric carries a particular type of gravitational charge named the NUT charge, which is analogous to the magnetic monopole in many respects. In theoretical physics, substantial work has been carried out to study the essential characteristics of the NUT parameter. Sharif and Wajiha [13] studied Hawking radiation as tunneling of charged fermions through event horizons of a pair of charged accelerating and rotating BHs with the NUT parameter. The same authors [14] evaluated thermodynamic quantities such as the Hawking temperature, entropy, and heat capacity in a charged rotating and accelerating BH with the NUT parameter. Jan and Gohar [15] found the exact expression of the Hawking temperature using the quantum tunneling approach in a rotating and accelerating NUT BH. Johnson [16] considered a cosmological constant as dynamical pressure and derived gravitational thermodynamics for the Taub–NUT geometry in AdS spacetime.

Liu and Lu [17] discussed the thermodynamics of a charged rotating AdS BH in conformal gravity. They derived all the thermodynamical quantities, including mass, angular momentum, electric/magnetic charges, and their thermodynamical conjugates. They verified that the first law of thermodynamics, as well as the Smarr relation, holds. In [18] the author derived the area product, entropy product, area sum, and entropy sum of the event horizon and Cauchy horizons for the Kerr–Newman–Taub–NUT BH in four-dimensional Lorentzian geometry. He observed that these thermodynamic products are not universal (mass-independence). He also examined the entropy sum and area sum. It is shown that they all depend on the mass, charge, and NUT parameter of the background spacetime. He concluded that the Kerr–Newman-Taub–NUT BH does not satisfy the first law of BH thermodynamics and Smarr–Gibbs–Duhem relations. Hennigar et al [19] discussed the thermodynamics of the Lorentzian Taub–NUT solution and formulated the first law of BH thermodynamics with a new NUT charge. Bordo et al [20] derived the thermodynamics of Taub–NUT spacetimes in the presence of magnetic as well as electric charge and showed that the NUT parameter can be varied independently without dependence on the event horizon.

This paper aims to study the impact of statistical perturbations on a charged rotating NUT–AdS BH. The paper is arranged as follows. The following section provides the fundamentals of spacetime and calculates the thermodynamic variables in extended phase space (EPS). In section 2, we provide the exact expression of corrected entropy, internal energy, modified mass, and Gibbs and Helmholtz free energies and graphically analyze their behavior. Moreover, we examine the stability of the BH through specific heat, and the final comments are summarized in the last section.

2. Kerr–Newman–NUT–AdS BH

In theoretical physics, the crucial discovery of BHs assists in the exploration of hidden characteristics of the Universe. The first-ever non-trivial spherically symmetric BH solution of the Einstein field equations is known as the Schwarzschild BH, which is extended to other BH geometries such as RN, Kerr, and Kerr–Newman by including the effects of electric charge and rotation parameters. Later, many BH solutions were developed by incorporating various sources, such as acceleration, magnetic charge, the NUT parameter as well as a cosmological constant in the usual mass of a BH. BHs with these extensions are categorized as a class of type-D spacetimes (proposed by Plebanski and Demianski [21]), which is represented by seven arbitrary parameters. The charged rotating NUT–AdS BH, in Boyer-Lindquist coordinates [22], is defined by(1)$\begin{eqnarray}\begin{array}{rcl}{{\rm{ds}}}^{2} & = & -\displaystyle \frac{\chi }{{\rm{\Omega }}}{\left[{\rm{d}}t-\left(a{\sin }^{2}\theta +4l{\sin }^{2}\displaystyle \frac{\theta }{2}\right){\rm{d}}\phi \right]}^{2}\\ & & +\displaystyle \frac{{\rm{\Omega }}}{\chi }{\rm{d}}{r}^{2}+\displaystyle \frac{{\rm{\Omega }}}{P}{\rm{d}}{\theta }^{2}\\ & & +\displaystyle \frac{P{\sin }^{2}\theta }{{\rm{\Omega }}}{a{\rm{d}}t-({r}^{2}+{a+l}^{2}){\rm{d}}\phi }^{2},\end{array}\end{eqnarray}$with(2)$\begin{eqnarray}\chi ={{kw}}^{2}+{q}^{2}-2{mr}-{r}^{4}\left(\displaystyle \frac{{\rm{\Lambda }}}{3}\right)+{r}^{2}\epsilon ,\end{eqnarray}$

(3)$\begin{eqnarray}P=1+\displaystyle \frac{4{al}{\rm{\Lambda }}}{3}\cos \theta +\displaystyle \frac{{a}^{2}{\rm{\Lambda }}}{3}{\cos }^{2}\theta ,\end{eqnarray}$

(4)$\begin{eqnarray}{\rm{\Omega }}={r}^{2}+{l+a\cos \theta }^{2},\quad \epsilon =\displaystyle \frac{{{kw}}^{2}}{{a}^{2}-{l}^{2}}-\displaystyle \frac{1}{3}{\rm{\Lambda }}{a}^{2}+3{l}^{2}.\end{eqnarray}$Here, a is the rotation parameter, w is proportional to twisting behavior of the sources, and q is defined as ${q}^{2}={q}_{m}^{2}+{q}_{e}^{2}$, where q_m and q_e denote the magnetic and electric charges, respectively. Also, m is the BH mass, ${\rm{\Lambda }}=-\tfrac{3}{{L}^{2}}$ defines the radius with Λ as the cosmological constant, l is the NUT parameter, and k can be specified as$\begin{eqnarray*}k=\displaystyle \frac{{a}^{2}-{l}^{2}1-{\rm{\Lambda }}{l}^{2}}{{w}^{2}}.\end{eqnarray*}$The line element (1) can be re-written as(5)$\begin{eqnarray}\begin{array}{rcl}{{\rm{ds}}}^{2} & = & -{ \mathcal F }(r,\theta ){\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{{ \mathcal G }(r,\theta )}+{\rm{\Sigma }}(r,\theta ){\rm{d}}{\theta }^{2}\\ & & +K(r,\theta ){\rm{d}}{\phi }^{2}+H(r,\theta ){\rm{d}}t{\rm{d}}\phi ,\end{array}\end{eqnarray}$where(6)$\begin{eqnarray}\begin{array}{rcl}{ \mathcal F }(r,\theta ) & = & \displaystyle \frac{\chi -{a}^{2}P{\sin }^{2}\theta }{{\rm{\Omega }}},\\ { \mathcal G }(r,\theta ) & = & \displaystyle \frac{\chi }{{\rm{\Omega }}},\quad {\rm{\Sigma }}(r,\theta )=\displaystyle \frac{{\rm{\Omega }}}{P},\end{array}\end{eqnarray}$

(7)$\begin{eqnarray}K(r,\theta )=\displaystyle \frac{P{\sin }^{2}\theta {{a+l}^{2}+{r}^{2}}^{2}-\chi {\left(a{\sin }^{2}\theta +4l{\sin }^{2}\tfrac{\theta }{2}\right)}^{2}}{{\rm{\Omega }}},\end{eqnarray}$

(8)$\begin{eqnarray}H(r,\theta )=\displaystyle \frac{{aP}{\sin }^{2}\theta {a+l}^{2}+{r}^{2}-\chi \left(a{\sin }^{2}(\theta )+4l{\sin }^{2}\tfrac{\theta }{2}\right)}{{\rm{\Omega }}}.\end{eqnarray}$The electromagnetic potential for the considered BH solution is given as$\begin{eqnarray*}\begin{array}{rcl}B & = & \displaystyle \frac{-{q}_{e}r[a{\rm{d}}t-({a+l}^{2}-{l+a\cos \theta }^{2}){\rm{d}}\phi ]}{a{\rm{\Omega }}\left(1+\tfrac{{a}^{2}{\rm{\Lambda }}}{3}\right)}\\ & & -\displaystyle \frac{{q}_{m}(l+a\cos \theta )[a{\rm{d}}t-({r}^{2}+{l+a}^{2}){\rm{d}}\phi ]}{a{\rm{\Omega }}\left(1+\tfrac{{a}^{2}{\rm{\Lambda }}}{3}\right)}.\end{array}\end{eqnarray*}$

Generally, the NUT parameter represents the twisting property of the spacetime or gravitomagnetic monopole parameter of the central mass. However, its exact physical interpretation could not be ascertained until a static Schwarzschild mass immersed in the stationary source-free electromagnetic universe is not considered. In this scenario, the NUT parameter is associated with the twist of the electromagnetic universe by excluding the other possibility. In the absence of an electromagnetic field, it relates to the twist of vacuum space. Thus, the NUT parameter is generated by the twist of the surrounding space coupled with the mass of the source. In the Kerr–Newman–NUT–AdS BH, if the NUT parameter dominates the rotation parameter, i.e. a < l, the curvature singularities of the spacetime must vanish and the corresponding solution results in the NUT-like solution. For a > l, a ring singularity appears and the respective solution corresponds to Kerr-like. These cases of curvature singularity have no dependence on the cosmological constant.

Now, we analyze the effects of the NUT parameter on the quantum level. We provide thermal properties of the Kerr–Newman-NUT–AdS BH within the context of the EPS, which correlates pressure with the cosmological constant and the conjugate factor with the BH volume [23]. In this scenario, the area of the event horizon is given by(9)$\begin{eqnarray}\begin{array}{rcl}A & = & {\displaystyle \int }_{0}^{2\pi }{\displaystyle \int }_{0}^{\pi }\sqrt{{g}_{\theta \theta }{g}_{\phi \phi }}{| }_{r={r}_{+}}{\rm{d}}\theta {\rm{d}}\phi \\ & = & 4\pi {a+l}^{2}+{r}_{+}^{2},\end{array}\end{eqnarray}$where r₊ denotes the event horizon of the BH, which is evaluated through χ(r₊) = 0. Using the horizon area, the entropy is defined as(10)$\begin{eqnarray}S=\pi {a+l}^{2}+{r}_{+}^{2}.\end{eqnarray}$For the considered BH, the Hawking temperature is calculated as $({T}_{k}=\tfrac{{\chi }^{{\prime} }(r)}{4\pi ({r}^{2}+{a+l}^{2})}{| }_{r={r}_{+}})$ [15] and is evaluated as(11)$\begin{eqnarray}\begin{array}{rcl}{T}_{k} & = & \displaystyle \frac{{a}^{2}{r}_{+}}{2\pi {L}^{2}{r}_{+}^{2}+{l+a}^{2}}+\displaystyle \frac{3{l}^{2}{r}_{+}}{\pi {L}^{2}{r}_{+}^{2}+{l+a}^{2}}\\ & & +\displaystyle \frac{{r}_{+}^{3}}{\pi {L}^{2}{r}_{+}^{2}+{l+a}^{2}}\\ & & -\displaystyle \frac{m}{2\pi {r}_{+}^{2}+{l+a}^{2}}+\displaystyle \frac{{r}_{+}}{2\pi {r}_{+}^{2}+{l+a}^{2}},\end{array}\end{eqnarray}$where the mass is given by(12)$\begin{eqnarray}m=\displaystyle \frac{3{a}^{2}{l}^{2}+{a}^{2}{L}^{2}+{a}^{2}{r}_{+}^{2}+{q}^{2}{L}^{2}-3{l}^{4}-{l}^{2}{L}^{2}+6{l}^{2}{r}_{+}^{2}+{L}^{2}{r}_{+}^{2}+{r}_{+}^{4}}{2{L}^{2}{r}_{+}}.\end{eqnarray}$

The angular velocity is evaluated as(13)$\begin{eqnarray}\begin{array}{l}{{\rm{\Pi }}}_{r}=-\displaystyle \frac{{g}_{t\phi }}{{g}_{\phi \phi }}\\ =\,\displaystyle \frac{{aP}{\sin }^{2}(\theta ){a+l}^{2}+{r}^{2}-\chi \left(a{\sin }^{2}(\theta )+4l{\sin }^{2}\left(\tfrac{\theta }{2}\right)\right)}{P{\sin }^{2}(\theta ){{a+l}^{2}+{r}^{2}}^{2}-\chi {\left(a{\sin }^{2}(\theta )+4l{\sin }^{2}\left(\tfrac{\theta }{2}\right)\right)}^{2}}.\end{array}\end{eqnarray}$The radial function χ becomes zero at horizon r = r₊, which yields(14)$\begin{eqnarray}{{\rm{\Pi }}}_{H}=\displaystyle \frac{a}{{r}_{+}^{2}+{a+l}^{2}},\end{eqnarray}$where Π_H is the angular velocity. From equations (10) and (11), we obtain(15)$\begin{eqnarray}{T}_{k}S=\displaystyle \frac{{r}_{+}^{2}{a}^{2}+6{l}^{2}+{L}^{2}-{a}^{2}3{l}^{2}+{L}^{2}+{L}^{2}{l}^{2}-{q}^{2}+3{l}^{4}+3{r}_{+}^{4}}{4{L}^{2}{r}_{+}},\end{eqnarray}$which, in terms of thermodynamic quantities such as Q = q, M = m, J = ma, $\psi =\tfrac{1}{8\pi l}$, and $P=\tfrac{3}{8\pi {L}^{2}}$, satisfies the Smarr relation given as(16)$\begin{eqnarray}M=2{T}_{k}S-{PV}+{\rm{\Pi }}J+\psi N+Q{\rm{\Phi }},\end{eqnarray}$where the new charge factor (N) (related to the NUT parameter), thermodynamical volume (V) and the electric potential (Φ) read$\begin{eqnarray*}\begin{array}{rcl}N & = & 2\pi {l}^{2}(2(2a+l){L}^{2}{a}^{2}+{q}^{2}-{l}^{2}\\ & & +3{l}^{2}(a-l)(a+l+{r}_{+}^{2}(2{a}^{3}\\ & & +{a}^{2}l+12{{al}}^{2}-2{{lL}}^{2})+6{{lr}}_{+}^{4}{{L}^{2}{r}_{+}{a+l}^{2}+{r}_{+}^{2}}^{-1},\\ V & = & {\left(\displaystyle \frac{\partial M}{\partial P}\right)}_{S,Q,J,N}=\displaystyle \frac{2}{3}\pi {r}_{+}\left(\displaystyle \frac{{a}^{2}{a}^{2}+{r}_{+}^{2}}{{r}_{+}^{2}+{a+l}^{2}}+2{r}_{+}^{2}+{a}^{2}+6{l}^{2}\right),\\ {\rm{\Phi }} & = & \displaystyle \frac{{{qr}}_{+}}{{r}_{+}^{2}+{l+a}^{2}}.\end{array}\end{eqnarray*}$For l = 0, the derived results reduce to a charged rotating AdS BH [24]. From equation (16), it can be seen that the NUT parameter is an independently varied function and can be introduced separately in the first law. The first law of thermodynamics, within the context of the EPS, is expressed as(17)$\begin{eqnarray}{\rm{d}}M={T}_{k}{\rm{d}}S+{\rm{\Pi }}{\rm{d}}J+{\rm{\Phi }}{\rm{d}}Q+V{\rm{d}}P+\psi {\rm{d}}N,\end{eqnarray}$where the corresponding potential functions are given by$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Pi }} & = & {\left(\displaystyle \frac{\partial M}{\partial J}\right)}_{S,Q,P,N},\quad {\rm{\Phi }}={\left(\displaystyle \frac{\partial M}{\partial Q}\right)}_{S,T,P,N},\\ \psi & = & {\left(\displaystyle \frac{\partial M}{\partial N}\right)}_{S,J,Q,P,},\quad {T}_{k}={\left(\displaystyle \frac{\partial M}{\partial S}\right)}_{J,Q,P,N}.\end{array}\end{eqnarray*}$

3. Thermal fluctuations

This section is devoted to examining the impact of thermal fluctuations on the thermodynamics of a charged rotating NUT–AdS BH. We firstly compute corrected entropy near the equilibrium position, which implies modification in other thermodynamic potentials. For this purpose, we consider the function(18)$\begin{eqnarray}{Z}(\beta )={\int }_{0}^{\infty }{{\rm{e}}}^{-\eta {E}}\sigma ({E}){\rm{d}}{E},\end{eqnarray}$where Σ(E) corresponds to the quantum density of the system, and E represents the average energy with ${T}_{k}=\tfrac{1}{\eta }$ [3]. Using inverse transformation, we have(19)$\begin{eqnarray}\begin{array}{rcl}\sigma ({E}) & = & \displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{b-{i}\infty }^{b+i\infty }{{\rm{e}}}^{\eta {E}}{Z}(\eta ){\rm{d}}\eta \\ & = & \displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{b-i\infty }^{b+i\infty }{{\rm{e}}}^{{S}_{0}(\eta )}{\rm{d}}\eta ,\end{array}\end{eqnarray}$where S₀ = lnZ + βE is the corrected entropy, and b > 0. Using the steepest descent approach near η, the above equation reduces to(20)$\begin{eqnarray}{S}_{0}(\eta )=S+\displaystyle \frac{1}{2}{\eta -b}^{2}\displaystyle \frac{{{\rm{d}}}^{2}S}{{\rm{d}}{\eta }^{2}}{| }_{\eta =b}+\mathrm{higher}-\mathrm{order}\ \mathrm{terms},\end{eqnarray}$where S = S₀(η) with $\tfrac{\partial S}{\partial \eta }=0$ and $\tfrac{{\partial }^{2}S}{\partial {\eta }^{2}}\gt 0$ at η = b. Using equations (19) and (20), we obtain(21)$\begin{eqnarray}\sigma ({\rm{E}})=\displaystyle \frac{{e}^{S}}{2\pi i}{\int }_{b-i\infty }^{b+i\infty }{{\rm{e}}}^{\tfrac{1}{2}{\eta -b}^{2}\tfrac{{{\rm{d}}}^{2}{S}_{0}}{{\rm{d}}{\eta }^{2}}}{\rm{d}}\eta ,\end{eqnarray}$which can be written as(22)$\begin{eqnarray}\sigma ({\rm{E}})=\displaystyle \frac{{{\rm{e}}}^{S}}{\sqrt{2\pi \tfrac{{{\rm{d}}}^{2}{S}_{0}}{{\rm{d}}{\eta }^{2}}}}.\end{eqnarray}$Eventually, this leads to$\begin{eqnarray*}{S}_{0}=S-\displaystyle \frac{1}{2}\mathrm{ln}({{ST}}_{k}^{2}).\end{eqnarray*}$

Without loss of generality, we can substitute a general correction parameter α in place of the factor $\tfrac{1}{2}$ to increase the participation of correction terms in the entropy. Around the equilibrium phase, the corrected entropy takes the form(23)$\begin{eqnarray}{S}_{0}=S-\alpha \mathrm{ln}({{ST}}_{k}^{2}).\end{eqnarray}$Notice that the above expression contains a logarithmic term which shows the small contribution of quantum corrections. It is known that statistical perturbations become efficient on the Planck scale, whereas the BHs are macroscopic stellar objects; therefore, the logarithmic corrections have little influence on the equilibrium entropy. From equations (10) and (11), the corrected entropy turns out to be(24)$\begin{eqnarray}\begin{array}{rcl}{S}_{0} & = & \pi {a+l}^{2}+{r}_{+}^{2}-\alpha \mathrm{ln}\pi {r}_{+}^{2}+{l+a}^{2}\\ & & \left(\displaystyle \frac{{a}^{2}{r}_{+}}{2\pi {L}^{2}{r}_{+}^{2}+{l+a}^{2}}+\displaystyle \frac{3{l}^{2}{r}_{+}}{\pi {L}^{2}{r}_{+}^{2}+{l+a}^{2}}\right.\\ & & +\displaystyle \frac{{r}_{+}^{3}}{\pi {L}^{2}{r}_{+}^{2}+{l+a}^{2}}-\displaystyle \frac{M}{2\pi {r}_{+}^{2}+{l+a}^{2}}\\ & & \left.{\left.+\displaystyle \frac{{r}_{+}}{2\pi {r}_{+}^{2}+{l+a}^{2}}\right)}^{2}\right).\end{array}\end{eqnarray}$

To study the effects of state parameters, we plot entropy (corrected and uncorrected) for different choices of NUT and rotation parameters. For graphical analysis, we have considered two cases, i.e. a > l (figure 1) and a < l (figure 2) with L = 2 and q = 0.5. It is found that for both cases, the equilibrium entropy (α = 0) behaves as a monotonically increasing function (with maximum value at the event horizon) and attains positive values in the considered domain. It shows that the uncorrected entropy meets the second law of BH thermodynamics, which states that the BH entropy always follows the increasing trend. However, due to quantum corrections, the small BH (for a > l) obtains the negative value of entropy against larger choices of correction parameter and shows decreasing behavior for a specific range of horizon radius. We observe that BH entropy increases for larger modes of NUT and rotation parameters, which correspondingly increases the area of the BH. For a larger horizon radius, the behavior of corrected entropy coincides with the uncorrected one, which implies that the thermodynamics of a large BH is not affected by thermal fluctuations. From equation (16), the corrected mass can be computed as(25)$\begin{eqnarray}\begin{array}{rcl}{M}_{0} & = & 2{a}^{2}({a}^{2}3{l}^{2}+{L}^{2}+{r}_{+}^{2}+{q}^{2}{L}^{2}\\ & & +{r}_{+}^{2}6{l}^{2}+{L}^{2}-{l}^{2}3{l}^{2}+{L}^{2}+{r}_{+}^{4}\\ & & {{a+l}^{2}+{r}_{+}^{2}}^{-1}-{a}^{2}3{l}^{2}+{L}^{2}-{r}_{+}^{2}\\ & & +{q}^{2}{L}^{2}-{l}^{2}+{r}_{+}^{2}\\ & \times & 3{l}^{2}+{L}^{2}+3{r}_{+}^{2})\alpha \\ & & -\mathrm{ln}{a}^{2}-3{l}^{2}-{L}^{2}+{r}_{+}^{2}-{q}^{2}{L}^{2}\\ & & {+{l}^{2}+{r}_{+}^{2}3{l}^{2}+{L}^{2}+3{r}_{+}^{2}}^{2}\\ & & {{L}^{4}{r}_{+}^{2}{a+l}^{2}+{r}_{+}^{2}}^{-1}\\ & & +\pi {r}_{+}^{2}+{a+l}^{2}+\alpha \mathrm{ln}(16\pi \\ & & {\pi {r}_{+}^{2}+{a+l}^{2}}^{-1}-{r}_{+}^{2}({a}^{2}\\ & & +6{l}^{2}+2{r}_{+}^{2}+\displaystyle \frac{{a}^{2}{a}^{2}+{r}_{+}^{2}}{{a+l}^{2}+{r}_{+}^{2}})\\ & & +l(2(2a+l)({L}^{2}{a}^{2}+{q}^{2}-{l}^{2}+3{l}^{2}({a}^{2}\\ & & -{l}^{2}))+{r}_{+}^{2}2{a}^{3}+{a}^{2}l+12{{al}}^{2}-2{{lL}}^{2}\\ & & +6{{lr}}_{+}^{4}{{a+l}^{2}+{r}_{+}^{2}}^{-1}\\ & \times & {2{L}^{2}{r}_{+}}^{-1}+\displaystyle \frac{{q}^{2}{r}_{+}}{{a+l}^{2}+{r}_{+}^{2}}.\end{array}\end{eqnarray}$

Figure 1.

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Figure 1.Corrected entropy versus event horizon for q = 0.5, L = 2 with a = 1 > l = 0.1 (left plot), 0.9 (right plot). Here, α = 0.9, 0.5, and 0 are represented by blue, green, and red curves, respectively.

Figure 2.

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Figure 2.Corrected entropy versus event horizon for q = 0.5, L = 2 with l = 1 > a = 0.1 (left plot), 0.9 (right plot).

Figures 3 and 4 represent the graphical behavior of corrected mass for different values of NUT and rotation parameters, respectively. We observe that for a > l, the mass of the BH remains positive while, in the case of a < l, the physical mass becomes negative for small BHs. Thus, one can conclude that the case a > l depicts a more proficient and realistic scenario in contrast to another possibility as the mass can never be a negative quantity. Figure 3 shows that BH mass decreases until the critical horizon; thereafter, it is an increasing function. We find that the correction parameter decreases and increases the corrected BH mass before and after the horizon radius, respectively. From figure 4, one can observe a continuous increase in the physical mass, and the critical horizon radius decreases for larger values of the rotation parameter. It is found that for larger values of l and a, the BH becomes more massive.

Figure 3.

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Figure 3.Corrected mass versus event horizon for q = 0.5, L = 2 with a = 1 > l = 0.1 (left plot), 0.9 (right plot).

Figure 4.

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Figure 4.Corrected mass versus event horizon for q = 0.5, L = 2 with l = 1 > a = 0.1 (left plot), 0.9 (right plot).

The internal energy, as the total energy of the BH, is directly proportional to the temperature. Using the definition, U = M₀ − PV − ψN, the internal energy is evaluated as(26)$\begin{eqnarray}\begin{array}{rcl}U & = & -\alpha \mathrm{ln}(16\pi ){a}^{2}3{l}^{2}+{L}^{2}-{r}_{+}^{2}\\ & & +{q}^{2}{L}^{2}-{l}^{2}+{r}_{+}^{2}3{l}^{2}+{L}^{2}+3{r}_{+}^{2}\\ & & +\alpha {a}^{2}-3{l}^{2}-{L}^{2}+{r}_{+}^{2}-{q}^{2}{L}^{2}+{l}^{2}+{r}_{+}^{2}\\ & & \times \,3{l}^{2}+{L}^{2}+3{r}_{+}^{2}\\ & & \times \mathrm{ln}{a}^{2}-3{l}^{2}-{L}^{2}+{r}_{+}^{2}-{q}^{2}{L}^{2}\\ & & {+{l}^{2}+{r}_{+}^{2}3{l}^{2}+{L}^{2}+3{r}_{+}^{2}}^{2}\\ & & \times {{L}^{4}{r}_{+}^{2}{a+l}^{2}+{r}_{+}^{2}}^{-1}\\ & & +\ \pi ({a}^{4}-3{l}^{2}+{L}^{2}+{a}^{2}(3{l}^{2}+{L}^{2}\\ & & \times {l}^{2}-2{r}_{+}^{2}-{q}^{2}{L}^{2})-2{{alL}}^{2}{r}_{+}^{2}\\ & & +{L}^{2}-{q}^{2}{r}_{+}^{2}-{r}_{+}^{2}{l}^{2}+{r}_{+}^{2}\\ & & \times {2\pi {L}^{2}{r}_{+}{a+l}^{2}+{r}_{+}^{2}}^{-1}.\end{array}\end{eqnarray}$Figure 5 provides evidence that higher modes of the NUT parameter increase the internal energy, indicating that BHs have a high temperature. Due to the fluctuation effect, the internal energy decreases and increases before and after the horizon radius, respectively. Figure 6 shows that the internal energy becomes negative for a small BH, which shows that the BH is releasing heat to its surroundings. However, for large BHs, it depicts increasing as well as positive behavior. It is observed that the system attains negative values corresponding to larger modes of rotation.

Figure 5.

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Figure 5.Internal energy versus event horizon for q = 0.5, L = 2 with a = 1 > l = 0.1 (left plot), 0.9 (right plot).

Figure 6.

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Figure 6.Internal energy versus event horizon for q = 0.5, L = 2 with l = 1 > a = 0.1 (left plot), 0.9 (right plot).

The Helmholtz free energy (F = M₀ − T_kS₀ − ψN) is the direct measure of work that can be extracted from a system. If the system achieves its reversible equilibrium state, the Helmholtz free energy becomes constant. The first-order corrected Helmholtz free energy is given by(27)$\begin{eqnarray}\begin{array}{rcl}F & = & \left(\pi {a}^{4}\left(9{l}^{2}+3{L}^{2}+{r}_{+}^{2}\right)+2\pi {a}^{3}l\left(3{l}^{2}+{L}^{2}\right)\right.\\ & & +\alpha ({a}^{2}\left(3{l}^{2}+{L}^{2}-{r}_{+}^{2}\right)\\ & & +{q}^{2}{L}^{2}-\left({l}^{2}+{r}_{+}^{2}\right)\left(3{l}^{2}+{L}^{2}+3{r}_{+}^{2}\right))\\ & & \times \,\mathrm{ln}\left(\left(\left({a}^{2}\left(-3{l}^{2}-{L}^{2}+{r}_{+}^{2}\right)-{q}^{2}{L}^{2}\right.\right.\right.\\ & & \left.{\left.+\left({l}^{2}+{r}_{+}^{2}\right)\left(3{l}^{2}+{L}^{2}+3{r}_{+}^{2}\right)\right)}^{2}\right)\\ & & \left.\times \,{\left({L}^{4}{r}_{+}^{2}\left({\left(a+l\right)}^{2}+{r}_{+}^{2}\right)\right)}^{-1}\right)+{a}^{2}\left(\pi ({L}^{2}(3{q}^{2}\right.\\ & & +4{r}_{+}^{2})-6{l}^{4}+{l}^{2}\left(15{r}_{+}^{2}-2{L}^{2}\right))\\ & & \left.+\alpha \mathrm{log}(16\pi )\left(-3{l}^{2}-{L}^{2}+{r}_{+}^{2}\right)\right)\\ & & -2\pi {al}\left(-{q}^{2}{L}^{2}+3{l}^{4}+{l}^{2}{L}^{2}-{L}^{2}{r}_{+}^{2}+{r}_{+}^{4}\right)\\ & & +\alpha \,\mathrm{log}(16\pi )\left(\left({l}^{2}+{r}_{+}^{2}\right)(3{l}^{2}\right.\\ & & \left.+{L}^{2}+3{r}_{+}^{2})-{q}^{2}{L}^{2}\right)-\pi ({r}_{+}^{2}\left(3{l}^{4}-3{q}^{2}{L}^{2}\right)\\ & & +{l}^{2}{L}^{2}\left({l}^{2}-{q}^{2}\right)+3{l}^{6}\\ & & \left.+{r}_{+}^{4}({l}^{2}-{L}^{2})+{r}_{+}^{6}\right)\left(4\pi {L}^{2}r\right.\\ & & \times \,{\left.\left({\left(a+l\right)}^{2}+{r}_{+}^{2}\right)\right)}^{-1}.\end{array}\end{eqnarray}$Figure 7 shows that the small BH has higher Helmholtz free energy, whereas for the large BH, the free energy gains negative values and observes the same trend as that of the equilibrium state. The negative behavior of F shows that entropy and NUT charge dominate the physical mass of the BH. It is noted that smaller values of l yield higher values of the Helmholtz free energy. Figure 8 shows that smaller values of the rotation than the NUT parameter (l > a) lead to negative values of the Helmholtz free energy, which becomes positive by considering larger values of the rotation and correction parameters. It is important to note that leading order correction terms play a critical part in the thermodynamics of small BHs, whereas the large BHs remain unaffected. The BH mass, within the context of the EPS, is named enthalpy, and Gibbs free energy is utilized to quantify the reversible work that might be carried out by a thermodynamic system. The Gibbs energy (G = M₀ − T_kS₀ − ΦQ − ψN) is derived to be$\begin{eqnarray}\begin{array}{rcl}G & = & \pi {a}^{4}9{l}^{2}+3{L}^{2}+{r}_{+}^{2}+2\pi {a}^{3}l3{l}^{2}+{L}^{2}\\ & & +\alpha {a}^{2}3{l}^{2}+{L}^{2}-{r}_{+}^{2}\\ & & +{q}^{2}{L}^{2}-{l}^{2}+{r}_{+}^{2}3{l}^{2}+{L}^{2}+3{r}_{+}^{2}\\ & & \times \,\mathrm{ln}{a}^{2}-3{l}^{2}-{L}^{2}+{r}_{+}^{2}\\ & & {-{q}^{2}{L}^{2}+{l}^{2}+{r}_{+}^{2}3{l}^{2}+{L}^{2}+3{r}_{+}^{2}}^{2}\\ & & {{L}^{4}{r}_{+}^{2}{a+l}^{2}+{r}_{+}^{2}}^{-1}+{a}^{2}(\pi \end{array}\end{eqnarray}$

(28)$\begin{eqnarray}\begin{array}{rll} & \,\times & {L}^{2}3{q}^{2}+4{r}_{+}^{2}-6{l}^{4}+{l}^{2}15{r}_{+}^{2}-2{L}^{2}\\ & & +\alpha \mathrm{ln}(16\pi )(-3{l}^{2}-{L}^{2}\\ & & +{r}_{+}^{2}))-2\pi {al}-{q}^{2}{L}^{2}+3{l}^{4}+{l}^{2}{L}^{2}-{L}^{2}{r}_{+}^{2}+{r}^{4}\\ & & +\alpha \mathrm{ln}(16\pi )({l}^{2}+{r}_{+}^{2}\\ & & \times \,3{l}^{2}+{L}^{2}+3{r}_{+}^{2}-{q}^{2}{L}^{2})-\pi ({r}_{+}^{2}{q}^{2}{L}^{2}+3{l}^{4}\\ & & +{l}^{2}{L}^{2}{l}^{2}-{q}^{2}+3{l}^{6}\\ & & +{r}_{+}^{4}({l}^{2}-{L}^{2})+{r}_{+}^{6}{4\pi {L}^{2}{r}_{+}{a+l}^{2}+{r}_{+}^{2}}^{-1}.\end{array}\end{eqnarray}$

Figure 7.

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Figure 7.Helmholtz energy versus event horizon for q = 0.5, L = 2 with a = 1 > l = 0.1 (left plot), 0.9 (right plot).

Figure 8.

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Figure 8.Helmholtz energy versus event horizon for q = 0.5, L = 2 with l = 1 > a = 0.1 (left plot), 0.9 (right plot).

Figures 9 and 10 indicate that the Gibbs free energy remains positive for small and medium BHs, while it becomes negative for larger values of the horizon radius. It is known that positive values of the Gibbs energy correspond to non-spontaneous reactions that require an external source of energy, whereas its negative values correspond to spontaneous reactions which can be driven without any external source. BHs with negative Gibbs energy are thermodynamically stable as they release their energy into the surroundings to acquire the low-energy state. It is seen that small and medium BHs are thermodynamically unstable as G > 0. It is also noted that correction terms increase the Gibbs free energy for small BHs but, for large BHs, its negative range increases corresponding to larger values of acceleration and rotation parameters. This indicates that larger values of state parameters yield the stable model. Figure 10 implies that the negative profile of the Gibbs energy decreases against the higher choices of rotation parameter, which shows that the smaller modes lead to the stable model.

Figure 9.

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Figure 9.Gibbs energy versus event horizon for q = 0.5, L = 2 with a = 1 > l = 0.1 (left plot), 0.9 (right plot).

Figure 10.

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Figure 10.Gibbs energy versus event horizon for q = 0.5, L = 2 with l = 1 > a = 0.1 (left plot), 0.9 (right plot).

To study the stability and phase transition, the specific heat is computed within the context of thermal fluctuations. The transition points are simply the divergence points of specific heat, whereas its positive range ensures the thermodynamically stable phase. The specific heat $(C=T\tfrac{\partial {S}_{0}}{\partial T})$ can be calculated as$\begin{eqnarray}\begin{array}{rcl}C & = & -2{a}^{4}\alpha 3{l}^{2}+{L}^{2}+{r}_{+}^{2}\\ & & +\pi {r}_{+}^{2}3{l}^{2}+{L}^{2}-{r}_{+}^{2}+2{a}^{3}l\alpha (3{l}^{2}\\ & & +{L}^{2}+{r}_{+}^{2})+\pi {r}_{+}^{2}3{l}^{2}+{L}^{2}-{r}_{+}^{2}\\ & & +{a}^{2}{q}^{2}{L}^{2}\alpha +\pi {r}_{+}^{2}+\alpha {r}_{+}^{2}(13{l}^{2}\\ & & +3{L}^{2}+9{r}_{+}^{2})-4\pi {r}_{+}^{4}{l}^{2}+{r}_{+}^{2}\end{array}\end{eqnarray}$

(29)$\begin{eqnarray}\begin{array}{rcl} & & \,+2{al}\alpha {q}^{2}{L}^{2}+{r}_{+}^{2}6{l}^{2}+{L}^{2}-{l}^{2}\\ & & \,\times \,3{l}^{2}+{L}^{2}+9{r}_{+}^{4}-\pi {r}_{+}^{2}{l}^{2}+{r}_{+}^{2}\\ & & \,3{l}^{2}+{L}^{2}+3{r}_{+}^{2}-{q}^{2}{L}^{2}\\ & & \,-\pi {r}_{+}^{2}{l}^{2}+{r}_{+}^{2}{l}^{2}+{r}_{+}^{2}3{l}^{2}+{L}^{2}+3{r}_{+}^{2}\\ & & \,-{q}^{2}{L}^{2}+\alpha {L}^{2}{r}_{+}^{2}2{q}^{2}-{l}^{2}\\ & & \,+{l}^{2}{L}^{2}{q}^{2}-{l}^{2}-3{l}^{6}+9{l}^{2}{r}_{+}^{4}+6{r}_{+}^{6}\\ & & \,{a+l}^{2}{L}^{2}{a}^{2}+{q}^{2}-{l}^{2}\\ & & \,+3{l}^{2}(a-l)(a+l+{r}_{+}^{4}8{a}^{2}+18{al}\\ & & \,+3{l}^{2}-{L}^{2}+{r}_{+}^{2}{a}^{4}+2{a}^{3}l\\ & & \,+16{a}^{2}{l}^{2}+{L}^{2}2(2a-l)(a+l)+3{q}^{2}\\ & & \,{+12{{al}}^{3}-3{l}^{4}+3{r}_{+}^{6}}^{-1}.\end{array}\end{eqnarray}$

The BH with larger choices of the NUT parameter yields larger heat capacity values (figure 11). From figure 12, one can observe that the specific heat diverges at critical radii r₊ = 1.3 and r₊ = 0.18 for a = 0.1 and a = 0.9, respectively, which shows that the BH experiences the first-order phase transition. Notably, the position, as well as the number of the transition, points rely on the considered choices of BH parameters. For the small BH, the uncorrected specific heat is negative, which indicates that the rotating BH is unstable even without consideration of any thermal fluctuation effects. However, for larger modes of rotation parameters, the corrected specific heat becomes more negative for small BHs without affecting the large BH’s thermodynamics. Thus, we can conclude that small BHs are thermodynamically unstable due to statistical perturbations, while this does not affect the stability of large-sized BHs.

Figure 11.

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Figure 11.Specific heat versus event horizon for q = 0.5, L = 2 with a = 1 > l = 0.1 (left plot), 0.9 (right plot).

Figure 12.

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Figure 12.Specific heat versus event horizon for q = 0.5, L = 2 with l = 1 > a = 0.1 (left plot), 0.9 (right plot).

4. Conclusions

In this paper, we have analyzed the influence of statistical fluctuations on the thermodynamics of the Kerr–Newman–NUT–AdS BH. For this purpose, the exact expression of the Hawking temperature, angular velocity, and entropy are computed. We have found that these variables meet the first law of thermodynamics as well as the Smarr relation in the presence of a new NUT charge in contrast to the Kerr–Newman–Taub–NUT BH [18]. To investigate the influence of fluctuations, we have computed corrected entropy, which modifies other thermodynamic quantities. We have plotted these thermodynamic potentials and compared their corrected and uncorrected forms for different choices of rotation and NUT parameters. Finally, we have studied the phase transition points as well as the stability of the BH through specific heat.

It is observed that the entropy of the BH increases against larger values of rotation and NUT parameters, which leads to the increase in the BH area. The leading order correction terms perturb the entropy of small BHs while, for BHs with a larger horizon radius, the corrected entropy observes the same behavior as that of equilibrium entropy, which implies that logarithmic corrections do not affect the thermodynamics of large BHs. For the two possibilities, i.e. a > l and l > a, the former represents the realistic scenario as it provides a positive range of the mass for small as well as large BHs. The profile of internal energy shows that the temperature of small BHs decreases for a > l, which indicates that the BH emits thermal radiation to its surroundings. However, the internal energy of the large BH increases due to quantum fluctuation effects. For smaller values of horizon radius, the Helmholtz free energy becomes positive against a > l, while it shows a negative as well as a decreasing trend for large BHs. It is noted that smaller values of l yield higher values of the Helmholtz free energy.

The Gibbs energy is negative (positive) for l > a (a > l) indicating a stable (unstable) phase of small BHs. For BHs with a larger horizon radius, the Gibbs energy is negative for both considered cases, which leads to stable BH geometries. The profile of specific heat is studied versus the horizon radius to analyze the local stability of the BH. We observe that for large BHs, the specific heat attains positive values, which indicate that large BHs are located in a thermally stable regime [11]. Moreover, the larger values of l lead the system towards stability. We observe that the BH experiences first-order phase transition due to divergence of the specific heat at r₊ = 1.3 against smaller values of the rotation parameter. It is concluded that thermal fluctuations (NUT parameter) induce more instability (stability) in small BHs. It is noteworthy that all the results reduce to rotating as well as charged AdS BHs [24] in the absence of the NUT parameter and, for q = a = 0, it leads to the NUT–AdS BH solution [19].

Acknowledgments

QM would like to thank the Higher Education Commission, Islamabad, Pakistan for its financial support through the Indigenous Ph.D. Fellowship, Phase-II, Batch-III.