Weak cosmic censorship conjecture in the nonlinear electrodynamics black hole under the charged scal
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Tianhao Bai,1, Wei Hong,1, Benrong Mu,2, Jun Tao,1,31Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu, 610065, China 2Physics Teaching and Research Section, College of Medical Technology, Chengdu University of Traditional Chinese Medicine, Chengdu 611137, China
First author contact:3Author to whom any correspondence should be addressed. Received:2019-08-12Revised:2019-09-26Accepted:2019-10-31Online:2020-01-07
Abstract In this paper, we study the thermodynamics and the weak cosmic censorship conjecture of the nonlinear electrodynamics black hole under the scattering of a charged complex scalar field. According to the energy and charge fluxes of the scalar field, the variations of this black hole’s energy and charge can be calculated during an infinitesimal time interval. With scalar field scattering, the variation of the black hole is calculated in the extended and normal phase spaces. In the normal phase space, the cosmological constant and the normalization parameter are fixed, and the first and second laws of thermodynamics can also be satisfied. In the extended phase space, the cosmological constant and the normalization parameter are considered as thermodynamic variables, and the first law of thermodynamics is valid, but the second law of thermodynamics is not valid. Furthermore, the weak cosmic censorship conjecture is both valid in the extended and normal phase spaces. Keywords：nonlinear electrodynamics black hole;weak cosmic censorship conjecture;complex scalar field;thermodynamics
PDF (338KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Tianhao Bai, Wei Hong, Benrong Mu, Jun Tao. Weak cosmic censorship conjecture in the nonlinear electrodynamics black hole under the charged scalar field. Communications in Theoretical Physics, 2020, 72(1): 015401- doi:10.1088/1572-9494/ab544b
1. Introduction
The research of black hole theory in general relativity reveals a deep and fundamental relationship between gravity, thermodynamics and quantum theory. Since Hawking and Bekenstein [1–3] did much of the work on the black holes and the Universe, the research on the thermodynamics of black holes has developed rapidly. The strong implication of the thermal properties of black holes comes from the behavior of their macroscopic properties. Similar to the laws of thermodynamics, the four laws of thermodynamics for black holes are established in [4]. Hawking proposed that the radiation from black holes can be found through quantum tunneling [5]. This idea provides a real link between quantum mechanics and gravity. The main consideration for black holes is that they have a certain geometrical boundary, which is called the event horizon. The event horizon is an important part of studying the nature of black holes. In fact, the thermodynamics of black holes is constructed by the thermodynamic parameters which are obtained at the horizon. There are several ways to calculate the temperature and entropy of black holes [6–8]. A general argument is that the Hawking temperature of a black hole is proportional to the surface gravity κ at the horizon, and the entropy is proportional to the area of the black hole. According to Penrose’s theory [9], all singularities in spacetime have to be behind event horizon. In other words, the spacetime singularities that arise in gravitational collapse are always hidden inside of black holes, invisible to distant observers, which is known as the weak cosmic censorship conjecture.
The thermodynamic laws and the weak cosmic censorship conjecture can be tested by the absorptions of a particle or by the scattering of an external field through a black hole. In a seminal work, Wald innovatively proposed a method to test this conjecture in the extreme Kerr-Newman black hole by absorbing a particle, which showed that the conjecture was satisfied [10]. However, there are some controversies about the weak cosmic censorship conjecture, as this conjecture is violated in the near-extreme Reissner-Nordström black hole [11] and the near-extreme Kerr black hole [12]. The validity of weak cosmic censorship conjecture in the context of various black holes via the absorption of a charged or rotating particle has received a lot of attention and was followed by extensive studies [13–25, 34, 26–33]. Therefore, it is important and meaningful to study the thermodynamics and the weak cosmic censorship conjecture of black holes.
Nonlinear electrodynamics is an effective model that combines the quantum correction theory with Maxwell electromagnetic theory. Coupling nonlinear electrodynamics to gravity, various nonlinear electrodynamics charged black holes were derived and discussed in many papers [35–41]. The thermodynamics of nonlinear electrodynamics black holes have been considered in literature [42–46]. The thermodynamics of the black holes with cosmological constant Λ as a thermodynamical variable have been studied in the references [47–49]. The physical interpretation of the conjugate variable associated with Λ is a point of interest till now. In the extended phase space, the cosmological constant and the normalization parameter are considered thermodynamic variables. Thermodynamics and weak cosmic censorship conjecture in nonlinear electrodynamics black holes via charged particle absorption has been analyzed by [50]. With these variations, they checked the first and second laws of thermodynamics and the weak cosmic censorship conjecture for the nonlinear electrodynamics black hole in the test particle limit.
In this paper, we discuss the thermodynamic laws and weak cosmic censorship conjecture for a D-dimensional AdS charged black hole in a general Einstein-nonlinear electrodynamics theory under the scattering of a charged complex scalar field. The paper is organized as follows. In section 2, we investigate the dynamical of the charged complex scalar field, and calculate the variations of this black hole’s energy and charge within the certain time interval. In section 3, the thermodynamics of the nonlinear electrodynamics black holes are discussed in the extended and normal phase space. Moreover, we test the validity of the weak cosmic censorship conjecture in those case. We summarize our results in section 4.
2. Complex scalar field in the nonlinear electrodynamics black hole
2.1. Black hole solution
For general nonlinear electrodynamics theories, the D-dimensional static and spherically symmetric black hole solution with an electric field has already been given by [50–52]. Consider a D-dimensional model of gravity coupled to a nonlinear electromagnetic field Aμ with the bulk action$ \begin{eqnarray}{{ \mathcal S }}_{\mathrm{Bulk}}=\int {{\rm{d}}}^{D}x\sqrt{-g}\left[\displaystyle \frac{R-2{\rm{\Lambda }}}{16\pi }+\displaystyle \frac{{ \mathcal L }\left(s;{a}_{i}\right)}{4\pi }\right],\end{eqnarray}$where l is the AdS radius, and the cosmological constant ${\rm{\Lambda }}=-\tfrac{(D-1)(D-2)}{2{l}^{2}}$. In the action, we assume that the generic nonlinear electrodynamics Lagrangian ${ \mathcal L }\left(s;{a}_{i}\right)$ is a function of s and the dimensionful parameters ai. With the calculations of [50], one can construct a charged AdS black hole solution$ \begin{eqnarray}{\rm{d}}{s}^{2}=-f(r){\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{f(r)}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{D-2}^{2}\end{eqnarray}$$ \begin{eqnarray}{A}_{\mu }{\rm{d}}{x}^{\mu }={A}_{t}(r){\rm{d}}t\end{eqnarray}$with$ \begin{eqnarray}\begin{array}{rcl}f(r) & = & 1-\displaystyle \frac{16\pi M}{(D-2){r}^{D-3}{{\rm{\Omega }}}_{D-2}}+\displaystyle \frac{{r}^{2}}{{l}^{2}}-\displaystyle \frac{4}{(D-2){r}^{D-3}}\\ & & \times {\displaystyle \int }_{r}^{\infty }{r}^{D-2}\left[{ \mathcal L }\left(s;{a}_{i}\right)-{A^{\prime} }_{t}(r)\displaystyle \frac{4\pi Q}{{r}^{D-2}{{\rm{\Omega }}}_{D-2}}\right]{\rm{d}}r.\end{array}\end{eqnarray}$Where M is the mass of the black hole and Q is the charge of the black hole, and ${{\rm{\Omega }}}_{D-2}$ being the volume of the unit (D−2)-sphere$ \begin{eqnarray}{{\rm{\Omega }}}_{D-2}=\displaystyle \frac{2{\pi }^{\tfrac{D-1}{2}}}{{\rm{\Gamma }}\left(\tfrac{D-1}{2}\right)},\end{eqnarray}$${A^{\prime} }_{t}(r)$ is determined by$ \begin{eqnarray}{{ \mathcal L }}^{{\prime} }\left(\displaystyle \frac{{A}_{t}^{{\prime} 2}(r)}{2};{a}_{i}\right){A^{\prime} }_{t}(r)=\displaystyle \frac{\tilde{Q}}{{r}^{D-2}}.\end{eqnarray}$The thermal properties can be defined on the horizon r+ of the black hole. The Hawking temperature of the black hole is given by$ \begin{eqnarray}T=\displaystyle \frac{{f}^{{\prime} }\left({r}_{+}\right)}{4\pi }.\end{eqnarray}$The electrical potential measured with respect to the horizon is$ \begin{eqnarray}\varphi ={\int }_{{r}_{+}}^{\infty }{A^{\prime} }_{t}(r){\rm{d}}r=-{A}_{t}\left({r}_{+}\right),\end{eqnarray}$where we fix the field At(r) at $r=\infty $ to be zero. The electrostatic potential Φ plays a role as the conjugated variable to Q in black hole thermodynamics. The entropy of a black hole can be expressed as$ \begin{eqnarray}S=\displaystyle \frac{{r}_{+}^{D-2}{{\rm{\Omega }}}_{D-2}}{4},\end{eqnarray}$and can be obtained as$ \begin{eqnarray}{\rm{d}}S=\displaystyle \frac{D-2}{4}{r}_{+}^{D-3}{{\rm{\Omega }}}_{D-2}{\rm{d}}{r}_{+}.\end{eqnarray}$
2.2. Complex scalar field in the black hole
Now we start with the dynamics of a charged complex scalar field and calculate energy and momentum changes by the energy-momentum tensor. The action of a complex scalar field in the gravitational and electromagnetic fields is$ \begin{eqnarray}\begin{array}{rcl}{ \mathcal S } & = & -\displaystyle \frac{1}{2}\displaystyle \int \sqrt{-g}\left[\left({\partial }^{\mu }-{\rm{i}}{{qA}}^{\mu }\right)\right.\\ & & \left.\times {{\rm{\Psi }}}^{* }\left({\partial }_{\mu }+{\rm{i}}{{qA}}_{\mu }\right){\rm{\Psi }}-{m}^{2}{{\rm{\Psi }}}^{* }{\rm{\Psi }}\right]{{\rm{d}}}^{D}x,\end{array}\end{eqnarray}$where Aμ is the electromagnetic potential, m is the mass, q is the charge, ψ denotes the wave function, and its conjugate is ψ*. Now we turn to investigate the dynamical of the charged complex scalar field. The field equation obtained from the action satisfies$ \begin{eqnarray}\left({{\rm{\nabla }}}^{\mu }-{\rm{i}}{{qA}}^{\mu }\right)\left({{\rm{\nabla }}}_{\mu }-{\rm{i}}{{qA}}_{\mu }\right){\rm{\Psi }}-{m}^{2}{\rm{\Psi }}=0.\end{eqnarray}$To solve this wave function, we carry out a separation of variables$ \begin{eqnarray}{\rm{\Psi }}={{\rm{e}}}^{-{\rm{i}}\omega t}R(r){Y}_{{lm}}\left({\theta }_{1},{\theta }_{2},\ldots {\theta }_{D-2}\right).\end{eqnarray}$In the above equation, ω is the energy of the particle, and ${Y}_{{lm}}\left({\theta }_{1},{\theta }_{2},\ldots {\theta }_{D-2}\right)$ is the hyperspherical harmonics on a (D−2)-dimensional sphere. Then, the field equation in equation (12) is separated into radial and angular parts. The radial equation is$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{{r}^{2}}{{r}^{D-2}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}r}\left[{r}^{D-2}f\displaystyle \frac{{\rm{d}}}{{\rm{d}}r}R(r)\right]\\ \ \ -\ \left[\lambda +{m}^{2}{r}^{2}-\displaystyle \frac{{r}^{2}}{f}{\left(\omega +{{qA}}_{t}\right)}^{2}\right]R(r)=0.\end{array}\end{eqnarray}$The radial solution can be obtained in simple form by the tortoise coordinate transformation$ \begin{eqnarray}{\rm{d}}{r}_{* }=\displaystyle \frac{1}{f}{\rm{d}}r,\end{eqnarray}$then, the radial equation is rewritten as$ \begin{eqnarray}\begin{array}{l}{r}^{2}\displaystyle \frac{{{\rm{d}}}^{2}}{{\rm{d}}{r}_{* }^{2}}R(r)+(D-2){rf}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}_{* }}R(r)\\ \ \ -\ \left[(\lambda +{m}^{2}{r}^{2})f-{r}^{2}{\left(\omega +{{qA}}_{t}\right)}^{2}\right]R(r)=0.\end{array}\end{eqnarray}$The fluxes of the charged scalar field entering inside of the black hole through its outer horizon, and the radial solution at the outer horizon provides the fluxes. Near the event horizon r+, we have $f({r}_{+})\to 0$, use the definition of (8), the above equation is reduced to$ \begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}}{{\rm{d}}{r}_{* }^{2}}R(r)+{\left[\omega +{{qA}}_{t}({r}_{+})\right]}^{2}R(r)=0,\end{eqnarray}$then we can obtain the radial wave function$ \begin{eqnarray}R(r)={{\rm{e}}}^{\pm {\rm{i}}\left[\omega +{{qA}}_{t}({r}_{+})\right]{r}_{* }},\end{eqnarray}$where ${\rm{d}}{r}_{* }=\tfrac{1}{f}{\rm{d}}r$, r* is a function of r, and +/−corresponds to the solution of the outgoing/ingoing radial wave. Since the thermodynamics and the validity of the weak cosmic censorship conjecture are discussed by the scattering of the ingoing wave at the event horizon in this paper, we focus our attention on the ingoing wave function. Therefore, the solutions of the scalar fields are obtained as$ \begin{eqnarray}{\rm{\Psi }}={{\rm{e}}}^{-{\rm{i}}\omega t}{e}^{-{\rm{i}}\left[\omega +{{qA}}_{t}({r}_{+})\right]{r}_{* }}{Y}_{{lm}}\left({\theta }_{1},{\theta }_{2},\ldots {\theta }_{D-2}\right).\end{eqnarray}$
The energy-momentum tensor is obtained as follows [25, 26, 53, 54]$ \begin{eqnarray}\begin{array}{rcl}{T}_{\nu }^{\mu } & = & \displaystyle \frac{1}{2}\left[({\partial }^{\mu }-{\rm{i}}{{qA}}^{\mu }){{\rm{\Psi }}}^{* }{\partial }_{\nu }{\rm{\Psi }}\right.\\ & & \left.+\left({\partial }^{\mu }+{\rm{i}}{{qA}}^{\mu }\right){\rm{\Psi }}{\partial }_{\nu }{{\rm{\Psi }}}^{* }\right]+{\delta }_{\nu }^{\mu }{ \mathcal L }.\end{array}\end{eqnarray}$The energy flux is the component ${T}_{t}^{r}$ integrated by a solid angle at the outer horizon, and we can obtain the electric charge flux from the energy flux [55]. The fluxes of energy and electric charge are given by [25]$ \begin{eqnarray}\begin{array}{rcl}{\rm{d}}E & = & {\rm{d}}t\displaystyle \int {T}_{t}^{r}\sqrt{-g}{\rm{d}}{{\rm{\Omega }}}_{D-2}=\omega \left[\omega +{{qA}}_{t}({r}_{+})\right]{r}_{+}^{D-2}{\rm{d}}t,\\ {\rm{d}}Q & = & \displaystyle \frac{q}{\omega }{\rm{d}}E=q\left[\omega +{{qA}}_{t}({r}_{+})\right]{r}_{+}^{D-2}{\rm{d}}t.\end{array}\end{eqnarray}$Since the transferred energy and charge are very small, the time interval dt must be small.
3. Thermodynamics and weak cosmic censorship conjecture
In this section, we test the thermodynamics laws and the weak cosmic censorship conjecture of the nonlinear electrodynamics black hole under the charged scalar field at the horizon. When the scalar field is scattered by the black hole, the mass and charge of the black hole are varied due to the energy and charge conservation. Other thermodynamic variables would change with it. The purpose of this section is to check whether the changes of the black hole thermodynamic variables satisfy the first and second law of thermodynamics and the validity of the weak cosmic censorship conjecture in the normal and extended phase spaces.
3.1. Normal phase space
In the normal phase space, the cosmological constant Λ and phase space variable ai are fixed and not treated as thermodynamic variables. The initial state of the black hole is represented by (M, Q, r+), after the black hole scatter a charged scalar field, the final state of the black hole changed to (M+dM, Q+dQ, r++dr+). The variation of the radius can be obtained from the variation of the metric function $f\left({r}_{+}\right)$. For the initial state (M, Q, r+), satisfies$ \begin{eqnarray}f(M,Q,{r}_{+})=0.\end{eqnarray}$We assume that the final state of the charged RN-AdS black hole is still a black hole in the nonlinear electrodynamics, which satisfies$ \begin{eqnarray}f(M+{\rm{d}}M,Q+{\rm{d}}Q,{r}_{+}+{\rm{d}}{r}_{+})=0.\end{eqnarray}$The functions f(M+dM, Q+dQ, r++dr+) and f(M, Q, r+) satisfy the following relation$ \begin{eqnarray}\begin{array}{l}f\left(M+{\rm{d}}M,Q+{\rm{d}}Q,{r}_{+}+{\rm{d}}{r}_{+}\right)\\ \ \ =\ f\left(M,Q,{r}_{+}\right)+{\left.\displaystyle \frac{\partial f}{\partial M}\right|}_{r={r}_{+}}{\rm{d}}M\\ \ \ \ \ \ +\ {\left.\displaystyle \frac{\partial f}{\partial Q}\right|}_{r={r}_{+}}{\rm{d}}Q+{\left.\displaystyle \frac{\partial f}{\partial r}\right|}_{r={r}_{+}}{\rm{d}}{r}_{+},\end{array}\end{eqnarray}$where we use φ instead of At(r+) for simplicity, and$ \begin{eqnarray}\begin{array}{rcl}{\left.\displaystyle \frac{\partial f(r)}{\partial r}\right|}_{r={r}_{+}} & = & 4\pi T,\quad {\left.\displaystyle \frac{\partial f(r)}{\partial M}\right|}_{r={r}_{+}}=-\displaystyle \frac{16\pi }{(D-2){{\rm{\Omega }}}_{D-2}{r}_{+}^{D-3}},\\ {\left.\displaystyle \frac{\partial f(r)}{\partial Q}\right|}_{r={r}_{+}} & = & \displaystyle \frac{16\pi \varphi }{(D-2){{\rm{\Omega }}}_{D-2}{r}_{+}^{D-3}},\end{array}\end{eqnarray}$Bring equations (10) and (25) into equation (24) leads to the first law of thermodynamics$ \begin{eqnarray}{dM}={TdS}+\varphi {dQ}.\end{eqnarray}$In the normal phase space, equation (21) gives the transferred energy and charge within the certain time interval$ \begin{eqnarray}\begin{array}{rcl}{\rm{d}}M & = & {\rm{d}}E=\omega \left(\omega -q\varphi \right){r}_{+}^{D-2}{\rm{d}}t,\\ {\rm{d}}Q & = & q\left(\omega -q\varphi \right){r}_{+}^{D-2}{\rm{d}}t.\end{array}\end{eqnarray}$Bring these results into the equation (24) we can obtain the variation of the radius at horizon$ \begin{eqnarray}{\rm{d}}{r}_{+}=\displaystyle \frac{4{r}_{+}{\left(\omega -q\varphi \right)}^{2}}{(D-2)T{{\rm{\Omega }}}_{D-2}}{\rm{d}}t.\end{eqnarray}$So the variation of the entropy is$ \begin{eqnarray}\begin{array}{rcl}{\rm{d}}S & = & \displaystyle \frac{1}{4}(D-2){r}_{+}^{D-3}{{\rm{\Omega }}}_{D-2}{\rm{d}}{r}_{+}\\ & = & \displaystyle \frac{{r}_{+}^{D-2}{\left(\omega -q\varphi \right)}^{2}}{T}{\rm{d}}t\gt 0,\end{array}\end{eqnarray}$which shows that the entropy of the black hole increases. This situation occurs in any D-dimensional case. So the second law of black hole thermodynamics is satisfied for the black hole.
Then we test the validity of the weak cosmic censorship conjecture in the extremal and near extremal nonlinear electrodynamics black hole. When the charge absorbed by the black hole is more enough, the black hole is overcharged and the weak cosmic censorship conjecture is violated. Therefore, we just need check the existence of the event horizon after the scattering. A simple method to check this existence is to evaluate the solution of the equation f(r)=0. If the solution exists, the metric function f(r) with a minimum negative value guarantees the existence of the event horizon. Suppose there exists one minimum point at r=r0 for f(r), and the minimum value of f(r) is not greater than zero$ \begin{eqnarray}\delta \equiv f\left({r}_{0}\right)\leqslant 0,\end{eqnarray}$where δ=0 corresponds to the extremal black hole. The near extremal condition of the initial state is given at the minimum point with a negative constant $| \delta | \ll 1$. We can estimate the infinitesimal change to the minimum value of the final state after an infinitesimal time interval dt. After the black hole scatters the scalar field, the minimum point would move to r0+dr0. For the final black hole solution, if the minimum value of f(r) at r=r0+dr0 is still not greater than zero, there exists an event horizon. For the final state, the minimum value of f (r) at r=r0+dr0 becomes$ \begin{eqnarray}\begin{array}{l}f\left({r}_{0}+{\rm{d}}{r}_{0},M+{\rm{d}}M,Q+{\rm{d}}Q\right)\\ \quad \ =\,\delta +{\left.\displaystyle \frac{\partial f}{\partial r}\right|}_{r={r}_{0}}{\rm{d}}{r}_{0}+{\left.\displaystyle \frac{\partial f}{\partial M}\right|}_{r={r}_{0}}{\rm{d}}M+{\left.\displaystyle \frac{\partial f}{\partial Q}\right|}_{r={r}_{0}}{\rm{d}}Q\\ \ \quad \ =\,\delta +{\delta }_{1},\end{array}\end{eqnarray}$where$ \begin{eqnarray}{\delta }_{1}=-\displaystyle \frac{16\pi {r}_{+}^{D-2}{r}_{0}^{3-D}\left[\omega +{{qA}}_{t}\left({r}_{+}\right)\right]\left[\omega +{{qA}}_{t}\left({r}_{0}\right)\right]}{(D-2){{\rm{\Omega }}}_{D-2}}{\rm{d}}t,\end{eqnarray}$and$ \begin{eqnarray}\begin{array}{rcl}{\left.\displaystyle \frac{\partial f(r)}{\partial r}\right|}_{r={r}_{0}} & = & 0,\ {\left.\displaystyle \frac{\partial f(r)}{\partial M}\right|}_{r={r}_{0}}=-\displaystyle \frac{16\pi }{(D-2){{\rm{\Omega }}}_{D-2}{r}_{0}^{D-3}},\\ {\left.\displaystyle \frac{\partial f(r)}{\partial Q}\right|}_{r={r}_{0}} & = & -\displaystyle \frac{16\pi {A}_{t}\left({r}_{0}\right)}{(D-2){{\rm{\Omega }}}_{D-2}{r}_{0}^{D-3}}.\end{array}\end{eqnarray}$
If the initial black hole is extremal, we have r+=r0, δ=0 and $T=f^{\prime} ({r}_{+})=f^{\prime} ({r}_{0})=0$. Substituting equations (26), (27) and (33) into equation (31), the minimum value of f(r) at the final black hole reduces to$ \begin{eqnarray}\begin{array}{l}f\left({r}_{0}+{\rm{d}}{r}_{0},M+{\rm{d}}M,Q+{\rm{d}}Q\right)\\ \ \ =\ -\displaystyle \frac{16\pi {r}_{+}{\left[\omega +{{qA}}_{t}\left({r}_{+}\right)\right]}^{2}}{(D-2){{\rm{\Omega }}}_{D-2}}{\rm{d}}t=0.\end{array}\end{eqnarray}$
This means that the extremal black hole becomes another extremal one with new mass and charge. The weak cosmic censorship conjecture is valid in the normal phase of extremal nonlinear electrodynamics black hole.
If the initial state is a near extremal black hole, the minimum location near the horizon. We can order r0=r+−ϵ, where $\epsilon \ll 1$. Moreover, We find δ1 is a quadratic function for ω, so we take the derivative and find its maximum point at $\omega =-\tfrac{1}{2}\left[{{qA}}_{t}\left({r}_{+}\right)+{{qA}}_{t}\left({r}_{0}\right)\right]$, where$ \begin{eqnarray}\begin{array}{rcl}{\delta }_{1\max } & = & \displaystyle \frac{4\pi {q}^{2}{r}_{+}^{D-2}{r}_{0}^{3-D}{\left[{A}_{t}\left({r}_{+}\right)-{A}_{t}\left({r}_{0}\right)\right]}^{2}}{(D-2){{\rm{\Omega }}}_{D-2}}{\rm{d}}t\\ & = & \displaystyle \frac{4\pi {q}^{2}{r}_{+}\,{A}_{t}^{\prime} \left({r}_{+}\right){}^{2}}{(D-2){{\rm{\Omega }}}_{D-2}}{\epsilon }^{2}{\rm{d}}t+O\left({\epsilon }^{3}\right),\end{array}\end{eqnarray}$where δ is suppressed by ϵ. In the test scalar field limit for the near-extremal black hole, ${\delta }_{1}\ll \delta $. So, the equation (31) can be written as$ \begin{eqnarray}f\left({r}_{0}+{{dr}}_{0},M+{dM},Q+{dQ}\right)\lt \delta +{\delta }_{1\max }\lt 0,\end{eqnarray}$which is less than zero and implies that horizon exists in the finial state. The black hole can not be overcharged by the scattering of the scalar filed. Therefore, the weak cosmic censorship conjecture is also valid in the normal phase space of the near extremal nonlinear electrodynamics black hole.
3.2. Extended phase space
In the extended phase space, cosmological constant and normalization parameter are considered as thermodynamic variables. Solving the equation f(r)=0 at the horizon radius r=r+, one can obtain the mass of the black hole$ \begin{eqnarray}\begin{array}{rcl}M & = & \displaystyle \frac{(D-2){r}_{+}^{D-3}{{\rm{\Omega }}}_{D-2}}{16\pi }+\displaystyle \frac{(D-2){r}_{+}^{D-1}{{\rm{\Omega }}}_{D-2}}{16\pi {l}^{2}}\\ & & -\displaystyle \frac{{{\rm{\Omega }}}_{D-2}}{4\pi }{\displaystyle \int }_{{r}_{+}}^{\infty }{r}^{D-2}\left[{ \mathcal L }\left(s;{a}_{i}\right)-{A}_{t}^{{\prime} }(r)\displaystyle \frac{4\pi Q}{{r}^{D-2}{{\rm{\Omega }}}_{D-2}}\right]{\rm{d}}r.\end{array}\end{eqnarray}$The parameter ai in nonlinear electrodynamics theory is considered as the thermodynamic phase space variable, and the associated conjugate ai is$ \begin{eqnarray}{{ \mathcal A }}_{i}=\displaystyle \frac{\partial M}{\partial {a}_{i}}=-\displaystyle \frac{{{\rm{\Omega }}}_{D-2}}{4\pi }{\int }_{{r}_{+}}^{\infty }{r}^{D-2}\displaystyle \frac{\partial { \mathcal L }\left(s;{a}_{i}\right)}{\partial {a}_{i}}{\rm{d}}r.\end{eqnarray}$We can regard the cosmological constant as thermodynamic pressure, and its conjugate regarded as thermodynamic volume[56–58]. Defined as follows$ \begin{eqnarray}\begin{array}{rcl}P & \equiv & -\displaystyle \frac{{\rm{\Lambda }}}{8\pi }=\displaystyle \frac{(D-1)(D-2)}{16\pi {l}^{2}},\\ V & \equiv & {\left(\displaystyle \frac{\partial M}{\partial P}\right)}_{S,Q}=\displaystyle \frac{{{\rm{\Omega }}}_{D-2}}{D-1}{r}_{+}^{D-1}.\end{array}\end{eqnarray}$The initial state of a black hole is represented by (M, Q, r+, l, ai), and the final state is represented by (M+dM, Q+dQ, r+ + dr+, l+dl, Ai+dai) indicates. The change in the horizon radius can be obtained by measuring the change of the function $f\left({r}_{+}\right)$. For the initial state (M, Q, r+, l, ai) satisfied$ \begin{eqnarray}f(M,Q,{r}_{+},l,{a}_{i})=0.\end{eqnarray}$When the mass and charge of the black hole changes, we assume that the final state of the nonlinear electrodynamics black hole is still the black hole, which satisfies the$ \begin{eqnarray}f\left(M+{\rm{d}}M,Q+{\rm{d}}Q,{r}_{+}+{\rm{d}}{r}_{+},l+{\rm{d}}l,{a}_{i}+{\rm{d}}{a}_{i}\right)=0.\end{eqnarray}$The functions f(M+dM, Q+dQ, r++dr+, l+dl, ai+dai) and f(M, Q, r+, l, ai) satisfy the following relation$ \begin{eqnarray}\begin{array}{l}f\left(M+{\rm{d}}M,Q+{\rm{d}}Q,{r}_{+}+{\rm{d}}{r}_{+},\right.\\ \left.l+{\rm{d}}l,{a}_{i}+{\rm{d}}{a}_{i}\right)=f\left(M,Q,{r}_{+},l,{a}_{i}\right)\\ \quad +\,{\left.\displaystyle \frac{\partial f}{\partial M}\right|}_{r={r}_{+}}{\rm{d}}M+{\left.\displaystyle \frac{\partial f}{\partial Q}\right|}_{r={r}_{+}}{\rm{d}}Q\\ \quad +\,{\left.\displaystyle \frac{\partial f}{\partial {r}_{+}}\right|}_{r={r}_{+}}{\rm{d}}{r}_{+}+{\left.\displaystyle \frac{\partial f}{\partial l}\right|}_{r={r}_{+}}{\rm{d}}l+{\left.\displaystyle \frac{\partial f}{\partial {a}_{i}}\right|}_{r={r}_{+}}{\rm{d}}{a}_{i},\end{array}\end{eqnarray}$where$ \begin{eqnarray}\begin{array}{l}{\left.\displaystyle \frac{\partial f(r)}{\partial r}\right|}_{r={r}_{+}}=4\pi T,{\left.\displaystyle \frac{\partial f(r)}{\partial M}\right|}_{r={r}_{+}}\\ \ \ \ \ =\ -\displaystyle \frac{16\pi }{(D-2){{\rm{\Omega }}}_{D-2}{r}_{+}^{D-3}},{\left.\displaystyle \frac{\partial f(r)}{\partial l}\right|}_{r=r+}=-\displaystyle \frac{2{r}_{+}^{2}}{{l}^{3}},\\ {\left.\displaystyle \frac{\partial f(r)}{\partial Q}\right|}_{r={r}_{+}}=-\displaystyle \frac{16\pi {A}_{t}({r}_{+})}{(D-2){{\rm{\Omega }}}_{D-2}{r}_{+}^{D-3}},{\left.\displaystyle \frac{\partial f(r)}{\partial {a}_{i}}\right|}_{r={r}_{+}}\\ \ \ \ \ =\ \displaystyle \frac{16\pi {{ \mathcal A }}_{i}}{(D-2){{\rm{\Omega }}}_{D-2}{r}_{+}^{D-3}}.\end{array}\end{eqnarray}$Bring equations (10) and (43) into equation (42) leads to the first law of thermodynamics$ \begin{eqnarray}{\rm{d}}M=\varphi {\rm{d}}Q+T{\rm{d}}S+V{\rm{d}}P+\displaystyle \sum _{i}{{ \mathcal A }}_{i}{\rm{d}}{a}_{i}.\end{eqnarray}$In the extended phase, the mass of the black hole is then interpreted as the enthalpy$ \begin{eqnarray}M=U+{PV},\end{eqnarray}$we have$ \begin{eqnarray}\begin{array}{rcl}{\rm{d}}M & = & {\rm{d}}(U+{PV})=-\displaystyle \frac{(D-2){r}_{+}^{D-1}{{\rm{\Omega }}}_{D-2}}{8\pi {l}^{3}}{\rm{d}}l\\ & & +\displaystyle \frac{(D-2)(D-1){r}_{+}^{D-2}{{\rm{\Omega }}}_{D-2}}{16\pi {l}^{2}}{\rm{d}}{r}_{+}\\ & & +\omega {r}_{+}^{D-2}(\omega -q\varphi ){\rm{d}}t.\end{array}\end{eqnarray}$Bring these results into the equation (44), we can obtain the variation of the radius in horizon$ \begin{eqnarray}{\rm{d}}{r}_{+}=\displaystyle \frac{16\pi {l}^{2}\left\{{r}_{+}{\left[{{qA}}_{t}\left({r}_{+}\right)+\omega \right]}^{2}{\rm{d}}t-{r}_{+}^{3-D}{{ \mathcal A }}_{i}{\rm{d}}{a}_{i}\right\}}{(D-2){{\rm{\Omega }}}_{D-2}\left[(1-D){r}_{+}+4\pi {l}^{2}T\right]}.\end{eqnarray}$Using the relation (10), one can get the variation of the entropy$ \begin{eqnarray}{\rm{d}}S=\displaystyle \frac{4\pi {l}^{2}\left\{{r}_{+}^{D-2}{\left[{{qA}}_{t}\left({r}_{+}\right)+\omega \right]}^{2}{\rm{d}}t-{{ \mathcal A }}_{i}{\rm{d}}{a}_{i}\right\}}{\left[(1-D){r}_{+}+4\pi {l}^{2}T\right]}.\end{eqnarray}$
It is not easy to determine the variation of entropy whether it increases or decreases. Taking the case D=4 as an example, we choose a special Lagrangian density ${ \mathcal L }\left(s;a\right)={ \mathcal L }(s)\,=(1/a)\left(1-\sqrt{1-2{as}}\right)$, where $s={A^{\prime} }_{t}(r)=Q/\sqrt{{r}^{4}+{{aQ}}^{2}}$, which is mentioned in [46]. Also, we fixed some parameters in equation (48) to calculate the relation between dS and r+, which are $a=Q=l={{ \mathcal A }}_{i}=1$, q=0.8, ω=0.5 and dt=0.001. Then, we plotted the results of the numerical analysis as figure 1. From figure 1, we can see that when dai<0 and dai=0, dS is always less than zero, and the second law of thermodynamics no longer valid in these cases. When dai>0, the value of dS depends on the horizon of the black hole. From the above, we find that the second law of thermodynamics is not always valid.
Figure 1.
New window|Download| PPT slide Figure 1.The relation between dS and r+ in extended phase space. Those curves increase from bottom to top and correspond to dai=−0.03, −0.02, −0.01, 0, 0.01, 0.02, 0.03. We fix $a=Q=l={ \mathcal A }=1$, q=0.8, ω=0.5 and dt=0.001.
Then we test the validity of the weak cosmic censorship conjecture in the extreme and near extreme nonlinear electrodynamics black hole. We assume that f (r) at r=r0 has a minimum point, and that the minimum value of f (r) is not greater than zero. After the black hole scatters the scalar field, the minimum point will move to r0+dr0, and the other parameters of the black hole will change from (M, Q, l, a) to (M+dM, Q+dQ, l+dl, a+da). The minimum value of f (r) of the final state r=r0+dr0 is given by the following formula$ \begin{eqnarray}\begin{array}{l}f\left({r}_{0}+{\rm{d}}{r}_{0},M+{\rm{d}}M,Q+{\rm{d}}Q,l+{\rm{d}}l,a+{\rm{d}}a\right),\\ \quad =\,\delta +{\left.\displaystyle \frac{\partial f}{\partial M}\right|}_{r={r}_{0}}{\rm{d}}M+{\left.\displaystyle \frac{\partial f}{\partial Q}\right|}_{r={r}_{0}}{\rm{d}}Q\\ \qquad +\ {\left.\displaystyle \frac{\partial f}{\partial l}\right|}_{r={r}_{0}}{\rm{d}}l+{\left.\displaystyle \frac{\partial f}{\partial {a}_{i}}\right|}_{r={r}_{0}}{\rm{d}}{a}_{i}\\ \quad =\,\delta +{\delta }_{1}+{\delta }_{2},\end{array}\end{eqnarray}$where δ=f(r0, M, Q, l, a), and$ \begin{eqnarray}\begin{array}{l}{\left.\displaystyle \frac{\partial f(r)}{\partial r}\right|}_{r={r}_{0}}=\,0,\\ {\left.\displaystyle \frac{\partial f(r)}{\partial M}\right|}_{r={r}_{0}}=\,-\displaystyle \frac{16\pi }{(D-2){{\rm{\Omega }}}_{D-2}{r}_{0}^{D-3}},\\ {\left.\displaystyle \frac{\partial f(r)}{\partial l}\right|}_{r={r}_{0}}=-\displaystyle \frac{2{r}_{0}^{2}}{{l}^{3}},\\ {\left.\displaystyle \frac{\partial f(r)}{\partial Q}\right|}_{r={r}_{0}}=\,-\displaystyle \frac{16\pi {A}_{t}\left({r}_{0}\right)}{(D-2){{\rm{\Omega }}}_{D-2}{r}_{0}^{D-3}},\\ {\left.\displaystyle \frac{\partial f(r)}{\partial {a}_{i}}\right|}_{r={r}_{0}}=\,\displaystyle \frac{16\pi \left({{ \mathcal A }}_{i}+\delta {{ \mathcal A }}_{i}\right)}{(D-2){{\rm{\Omega }}}_{D-2}{r}_{0}^{D-3}},\end{array}\end{eqnarray}$and$ \begin{eqnarray}\begin{array}{rcl}{\delta }_{1} & = & -\displaystyle \frac{16\pi {{Tr}}_{0}^{3-D}}{(D-2){{\rm{\Omega }}}_{D-2}}{\rm{d}}S+\displaystyle \frac{16\pi {r}_{0}^{3-D}\delta {{ \mathcal A }}_{i}}{(D-2){{\rm{\Omega }}}_{D-2}}{\rm{d}}{a}_{i}\\ & & +\displaystyle \frac{2{r}_{0}^{2}\left({r}_{+}^{D-1}{r}_{0}^{1-D}-1\right)}{{l}^{3}}{\rm{d}}l,\\ {\delta }_{2} & = & \displaystyle \frac{16\pi {{qr}}_{+}^{D-2}{r}_{0}^{3-D}\left[{A}_{t}\left({r}_{+}\right)-{A}_{t}\left({r}_{0}\right)\right]\left[{{qA}}_{t}\left({r}_{+}\right)+\omega \right]}{(D-2){{\rm{\Omega }}}_{D-2}}{\rm{d}}t.\end{array}\end{eqnarray}$If the initial black hole is extremal, one has r+=rmin, T=0, $\delta {{ \mathcal A }}_{i}=0$ and δ=0. So the minimum value of f (r) at the final black hole is$ \begin{eqnarray}f\left({r}_{0}+{\rm{d}}{r}_{0},M+{\rm{d}}M,Q+{\rm{d}}Q,l+{\rm{d}}l,a+{\rm{d}}a\right)=0,\end{eqnarray}$which means that the extremal black hole still the extremal black hole. The weak cosmic censorship conjecture is valid in the extremal nonlinear electrodynamics black hole in the extended phase space.
For the near-extremal black hole, we define ϵ as r0=r+−ϵ, we can obtain$ \begin{eqnarray}\begin{array}{rcl}{\delta }_{1} & = & \displaystyle \frac{16\pi {r}_{+}^{3-D}}{(D-2){{\rm{\Omega }}}_{D-2}}T{\rm{d}}S+\displaystyle \frac{2(D-1){r}_{+}}{{l}^{3}}\epsilon {\rm{d}}l\\ & & +\displaystyle \frac{16\pi {r}_{+}^{3-D}}{(D-2){{\rm{\Omega }}}_{D-2}}{\rm{d}}a\delta {{ \mathcal A }}_{i}+O({\epsilon }^{3}),\\ {\delta }_{2} & = & \displaystyle \frac{16\pi {{qr}}_{+}{A^{\prime} }_{t}\left({r}_{+}\right)\left({{qA}}_{t}\left({r}_{+}\right)+\omega \right)}{(D-2){{\rm{\Omega }}}_{D-2}}\epsilon {\rm{d}}t+O({\epsilon }^{3}),\end{array}\end{eqnarray}$where ϵ ≪ 1, and δ is suppressed by ϵ in the near extremal limit. In the test scalar field limit for the near-extremal black hole, δ1+δ2≪ δ. So, the equation (49) can be written as$ \begin{eqnarray}\begin{array}{l}f\left({r}_{0}+{\rm{d}}{r}_{0},M+{\rm{d}}M,Q+{\rm{d}}Q,l+{\rm{d}}l,a+{\rm{d}}a\right)\\ \ \ \ =\ \delta +{\delta }_{1}+{\delta }_{2}\lt 0,\end{array}\end{eqnarray}$which is less than zero and indicates that the event horizon exists in the finial state. The black hole can not be overcharged by the scattering of the scalar filed. Therefore, the weak cosmic censorship conjecture is valid in the near extremal nonlinear electrodynamics black hole in the extended phase space.
4. Conclusion
In this paper, we first derived the D-dimensional asymptotically AdS charged black hole solution in general nonlinear electrodynamics. The black hole can change its states while interacting with the external field. We investigated the dynamics of a charged complex scalar field and obtain its energy-momentum tensor. With this relation, the variations of this black hole’s energy and charge can be calculated during an infinitesimal time interval. Then we investigated the validity of the thermodynamic laws in the extended and normal phase spaces with these variations. The first law of thermodynamics is always satisfied. The second law of thermodynamics is satisfied in the normal phase space, but is not valid in the extended phase space.
Moreover, we test the validity of the weak cosmic censorship conjecture in the extremal and near extremal nonlinear electrodynamics black hole. In the normal phase space and the extended phase space, not only the extremal black hole remains extremal under the scattering of the field, but the weak cosmic censorship conjecture is also valid in them.
Acknowledgments
We are grateful to Peng Wang and Haitang Yang for useful discussions. This work is supported in part by NSFC (Grant No. 11375121, 11747171, 11747302 and 11847305). Natural Science Foundation of Chengdu University of TCM (Grants No. ZRYY1729 and ZRQN1656). Discipline Talent Promotion Program of Xinglin Scholars (Grant No. QNXZ2018050) and The Key Fund Project for Education Department of Sichuan (Grant No. 18ZA0173). Sichuan University Students Platform for Innovation and Entrepreneurship Training Program (Grant No. C2019104639).
,1, Wei Hong
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