# Critical Phenomena of Charged AdS Black Holes in Rastall Gravity

Authors: De-Cheng Zou; Ming Zhang; Chao Wu; Rui-Hong Yue
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

We construct analytical charged anti-de Sitter (AdS) black holes surrounded by perfect fluids in four dimensional Rastall gravity. Then, we discuss the thermodynamics and phase transitions of charged AdS black holes immersed in regular matter like dust and radiation, or exotic matter like quintessence,ΛCDM type, and phantom fields. Surrounded by phantom field, the charged AdS black hole demonstrates a new phenomenon of reentrant phase transition (RPT) when the parameters Q, Np, and ψ satisfy some certain condition, along with the usual small/large black hole (SBH/LBH) phase transition for the surrounding dust, radiation, quintessence, and cosmological constant fields.


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## Body

## 1. Introduction

As well known,  Einstein’s theory of general relativity (GR) has made a lot of interesting predictions for solar system experiments, which are in perfect agreement with the observational data. On larger (cosmological) scales, there are lines of strong observational evidence implying that dark matter (DM) and dark energy (DE) account for about95% of the universe [1, 2]. From these observations, many theoretical models have been presented in Refs. [3–5]. In 1972, one of the potential gravity theories was proposed by Rastall [6, 7]. In the Rastall gravity theory, the usual conservation law of the energy–momentum tensor Tμν;μ=0 is not obeyed, and the energy–momentum tensor satisfies(1)Tμν;μ=λRν,whereR is the Ricci scalar, and the parameter λ measures the potential deviations of Rastall theory from GR. It is interesting that all electrovacuum solutions in GR are also solutions of Rastall gravity. However, if matter field (nonvanishing trace) is introduced, the spacetime depends on the Rastall parameter λ. As the simplest modified gravity scenario, it provides an explanation of the inflation problem, and realizes the late-time acceleration and other cosmological problems [8–11]. Until now, many works on the various black hole solutions have been also investigated in Rastall theory. The spherically symmetric black hole solutions were constructed in Refs. [12–16], the rotating black holes in Refs. [17, 18], the thermodynamics of black holes in Refs. [19–22], and also instability of black holes in Refs. [23, 24].According to the AdS/CFT correspondence, the bulk AdS black hole spacetimes admit a gauge duality description by thermal conformal field theory living on the AdS boundary [25–27]. For example, the usual Hawking-Page phase transition [28] in four dimensional GR can be interpreted as a confinement/deconfinement phase transition in the dual quark gluon plasma [29]. Recently, the study of thermodynamics of AdS black holes has been extended to the extended phase space, where the cosmological constant is treated as the pressure of the black hole [30, 31]. In the extended phase space, charged AdS black hole admits a more direct and precise coincidence between the first order small/large black holes (SBH/LBH) phase transition and Van der Waals (VdW) liquid–gas phase transition [32], and both systems share the same critical exponents near the critical point. More discussions in this direction can be found as well in Refs. [33–61], including reentrant phase transitions and some other phase transitions. In the Rastall theory, Ali recovered the existence of VdW like SBH/LBH phase transition for AdS black holes in four dimensional spacetimes [62]. Motivated by this result, it is interesting to explore the effect of charge Q on the critical behaviors for charged AdS black holes in four dimensional Rastall theory. We hope to reveal a more fascinating phenomenon.This paper is organized as follows. In Section2, the analytical solution for static spherical symmetric charged AdS black holes are obtained in four dimensional Rastall theory. In Section 3, we study the critical behaviors of the charged AdS black holes in context of P−V criticality and phase diagrams. We end the paper with conclusion and discussion in Section 4.


## 2. Charged AdS Black Hole Solutions in Rastall Theory

Considering Equation (1), the field equation including the negative cosmological constant Λ can be written as [62](2)Gμν+Λgμν+κλgμνR=Tμν,whereκ equals to 8πGN and GN is the Newton gravitational coupling constant.In order to derive the black hole solutions, we consider the general spherical symmetric spacetime metric(3)ds2=−frdt2+1frdr2+r2dθ2+sin2θdϑ2,wherefr is a generic metric function depending on the radial coordinate r. Using this metric, the nonvanishing components of right hand side of Equation (2) are(4)H00=H11=G00+Λ+κλR=1r2f′r−1+f+Λ+κλR,H22=H33=G22+Λ+κλR=1r2rf′+12r2f″+Λ+κλR,with the Ricci scalarR=−1/r2r2f′′+4rf′−2+2  f.On the other hand, the total energy–momentum tensorTμν supporting this spacetime takes the following form [13, 14](5)Tμν=Eμν+Tμν,whereEμν is given by(6)Eμν=2κFμαFνα−14gμνFαβFαβandFμν satisfies the Maxwell equations Fμν;ν=0. Using this Maxwell equation, we can derive(7)F01=Qr2,whereQ is an integration constant playing the role of a electrostatic charge. Then the nonvanishing components of the Maxwell tensor Eμν read as(8)Eμν=Q2κr4diag−1,−1,1,1.In addition,Tνμ describes the energy–momentum tensor of surrounding field defined as [63](9)T00=T11=−ρsr,T22=T33=121+3ωsρsr,whereρsr is the energy density and satisfies barotropic equation of state ps=ωsρs with the pressure psr and state parameter ωs. The subscript “s” denotes the surrounding field which can be a dust (ωd=0), radiation (ωr=1/3), ΛCDM type (ωΛ=−1), quintessence and phantom field constructed by the combination of these fields.FromH00=T00 and H22=T22 components of Rastall field equations, the solution of four dimensional charged AdS black hole can be obtained as(10)ds2=−frdt2+1frdr2+r2dθ2+sin2θdϑ2,fr=1−2Mr+Q2r2−Nsrξ+r2lΛ2,whereM and Ns are two integration constants representing the black hole mass and surrounding field structure parameter, respectively. The parameters ξ and lΛ are given by(11)ξ=1+3ωs−6ψ1+ωs1−3ψ1+ωs,1lΛ2=−Λ31−4ψ,ψ≡κλ,wherelΛ is the curvature radius, and Ns is related to the energy density ρs(12)ρs=−3WsNsκr31+ωs−12ψ1+ωs/1−3ψ1+ωs,whereWs is(13)Ws=−1−4ψψ1+ωs−ωs1−3ψ1+ωs2.Regarding the weak energy condition representing the positivity of any kind of energy density of the surrounding field, i.e.,ρs≥0, imposes the following conditions(14)WsNs≤0,which implies that for the surrounding field withWs>0, we need Ns<0 and conversely for Ws<0, we need Ns>0. If ωs≠−1, the constraint conditions for ωs and ξ can be derived, see Table 1. For the cosmological constant field ωs=ωΛ=−1, ξ equals to −2 which is independent of ψ and WΛ is −1+4ψ. Then, we find(15)NΛ>0,0<ψ<14,orNΛ<0,ψ>14Table 1
The condition forρs≥0.
N
s
>
0
∪
W
s
<
0
ω
s
>
−
1
∪
−
2
<
ξ
<
1
ω
s
<
−
1
∪
ξ
>
1
ω
s
<
−
1
∪
ξ
<
−
2
N
s
<
0
∪
W
s
>
0
ω
s
<
−
1
∪
−
2
<
ξ
<
1
ω
s
>
−
1
∪
ξ
>
1
ω
s
>
−
1
∪
ξ
<
−
2from the weak energy condition Equation (11).


## 3. Critical Behaviors of Charged AdS Black Holes

In the extended phase space, we interpret the cosmological constantΛ as a positive thermodynamic pressure P in the geometric units GN=ћ=c=1(16)P=−Λ8π=31−4ψ8πlΛ2.In terms of horizon radiusr+, mass M, Hawking temperature T, entropy S, and electromagnetic potential Φ of the charged AdS black holes can be written as(17)M=12Q2r++r+−Nsr+ξ−1+8πPr+331−4ψ,S=πr+2,T=14π1r++8πPr+1−4ψ−Q2r+3+ξ−1Nsr+1+ξ,Φ=Qr+.Here the black hole massM should be considered as the enthalpy rather than the internal energy of the gravitational system. Moreover, those thermodynamic quantities should satisfy first law of black hole thermodynamics(18)dM=TdS+VdP+ΘsdNs+ΦdQ,where the thermodynamic volumeV conjugate to P equals to 4πr+3/31−4ψ and Θs as a generalized force conjugate to the parameter Ns reads as −r+1−ξ/2.Using Equation (17), the equation of state PV,  T can be written as(19)P=1−4ψ2r+T−14πr++Q24πr+3−ξ−1Ns4πr+1+ξ.As usual, a critical point occurs whenP has an inflection point(20)∂P∂r+T=Tc,r+=rc=∂2P∂r+2T=Tc,r+=rc=0,where the subscript “c” denotes the values of the physical quantities at the critical point. When ωs≠−1, we can obtain(21)Tc=ξrc2−2Q2ξ−22π1+ξrc3,(22)Pc=ξrc2−3Q2ξ−23ωs−124πrc4ξ−2ωs+1,and the equation for critical horizon radiusrc(23)Frc=1−ξ1+ξ2+ξNs−2rcξ−2rc2−6Q2=0with Equation (11). Evidently, it is hard to derive any analytic solution for the critical quantities rc, Tc, and Pc from Equations (21)–(23). By analyzing the asymptotical behavior of function Frc at the horizon and infinity, we summarize the number of real root rc of Equation (23) in Table 2. The discussion for cosmological constant field ωs=−1 will be shown in the last part of this section.Table 2
Real rootrc of Equation (23).
r
c
ω
s
>
−
1
ω
s
<
−
1
N
s
>
0
N
s
<
0
N
s
>
0
N
s
<
0
ξ
<
−
2
None
1
0,1,2
None
−
2
<
ξ
<
−
1
1
None
None
0,1,2
−
1
<
ξ
<
0
0,1,2
None
None
1
0
<
ξ
≤
2
1
2
<
ξ
None
1
0,1,2
NoneTaking the region ofωs>−1, −1<ξ<0 and Ns>0 for instance, the function Frc [Equation (23)] approaches +∞ near the origin r→0 and tends to −Ns1−  ξ1+  ξ2+  ξ>  0 when r→+∞. For the critical temperature Tc [Equation (21)], it approaches +∞ near the origin r→0, Tc disappears at rc=rc0=2ξ−  2/ξQ and tends to 0 when r→+∞, see Figures 1(a) and 1(b). On the other hand, the derivative of Frc, Tc, and Pc leads to a same expression(24)F′rc=Tc′=Pc′=−6Q2−2+ξ+rc2ξ,which vanishes atrc≡rg=6ξ−  2/ξQ. From Frc=0, the two real positive critical radius rc1 and rc2 (if exist) satisfy rc1<rg<rc2. We have Tcrc2<0 because of rc0<rg<rc2 and Tcrc1>0 if rc1<rc0, see Figure 1(c). In general, a black hole should possess positive temperature T>0, and reduces to extremal one when T=0. Moreover, holding positive temperature T>0 makes it possible to set up an analogy between the realistic Van der Waals phase transition for liquid/gas system [64] and SBH/LBH phase transition for the charged AdS black hole in the Rastall gravity. For the four dimensional Reissner–Nordström AdS black hole, Kubiznak and Mann derived one set of positive critical quantities rc, Pc, and Tc by utilizing Equation (20) in the extended phase space, and further recovered that Reissner–Nordström AdS black hole admits VdW-like SBH/LBH phase transition in the Einstein gravity [32].Figure 1
TheFrc−lnrc and Tc−lnrc diagrams with Q=0.1 and Ns=0.1113 for different values of ξ.
(a)
(b)
(c)Besides the VdW-like SBH/LBH phase transition, Ref. [48] firstly proposed the reentrant phase transition (RPT) for Born–Infeld AdS black hole in GR, if there exist more than one positive critical points. Nevertheless, the RPT does not happen in this region (ωs>−1, −1<ξ<0 and Ns>0) for charged AdS black hole in Rastall gravity because of the unphysical critical temperature Tcrc2<0, even though there could exist two positive real critical radius rc from Table 2.In addition, to study the possible phase transitions in the system, we need to discuss the Gibbs free energy, which obeys the following thermodynamic relationG=M−TS with(25)G=r+4+3Q24r+−2Pπr+331−4ψ−1+ξNs4r+ξ−1,wherer+ is understood as a function of pressure and temperature, r+=r+P,  T, via equation of state [Equation (19)]. In the rest of this section, we mainly focus on that regions possessing more than one real roots of rc from Table 2, and then consider the critical behaviors of charged AdS black holes surrounded by dust, radiation, quintessence, phantom, or ΛCDM type fields.


### 3.1. Black Hole Surrounded by Dust or Quintessence Fields

For the dust surrounding field, we setωd=0. Taking ξ=−0.8, Q=0.1 and Nd=0.1 for instance, one can obtain two critical radius rc1=0.245 and rc2=120.762, and corresponding critical pressure and temperature Pc1≈0.0463, Tc1≈0.4104 and Pc2≈−2.5984∗10−7, Tc2≈−0.00527166. So, there only exists one physical critical point, which has been verified in the above part. As shown in Figure 2, P−r+ diagram displays that the solid curve represents critical isotherm at T=Tc1 and the dashed and dotted curves correspond to T>Tc1 and T<Tc1, respectively. In the G−T diagram, the solid curve represents P<Pc1, the dotted curve corresponds to P>Pc1, and the dashed curve is for P=Pc1. We observe standard swallowtail behavior. Moreover, the P−T diagram shows the coexistence line of the first-order SBH/LBH phase transition terminating at a critical point. These plots are analogous to typical behavior of the liquid–gas phase transition of the VdW fluid.Figure 2
TheP−r+, G−T, and P−T diagrams of four-dimensional charged AdS black holes with ωd=0, ξ=−0.8, Q=0.1, and Nd=0.1. There is one critical point, which corresponds to VdW-like SBH/LBH phase transition when T<Tc1≈0.4104 and P<Pc1≈0.0463.
(a)
(b)
(c)We turn to discuss the quintessence field. The quintessential dark energy is governed by an equation of state of the formpq=ωqρq with the state parameter ωq∈−1,−1/  3 [63]. Here we choose a ordinary example ωq=−2/3 [63, 65]. Taking ξ=−0.7, Q=0.4 and Nq=0.4, one can obtain two critical radius rc1=1.055 and rc2=17.85, and corresponding critical pressure and temperature Pc1≈0.0184, Tc1≈0.0382 and Pc2≈−0.000096, Tc2≈−0.0207. Therefore, there only exists one physical critical point, which implies the appearance of VdW-like SBH/LBH phase transition. The P−r+, G−T, and P−T diagrams are similar with Figure 2, so we do not plot these figures in this case.


### 3.2. Black Hole Surrounded by Radiation Field

With regard to the case ofωr=1/3, one have ξ=2, which is independent of ψ from Equation (11). Then, the black hole solution (10) becomes(26)fr=1−2Mr+Q2−Nrr2+r2lΛ2,Obviously, the parameterNr plays a shift role in the charge Q. It’s worthy to point out that the condition Q2>Nr needs to be satisfied so that we can derive the possible phase transition. Then, the equation of state PV,  T reduces to(27)P=1−4ψ2r+T−14πr++Q24πr+3−Nr4πr+3.According the judgement condition (20), corresponding critical values read as(28)rc=6Q2−Nr,Pc=1−4ψ96πQ2−Nr,Tc=136Q2−Nr,Q2>Nr,ψ<14.Then, VdW-like SBH/LBH phase transition occurs for charged AdS black hole in the region ofQ2>Nr and ψ<1/4, which is similar as that for charged AdS black hole in Einstein gravity [32].


### 3.3. Black Hole Surrounded by Phantom Field

For the phantom field, it possesses the supernegative equation of stateωp<−1 [65]. Nevertheless, when a black hole immersed this phantom field, it gives rise to pp+ρp<0 and Tpsp<0, and leads to two alternatives: either the entropy density sp is negative with positive temperature Tp or the entropy density sp is positive but the temperature is negative. With Tp>0, the entropy of black hole decreases as it accretes the phantom energy, meanwhile a decrease in the energy density of phantom field decreases the entropy of phantom field [66, 67]. Finally, the total entropy of thermodynamic system consisting of phantom field entropy and black holes entropy will decrease simultaneously, which violates the generalized second law (GSL) of thermodynamics, and then the accretion process could never occur in this case. In terms of Tp<0, the accretion of phantom field by this black hole is also not possible if the mass of black hole is above from corresponding critical value [68–71]. Here we only discuss thermodynamics and phase transitions of charged AdS black holes in the Rastall gravity. Therefore, we can assume the phantom field with Tp>0 for simplicity, so that the thermodynamic system consisting of charged AdS black hole and phantom field is always stable.Recently, Kumar and Xu adopted the observational data sets of SNLS3, BAO,Planck+WMAP9+WiggleZ to determine the parameter ωp=−1.06−0.13+0.11 [72]. Moreover, the Planck 2018 results recovers ωp=−1.03±0.03 [73]. In view of these constraints on ωp from observations, we consider the phantom field with ωp=−1.03. With Np>0 and ξ>2 in Table 2, the function Frc [Equation (23)] tends to −Npξ−  1ξ+  1ξ+2<  0 as rc→0 and −∞ when rc→−∞. Based on Frc=0 and Equations (21) and (22), we can obtain two positive critical points when the parameters Q, ξ, and Np satisfy(29)23ξ−2/ξQξξ−2ξ−1ξ+1<Ns<46ξ−2/ξQξξ−2ξ−1ξ+1ξ+2.By using the graphical and numerical method,P−r+ and G−T diagrams of charged AdS black hole with ξ=2.1, Q=0.4, and Np=0.1113 are displayed in Figure 3. One can find two physical critical points [Figure 3(a)]. For P<Pc2T<Tc2, the VdW-like SBH/LBH phase transition occurs, and terminates at P=PτT=Tτ. At the same time, there also exists a certain range of P∈Pτ,Pz, namely T∈Tτ,Tz, for which the global minimum of G is discontinuous; see Figure 3(c). In this range of temperatures, two separate branches of intermediate size and small size black holes co-exist. They are separated by a finite jump in G, which is so-called “zeroth-order phase transition”.Figure 3
TheP−r+ and G−T diagrams of four-dimensional charged AdS black holes with ωp=−1.03, ξ=2.1, Q=0.4, and Np=0.1113. It shows the existence of two critical points Pc1=165.8340,Tc1=  0.2743 and Pc2=356.10,Tc2=  0.3153. For P∈Pτ,Pz, we observe a “zeroth-order phase transition” at T0 signifying the onset of an RPT in c.
(a)
(b)
(c)Figure 4
TheP−T diagram of four-dimensional charged AdS black holes with ωp=−1.03, ξ=2.1, Q=0.4, and Np=0.1113. The thick solid line initiating from Pc2,Tc2 and terminates at Pτ,Tτ corresponds to the co-existence line of VdW-like SBH/LBH phase transition. The solid line commencing from Pz,Tz and terminates at Pτ,Tτ denotes the coexistence line of small and intermediate black holes indicating the RPT.
(a)
(b)This novel situation can be also clearly illustrated in theP−T diagrams in Figure 4. There is the expected SBH/LBH line of co-existence, which initiates from the critical point Tτ,Pτ and terminates at Tc2,Pc2. Especially, a “triple point” between the small (SBH), intermediate (IBH), and large black holes (LBH) emerges in the point Tτ,Pτ. In the region of T∈Tτ,Tz, a new IBH/SBH line of coexistence appears and then it terminates in another critical point Tz,Pz. These values are obtained as(30)Tτ,Tz,Tc2≈0.2938,0.2954,0.3153,Pτ,Pz,Pc2≈289.8545,300.2650,356.10.Considering the other regions ofNp>0, ξ<−2, or Np<0, −2<ξ<−1 in Table 2, two positive roots of rc1 and rc2 could appear from equation Frc=0. Taking Np>0 and ξ<−2 for instance, we choose ωp=−1.03, ξ=−2.4, Q=0.2, and Np=0.4, and find that there only exists the VdW-like SBH/LBH phase transition in the region of Tc1<Tc<Tc2 and Pc1<Pc<Pc2 with(31)Pc1,Pc1≈0.17836,0.43678,Tc2,Tc2≈0.23295,0.57949,see Figure5. Similar discussion can be done in the region of Np<0, −2<ξ<−1 and ωp=−1.03, we find there only is the VdW-like SBH/LBH phase transition, even though there also exist two different physical critical points: rc1,Tc1,Pc1>0 and rc2,Tc2,Pc2>0.Figure 5
TheP−r+, G−T, and P−T diagrams of four-dimensional charged AdS black holes with ωp=−1.03, ξ=−2.4, Q=0.2, and Np=0.4. The VdW-like SBH/LBH phase transition happens in the region of Tc1<T<Tc2 and Pc1<P<Pc2.
(a)
(b)
(c)


### 3.4. Black Hole SurroundedΛCDM Type Field

In case ofωΛ=−1, namely ξ=−2, the critical horizon radius, temperature, and pressure are obtained as(32)rc=6Q,Tc=136πQPc=1+36NΛQ21−4ψ96πQ2.HoldingPc>0, the parameters NΛ and ψ should satisfy NΛ<−1/36Q2 for ψ>1/4 and NΛ>0 for 0<ψ<1/4 in terms of the condition (15). Finally, VdW-like SBH/LBH phase transition occurs in these regions, which is similar as that for charged AdS black hole in Einstein gravity [32].


## 3.1. Black Hole Surrounded by Dust or Quintessence Fields

For the dust surrounding field, we setωd=0. Taking ξ=−0.8, Q=0.1 and Nd=0.1 for instance, one can obtain two critical radius rc1=0.245 and rc2=120.762, and corresponding critical pressure and temperature Pc1≈0.0463, Tc1≈0.4104 and Pc2≈−2.5984∗10−7, Tc2≈−0.00527166. So, there only exists one physical critical point, which has been verified in the above part. As shown in Figure 2, P−r+ diagram displays that the solid curve represents critical isotherm at T=Tc1 and the dashed and dotted curves correspond to T>Tc1 and T<Tc1, respectively. In the G−T diagram, the solid curve represents P<Pc1, the dotted curve corresponds to P>Pc1, and the dashed curve is for P=Pc1. We observe standard swallowtail behavior. Moreover, the P−T diagram shows the coexistence line of the first-order SBH/LBH phase transition terminating at a critical point. These plots are analogous to typical behavior of the liquid–gas phase transition of the VdW fluid.Figure 2
TheP−r+, G−T, and P−T diagrams of four-dimensional charged AdS black holes with ωd=0, ξ=−0.8, Q=0.1, and Nd=0.1. There is one critical point, which corresponds to VdW-like SBH/LBH phase transition when T<Tc1≈0.4104 and P<Pc1≈0.0463.
(a)
(b)
(c)We turn to discuss the quintessence field. The quintessential dark energy is governed by an equation of state of the formpq=ωqρq with the state parameter ωq∈−1,−1/  3 [63]. Here we choose a ordinary example ωq=−2/3 [63, 65]. Taking ξ=−0.7, Q=0.4 and Nq=0.4, one can obtain two critical radius rc1=1.055 and rc2=17.85, and corresponding critical pressure and temperature Pc1≈0.0184, Tc1≈0.0382 and Pc2≈−0.000096, Tc2≈−0.0207. Therefore, there only exists one physical critical point, which implies the appearance of VdW-like SBH/LBH phase transition. The P−r+, G−T, and P−T diagrams are similar with Figure 2, so we do not plot these figures in this case.


## 3.2. Black Hole Surrounded by Radiation Field

With regard to the case ofωr=1/3, one have ξ=2, which is independent of ψ from Equation (11). Then, the black hole solution (10) becomes(26)fr=1−2Mr+Q2−Nrr2+r2lΛ2,Obviously, the parameterNr plays a shift role in the charge Q. It’s worthy to point out that the condition Q2>Nr needs to be satisfied so that we can derive the possible phase transition. Then, the equation of state PV,  T reduces to(27)P=1−4ψ2r+T−14πr++Q24πr+3−Nr4πr+3.According the judgement condition (20), corresponding critical values read as(28)rc=6Q2−Nr,Pc=1−4ψ96πQ2−Nr,Tc=136Q2−Nr,Q2>Nr,ψ<14.Then, VdW-like SBH/LBH phase transition occurs for charged AdS black hole in the region ofQ2>Nr and ψ<1/4, which is similar as that for charged AdS black hole in Einstein gravity [32].


## 3.3. Black Hole Surrounded by Phantom Field

For the phantom field, it possesses the supernegative equation of stateωp<−1 [65]. Nevertheless, when a black hole immersed this phantom field, it gives rise to pp+ρp<0 and Tpsp<0, and leads to two alternatives: either the entropy density sp is negative with positive temperature Tp or the entropy density sp is positive but the temperature is negative. With Tp>0, the entropy of black hole decreases as it accretes the phantom energy, meanwhile a decrease in the energy density of phantom field decreases the entropy of phantom field [66, 67]. Finally, the total entropy of thermodynamic system consisting of phantom field entropy and black holes entropy will decrease simultaneously, which violates the generalized second law (GSL) of thermodynamics, and then the accretion process could never occur in this case. In terms of Tp<0, the accretion of phantom field by this black hole is also not possible if the mass of black hole is above from corresponding critical value [68–71]. Here we only discuss thermodynamics and phase transitions of charged AdS black holes in the Rastall gravity. Therefore, we can assume the phantom field with Tp>0 for simplicity, so that the thermodynamic system consisting of charged AdS black hole and phantom field is always stable.Recently, Kumar and Xu adopted the observational data sets of SNLS3, BAO,Planck+WMAP9+WiggleZ to determine the parameter ωp=−1.06−0.13+0.11 [72]. Moreover, the Planck 2018 results recovers ωp=−1.03±0.03 [73]. In view of these constraints on ωp from observations, we consider the phantom field with ωp=−1.03. With Np>0 and ξ>2 in Table 2, the function Frc [Equation (23)] tends to −Npξ−  1ξ+  1ξ+2<  0 as rc→0 and −∞ when rc→−∞. Based on Frc=0 and Equations (21) and (22), we can obtain two positive critical points when the parameters Q, ξ, and Np satisfy(29)23ξ−2/ξQξξ−2ξ−1ξ+1<Ns<46ξ−2/ξQξξ−2ξ−1ξ+1ξ+2.By using the graphical and numerical method,P−r+ and G−T diagrams of charged AdS black hole with ξ=2.1, Q=0.4, and Np=0.1113 are displayed in Figure 3. One can find two physical critical points [Figure 3(a)]. For P<Pc2T<Tc2, the VdW-like SBH/LBH phase transition occurs, and terminates at P=PτT=Tτ. At the same time, there also exists a certain range of P∈Pτ,Pz, namely T∈Tτ,Tz, for which the global minimum of G is discontinuous; see Figure 3(c). In this range of temperatures, two separate branches of intermediate size and small size black holes co-exist. They are separated by a finite jump in G, which is so-called “zeroth-order phase transition”.Figure 3
TheP−r+ and G−T diagrams of four-dimensional charged AdS black holes with ωp=−1.03, ξ=2.1, Q=0.4, and Np=0.1113. It shows the existence of two critical points Pc1=165.8340,Tc1=  0.2743 and Pc2=356.10,Tc2=  0.3153. For P∈Pτ,Pz, we observe a “zeroth-order phase transition” at T0 signifying the onset of an RPT in c.
(a)
(b)
(c)Figure 4
TheP−T diagram of four-dimensional charged AdS black holes with ωp=−1.03, ξ=2.1, Q=0.4, and Np=0.1113. The thick solid line initiating from Pc2,Tc2 and terminates at Pτ,Tτ corresponds to the co-existence line of VdW-like SBH/LBH phase transition. The solid line commencing from Pz,Tz and terminates at Pτ,Tτ denotes the coexistence line of small and intermediate black holes indicating the RPT.
(a)
(b)This novel situation can be also clearly illustrated in theP−T diagrams in Figure 4. There is the expected SBH/LBH line of co-existence, which initiates from the critical point Tτ,Pτ and terminates at Tc2,Pc2. Especially, a “triple point” between the small (SBH), intermediate (IBH), and large black holes (LBH) emerges in the point Tτ,Pτ. In the region of T∈Tτ,Tz, a new IBH/SBH line of coexistence appears and then it terminates in another critical point Tz,Pz. These values are obtained as(30)Tτ,Tz,Tc2≈0.2938,0.2954,0.3153,Pτ,Pz,Pc2≈289.8545,300.2650,356.10.Considering the other regions ofNp>0, ξ<−2, or Np<0, −2<ξ<−1 in Table 2, two positive roots of rc1 and rc2 could appear from equation Frc=0. Taking Np>0 and ξ<−2 for instance, we choose ωp=−1.03, ξ=−2.4, Q=0.2, and Np=0.4, and find that there only exists the VdW-like SBH/LBH phase transition in the region of Tc1<Tc<Tc2 and Pc1<Pc<Pc2 with(31)Pc1,Pc1≈0.17836,0.43678,Tc2,Tc2≈0.23295,0.57949,see Figure5. Similar discussion can be done in the region of Np<0, −2<ξ<−1 and ωp=−1.03, we find there only is the VdW-like SBH/LBH phase transition, even though there also exist two different physical critical points: rc1,Tc1,Pc1>0 and rc2,Tc2,Pc2>0.Figure 5
TheP−r+, G−T, and P−T diagrams of four-dimensional charged AdS black holes with ωp=−1.03, ξ=−2.4, Q=0.2, and Np=0.4. The VdW-like SBH/LBH phase transition happens in the region of Tc1<T<Tc2 and Pc1<P<Pc2.
(a)
(b)
(c)


## 3.4. Black Hole SurroundedΛCDM Type Field

In case ofωΛ=−1, namely ξ=−2, the critical horizon radius, temperature, and pressure are obtained as(32)rc=6Q,Tc=136πQPc=1+36NΛQ21−4ψ96πQ2.HoldingPc>0, the parameters NΛ and ψ should satisfy NΛ<−1/36Q2 for ψ>1/4 and NΛ>0 for 0<ψ<1/4 in terms of the condition (15). Finally, VdW-like SBH/LBH phase transition occurs in these regions, which is similar as that for charged AdS black hole in Einstein gravity [32].


## 4. Conclusion and Discussion

We have obtained charged AdS black holes solutions surrounded by perfect fluid in the context of four dimensional Rastall theory. Regarding the weak energy condition of energy density and asymptotic behavior of functionFrc, we derived the constraint conditions for the parameters Ns, ωs and ξ or ψ, as shown in Tables 1 and 2. Later, we chose some special surrounding fields which can be a dust, radiation, quintessence, ΛCDM type and phantom field, and found that the Van der Waals like SBH/LBH phase transition happens for the charged AdS black holes with dust, radiation or quintessence and Λ CDM type fields. In particular, the charged AdS black hole surrounded by phantom field showed more interesting phenomenon in the region of Np>0 and ξ>2. When the parameters Q, ξ, and Np satisfy 23ξ−2/ξQξ/ξ−2ξ−1ξ+1<Np<46ξ−2/ξQξ/ξ−2ξ−1ξ+1ξ+2 and ξ>2, we obtained two positive critical points and so-called RPT phase transition occurs, besides the VdW-like SBH/LBH phase transition. However, in the region of Np>0, ξ<−2, or Np<0, −2<ξ<−1, there only exist the VdW-like SBH/LBH phase transition, even though it has two physical critical points.Comparing with the Maxwell field, the Born–Infeld AdS black hole, and charged AdS black hole with Yang–Mills field in the Einstein gravity have proposed more interesting thermodynamical properties. In the Rastall gravity, one can also construct new black holes with nonlinear electrodynamics, and then explore whether the Van der Waals or reentrant phase transition can appear.



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