Thermodynamics and weak cosmic censorship conjecture in the charged RN-AdS black hole surrounded by quintessence under the scalar field

TL;DR

This study investigates the thermodynamic behavior and the validity of the weak cosmic censorship conjecture for RN-AdS black holes surrounded by quintessence under scalar field scattering, revealing phase space-dependent thermodynamic laws.

Contribution

It analyzes the thermodynamics and censorship conjecture in both extended and normal phase spaces, highlighting differences in thermodynamic law validity.

Findings

First law holds in both phase spaces.

Second law valid only in normal phase space.

Weak cosmic censorship conjecture remains valid in both phases.

Abstract

In this paper, we study the thermodynamics and the weak cosmic censorship conjecture of the RN-AdS black hole surrounded by the quintessence under the scattering of a charged complex scalar field. With scalar field scattering, the variation of the black hole is calculated in the extended and normal phase spaces. 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. 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. Furthermore, the weak cosmic censorship conjecture is both valid in the extended and normal phase spaces.

Equations

$$\mathcal{S}=\frac{1}{16\pi G}\int d^{4}x(\sqrt{-g}\left[R-2\Lambda -F^{\mu \nu }{F}_{\mu \nu }\right]+{\mathcal{L}}_q).$$

$${\mathcal{L}}_q=-{\rho }_q\left[1+{\omega }_q\ln \left(\frac{{\rho }_q}{{\rho }_0}\right)\right],$$

$$d{s}^2$$

$$f(r)$$

$${\rho }_q=-\frac{a}{2}\frac{3{\omega }_q}{r^3\left({\omega }_q+1\right)}.$$

$$A_{\mu }\left(r\right)=(-\frac{Q}{r},0,0,0).$$

$$\phi =-A_t\left(r_{+}\right)=\frac{Q}{r_{+}}.$$

$$T=\frac{f^{\prime }(r_{+})}{4\pi }=\frac{1}{4\pi }\left[\left(3{\omega }_q+1\right)r_{+}^{-3{\omega }_q-2}+\frac{2M}{r_{+}^2}+\frac{2r_{+}}{l^2}-\frac{2Q^2}{r_{+}^3}\right],$$

$$S=\pi r_{+}^2.$$

$$M=U+PV,$$

$$\mathcal{S}=-\frac{1}{2}\int \sqrt{-g}\left[\left({\partial }^{\mu }-iq{A}^{\mu }\right){\Psi }^{\ast }\left({\partial }_{\mu }+iq{A}_{\mu }\right)\Psi -m^2{\Psi }^{\ast }\Psi \right]d^{4}x,$$

$$\left({\nabla }^{\mu }-iq{A}^{\mu }\right)\left({\nabla }_{\mu }-iq{A}_{\mu }\right)\Psi -m^2\Psi =0.$$

$$\Psi =e^{-i\omega t}R(r)\Phi (\theta ,\varphi ).$$

$$R(r)=e^{\pm i(\omega -qQ/r)r_{\ast }},$$

$$T_{\nu }^{\mu }=\frac{1}{2}[({\partial }^{\mu }-iq{A}^{\mu }){\Psi }^{\ast }{\partial }_{\nu }\Psi +({\partial }^{\mu }+iq{A}^{\mu })\Psi {\partial }_{\nu }{\Psi }^{\ast }]+{\delta }_{\nu }^{\mu }\mathcal{L}.$$

$$\frac{dE}{dt}=\int T_t^r\sqrt{-g}d\theta d\varphi =\omega (\omega -q\phi )r_{+}^2.$$

$$j^{\mu }=\frac{\partial \mathcal{L}}{\partial A_{\mu }}=-\frac{1}{2}iq[{\Psi }^{\ast }({\partial }^{\mu }+iq{A}^{\mu })\Psi -\Psi ({\partial }^{\mu }-iq{A}^{\mu }){\Psi }^{\ast }].$$

$$\frac{dQ}{dt}=-\int j^r\sqrt{-g}d\theta d\phi =q(\omega -q\phi )r_{+}^2.$$

$$dU=dE=\omega (\omega -q\phi )r_{+}^{2}dt,dQ=q(\omega -q\phi )r_{+}^{2}dt,$$

$$\begin{gathered} P=-\frac{\Lambda }{8\pi }=\frac{3}{8\pi l^2}, \\ V={\left(\frac{\partial M}{\partial P}\right)}_{S,Q}, \end{gathered}$$

$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{a}{r^{3{\omega }_q+1}}+\frac{8}{3}\pi P{r}^2.$$

$$M=\frac{1}{6}\left(-3a{r}_{+}^{-3{\omega }_q}+8\pi P{r}_{+}^3+\frac{3Q^2}{r_{+}}+3r_{+}\right).$$

$$V=\frac{4\pi r_{+}^3}{3}.$$

$$f(M,Q,r_{+},P,a)=0.$$

$$f(M+dM,Q+dQ,r_{+}+d{r}_{+},P+dP,a+da)=0.$$

$$f\left(M+dM,Q+dQ,r_{+}+d{r}_{+},P+dP,a+da\right)=f\left(M,Q,r_{+},P,a\right)$$

$$+{\frac{\partial f}{\partial M}|}_{r=r_{+}}dM+{\frac{\partial f}{\partial Q}|}_{r=r_{+}}dQ+{\frac{\partial f}{\partial r_{+}}|}_{r=r_{+}}d{r}_{+}+{\frac{\partial f}{\partial P}|}_{r=r_{+}}dP+{\frac{\partial f}{\partial a}|}_{r=r_{+}}da,$$

$${\frac{\partial f}{\partial M}|}_{r=r_{+}}=-\frac{2}{r_{+}},{\frac{\partial f}{\partial Q}|}_{r=r_{+}}=\frac{2Q}{r_{+}^2},{\frac{\partial f}{\partial r_{+}}|}_{r=r_{+}}=4\pi T,{\frac{\partial f}{\partial P}|}_{r=r_{+}}=\frac{8\pi r_{+}^2}{3},{\frac{\partial f}{\partial a}|}_{r=r_{+}}=-\frac{1}{r_{+}^{3{\omega }_q+1}}.$$

$$dM=2\pi r_{+}Td{r}_{+}+\frac{Q}{r_{+}}dQ+\frac{4}{3}\pi r_{+}^{3}dP-\frac{1}{2r_{+}^{3{\omega }_q}}da.$$

$$dM=TdS+\phi dQ+VdP+\eta da,$$

$$\eta =-\frac{1}{2r_{+}^{3{\omega }_q}}.$$

Full text

††preprint: CTP-SCU/2019007

1. I Introduction

2. II Complex scalar field in RN-AdS black hole surrounded by quintessence

3. II.1 Black hole solution

4. II.2 Complex scalar field

5. III Thermodynamics and weak cosmic censorship conjecture

6. III.1 Extended phase space

7. III.2 Normal phase space

8. IV Conclusion

I Introduction

Since Hawking and Bekenstein Hawking:1974sw ; Bekenstein:1974ax ; Bekenstein:1973ur proposed the black hole thermodynamics, the research on the thermodynamics of black holes has developed rapidly. Analogous to the laws of thermodynamics, the four laws of black hole thermodynamics were established in Bardeen . There are several ways to calculate the temperature and entropy of black holes tHooft:1984kcu ; Parikh:1999mf ; Hartle:1976tp . A general argument gave that the Hawking temperature of a black hole is proportional to the surface gravity at the horizon, and the black hole entropy is proportional to the area of the horizon. The black hole’s horizon is an important part for the study of the nature of black holes. According to Penrose’s theory Penrose:1969pc , all singularities caused by gravitational collapse must be hidden in the black hole. In other words, singularities need to be hidden from an observer at infinity by the event horizon of black hole, which is the weak cosmic censorship conjecture.

The thermodynamic laws and the weak cosmic censorship conjecture can be tested by the scattering of an external field or by the absorptions of a particle through a black hole. As the particles fall into the black hole, it has been proven that the first law and the second law of black hole thermodynamics are still established Gwak:2016gwj ; Gwak:2015fsa . On the other hand, there are some controversies about the weak cosmic censorship conjecture. 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 Wild:1974gx . However, this conjecture is violated in the near-extreme Reissner-Nordström black hole Hubeny:1998ga and the near-extreme Kerr black hole Jacobson:2009kt . The validity of weak cosmic censorship conjecture in the context of various black holes via the absorption of a charged or rotating particle has been discussed by many work Rocha:2011wp ; Rocha:2014jma ; Colleoni:2015afa ; BouhmadiLopez:2010vc ; Gao:2012ca ; Gwak:2017icn ; Duztas:2016xfg ; Hod:2016hqx ; Natario:2016bay ; Horowitz:2016ezu ; Gwak ; Chen01 ; Chen02 ; Gwak:2017kkt ; Wang:2019dzl ; Kastor:2009wy ; Chen:2018yah ; Zeng:2019jrh ; Zeng:2019jta . Although a lot of work has been done, no consistent conclusion has been reached.

In this paper, we study the thermodynamics and weak cosmic censorship conjecture of the RN-AdS black hole surrounded by the quintessence under the scattering of a charged complex scalar field. The first law of thermodynamics is always satisfied, but the second law of thermodynamics is not always satisfied. And the weak cosmic censorship conjecture does not violate. The paper is organized as follows. In section II, we investigate the dynamical of the charged complex scalar field, and calculated the variations of this black hole’s energy and charge within the certain time interval. In section III, the thermodynamics of the black hole 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 IV.

II Complex scalar field in RN-AdS black hole surrounded by quintessence

In this section, we first briefly review the RN-AdS black hole solution with quintessence and then investigate the complex scalar field in the black hole background.

II.1 Black hole solution

The bulk action for a RN-AdS black hole surrounded by quintessence dark energy in four dimensional curved space time could be described as follows kvv2003 ; Ghaffarnejad:2018bsd

[TABLE]

In the above action, the cosmological constant is related to the AdS space radius $l$ by $\Lambda=-3/l^{2}$. The last term $\mathcal{L}_{q}$ in the action is the Lagrangian of quintessence as a barotropic perfect fluid, which is given by Olivier

[TABLE]

where $\rho_{q}$ is energy density, $\rho_{0}$ is the constant of integral, and the barotropic index $\omega_{q}$. One has that $-1<\omega_{q}<-1/3$ for the quintessence dark energy and $\omega_{q}<-1$ for the phantom dark energy. The metric of the charged RN-AdS black hole surrounded by quintessence is

[TABLE]

where $M$ and $Q$ are the mass and electric charge of the black hole, respectively, and $a$ is the normalization factor related to the density of quintessence as AzregAinou:2012hy ; Azreg-Ainou:2014lua

[TABLE]

The electromagnetic potential of the black hole is

[TABLE]

The potential of the black hole is

[TABLE]

Moreover, the Hawking temperature can be derived as

[TABLE]

and the entropy of the black hole is

[TABLE]

In the extended phase space where the cosmological constant is treated as a thermodynamic variable, the first law of thermodynamics via the RN-AdS black hole surrounded by quintessence have been studied in Li:2014ixn . The mass of the black hole is then interpreted as the enthalpy

[TABLE]

where $U$ is internal energy.

II.2 Complex scalar field

The action of a complex scalar field in the fixed RN-AdS gravitational and electromagnetic fields is

[TABLE]

where $A_{\mu}$ is the electromagnetic potential, $m$ is the mass, $q$ is the charge, $\Psi$ denotes the wave function, and its conjugate is $\Psi^{\ast}$. Now we turn to investigate the dynamical of the charged complex scalar field. The field equation obtained from the action satisfies

[TABLE]

To solve this wave function, we carry out a separation of variables

[TABLE]

In the above equation, $\omega$ is the energy of the particle, and $\Phi(\theta,\phi)$ is the scalar spherical harmonics. We put Eq. $\left(\ref{e19}\right)$ into Eq. $\left(\ref{e18}\right)$ and obtain the radial wave function

[TABLE]

where $dr_{\ast}=\frac{1}{f}dr$, $r_{\ast}$ 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.

From the action $\left(\ref{eAction}\right)$, the energy-momentum tensor is obtained as follows

[TABLE]

Combining the ingoing wave function and its conjugate with the energy-momentum tensor yields the energy flux

[TABLE]

The electric current is obtained from the action $\left(\ref{eAction}\right)$

[TABLE]

For the ingoing wave function $\left(\ref{e24}\right)$, the charge flux is

[TABLE]

When the complex scalar field is scattered off the black hole, the decreases of the energy and charge of the scalar field are equal to the increases of these of the black hole due to the energy and charge conservation. From Eqs. $\left(\ref{e27}\right)$ and $\left(\ref{e29}\right)$, the transferred energy and charge within the certain time interval are

[TABLE]

respectively. Since the transferred energy and charge are very small, the time $dt$ must be also very small. The increase or decrease of $dU$ and $dQ$ depend on the relation between $\omega$ and $q\varphi$.

III Thermodynamics and weak cosmic censorship conjecture

In this section, we test the thermodynamics laws and the weak cosmic censorship conjecture of the black hole by the scattering of the ingoing wave at the event horizon.

III.1 Extended phase space

In the extended phase space, the cosmological constant and normalization parameters are considered as thermodynamic variables. One can treat the cosmological constant as thermodynamic pressure and its conjugate quantity as thermodynamic volume.Dolan:2011xt ; Cvetic:2010jb ; Kubiznak:2012wp ; Hendi:2012um ; Azreg-Ainou:2014twa ; Caceres:2015vsa ; Pedraza:2018eey The definitions are as follows

[TABLE]

where $M$ is the mass of the black hole. So, the metric function $f(r)$ can be rewritten as

[TABLE]

Solving the equation $f(r)=0$ at the horizon radius $r=r_{+}$, one can obtain the mass of the black hole

[TABLE]

Using the Eqs. $\left(\ref{e6}\right)$ and $\left(\ref{e8}\right)$, we can obtain the thermodynamic volume as

[TABLE]

The initial state of the black hole is represented by $(M,Q,r_{+},P,a)$, and the final state is represented by $(M+dM,Q+dQ,r_{+}+dr_{+},P+dP,a+da)$. The variation of the horizon radius can be obtained from the variation of metric function $f\left(r_{+}\right)$. For the initial state $(M,Q,r_{+},P,a)$ satisfies

[TABLE]

When the black hole mass and charge are varied, we assume that the final state of the charged RN-AdS black hole surrounded by quintessence is still a black hole, which satisfies

[TABLE]

The functions $f(M+dM,Q+dQ,r_{+}+dr_{+},P+dP,a+da)$ and $f(M,Q,r_{+},P,a)$ satisfy the following relation

[TABLE]

where

[TABLE]

Bring Eqs. (23), (24) and (26) to Eq. $\left(\ref{fextend}\right)$ leads to

[TABLE]

Using the Eqs. $\left(\ref{varphi}\right)$, $\left(\ref{e11}\right)$ and $\left(\ref{e9}\right)$, gives the first law of thermodynamics

[TABLE]

where $\eta$ is the physical quantity conjugate to the parameter $a$

[TABLE]

And the Smarr relation Villalba:2019sxl can be written as

[TABLE]

For simplicity we use AdS radius $l$ instead of pressure $P$ do the next calculation. Inserting Eqs. (19) and (22) into Eq. (9) yields

[TABLE]

Bring these results into the Eq. (27), we can obtain the variation of the radius in horizon

[TABLE]

The variation of the entropy then takes on the form

[TABLE]

It is not easy to determine the variation of entropy whether increase or decrease. We can consider the extremal black hole which satisfies the condition $T=0$. Suppose in the restricted extended phase space with $da>0$, the change of the black hole entropy becomes

[TABLE]

which shows that the entropy decreases with time. So the second law of black hole thermodynamics is not always valid for this situation.

Then we test the validity of the weak cosmic censorship conjecture in the extremal and near-extremal RN-AdS black hole surrounded by quintessence. 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.

We suppose that there exists one minimum point at $r=r_{0}$ for $f(r)$, and the minimum value of $f(r)$ is not greater than zero,

[TABLE]

where $\delta=0$ corresponds to the extremal black hole. After the black hole scatters the scalar field, the minimum point would move to $r_{0}+dr_{0}$, the other parameters of the black hole change from $(M,Q,l,a)$ to $(M+dM,Q+dQ,l+dl,a+da)$. The minimum value of $f(r)$ at $r=r_{0}+dr_{0}$ of the final state is given by

[TABLE]

where

[TABLE]

Bring Eps. (37) and (27) to Eq. (36) leads to

[TABLE]

When it is an extreme black hole, $r_{0}=r_{+}$ and $T=0$, we can obtain $\delta=0$, $\delta_{1}=0$ and $\delta_{2}=0$. Hence Eq. $\left(\ref{houwen}\right)$ can be rewritten as

[TABLE]

which shows that the scattering does not change the minimum value. This implies that the final state of the extremal black hole is still an extremal black hole, the weak cosmic censorship conjecture is valid in the extremal charged RN-AdS black hole surrounded by quintessence in the extended phase space.

When it is a near-extremal black hole, the order of the variables becomes important, $f^{\prime}(r_{+})$ is very close to zero. To evaluate the value of the above equation, we can let $r_{+}=r_{0}+\epsilon$, where $0<\epsilon\ll 1$ and the relation $f(r_{+})=0$ and $f^{\prime}(r_{0})=0$. And then, the Eq. $\left(\ref{houwen}\right)$ can be written as

[TABLE]

where $dt$ is an infinitesimal scale and is set as $dt\sim\epsilon$. If the initial black hole is near extremal, we have $dS\sim\epsilon$, $dl\sim\epsilon$ and $da\sim\epsilon$. So $\delta_{1}+\delta_{2}\ll\delta$, the final black hole has

[TABLE]

This 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 charged RN-AdS black hole surrounded by quintessence in the extended phase space.

III.2 Normal phase space

In the normal phase space, the cosmological constant and dimensional parameters are fixed. The initial state of the black hole is represented by $(M,Q,r_{+})$, and the final state is represented by $(M+dM,Q+dQ,r_{+}+dr_{+})$. 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

[TABLE]

We assume that the final state of the charged RN-AdS black hole is still a black hole surrounded by quintessence, which satisfies

[TABLE]

The functions $f(M+dM,Q+dQ,r_{+}+dr_{+})$ and $f(M,Q,r_{+})$ satisfy the following relation

[TABLE]

Bring Eqs. (43), (42), (26) into Eq. $\left(\ref{e32}\right)$ leads to

[TABLE]

Using the Eqs. $\left(\ref{varphi}\right)$ and $\left(\ref{e11}\right)$ , gives the first law of thermodynamics

[TABLE]

In the normal phase space, Eq. $\left(\ref{e30}\right)$ gives the transferred energy and charge within the certain time interval

[TABLE]

Bring these results into the Eq. $\left(\ref{e32}\right)$, we can obtain the variation of the radius at horizon

[TABLE]

So the variation of the entropy takes on the form

[TABLE]

which shows that the entropy of the black hole increases. 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 via the black hole in this case. We suppose that there exists one minimum point at $r=r_{0}$ for $f(r)$, and the minimum value of $f(r)$ is not greater than zero

[TABLE]

where $\delta=0$ corresponds to the extremal black hole. After the black hole scatters the scalar field, the minimum point would move to $r_{0}+dr_{0}$. For the final black hole solution, if the minimum value of $f(r)$ at $r=r_{0}+dr_{0}$ is still not greater than zero, there exists an event horizon. Otherwise, the final black hole solution is over the extremal limit, and the weak cosmic censorship conjecture is violated. For the final state, the minimum value of $f(r)$ at $r=r_{0}+dr_{0}$ becomes

[TABLE]

Bring Eqs. (47) and (37) into Eq. (51), we can obtain

[TABLE]

When it is an extreme black hole, $r_{0}=r_{+}$ and $\delta=0$. So the above equation becomes

[TABLE]

which indicates that the horizon exists in the finial state. The black hole can not be overcharged by the scattering of the scalar filed. The extremal black hole still extremal black hole if $\omega=q\varphi$, but the extremal black hole becomes non-extremal black hole if $\omega\neq q\varphi$.

When it is a near-extremal black hole, the Eq. (52) above equation can be regarded as a quadratic function of $\omega$. When $\omega=2qQ(\frac{1}{r_{+}}+\frac{1}{r_{0}})$, we find an maximum value on the function of $f\left(r_{0}+dr_{0},M+dM,Q+dQ\right)$

[TABLE]

This implies metric function has a minimum negative value, which indicates that the event horizon also 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 charged RN-AdS black hole surrounded by quintessence in this case.

IV Conclusion

In this paper, we first derived the RN-AdS black hole surrounded by quintessence via the scattering of a complex scalar field. The variations of this black hole’s energy and charge within the certain time interval can be calculated. 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 RN-AdS black hole surrounded by quintessence. The weak cosmic censorship conjecture is valid both in the extended and normal phase spaces. In the extended phase space, the extremal black hole stays extremal under the scattering of the field. But in the normal phase space, the extremal black hole becomes non-extremal black hole if $\omega\neq q\varphi$, and the extremal black hole stays extremal if $\omega=q\varphi$.

Acknowledgements.

We are grateful to Deyou Chen, Haitang Yang and Peng Wang 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).