Thermodynamic geometry of a black hole surrounded by perfect fluid in Rastall theory
aa r X i v : . [ g r- q c ] O c t Thermodynamic geometry of a black hole surrounded byperfect fluid in Rastall theory
Saheb Soroushfar, a Reza Saffari, b Sudhaker Upadhyay c,d a Faculty of Technology and Mining, Yasouj University, Choram 75761-59836, Iran a Department of Physics, University of Guilan, 41335-1914, Rasht, Iran c Department of Physics, K.L.S. College, Nawada-805110, India d Visiting Associate, Inter-University Centre for Astronomy and Astrophysics (IUCAA) Pune, Maharashtra-411007
E-mail: [email protected] , [email protected] , [email protected] Abstract:
In this paper, we study thermodynamics and thermodynamic geometry of a blackhole surrounded by the perfect fluid in Rastall theory. In particular, we calculate the physicalquantity like mass, temperature and heat capacity of the system for two different cases. Fromthe resulting heat capacity, we emphasize stability of the system. Following Weinhold, Ruppiner,Quevedo and HPEM formalism, thermodynamic geometry of this black hole in Rastall gravity isalso analyzed. We find that the singular points of the curvature scalar of Ruppeiner and HPEMmetrics entirely coincides with zero points of the heat capacity. But there is another divergenceof HPEM metric which coincides with the singular points of heat capacity, so we can extractmore information of HPEM metric compared with Ruppeiner metric. However, we are unableto find any physical data about the system from the Weinhold and Quevedo formalism. ontents
In understanding theory of gravitation, general relativity (GR) follows the covariant conserva-tion of matter energy-momentum tensor. The limitation of such idea is that conservation ofenergy-momentum tensor has been probed only in the flat or weak-field arena of spacetime. Ageneralization of this theory, so-called Rastall theory, has recently been proposed, which relaxesthe necessity of covariant conservation of the energy-momentum tensor by adding some newterms to the Einstein’s equation [1]. Recently, this idea which has been supported in Ref. [2],which shows that the divergence of the energy-momentum tensor can be non-zero in a curvedspacetime. Rastall theory provides an explanation for inflation problem and other cosmologicalproblems [3–8]. In order to explain accelerated expansion of universe, one may think that thenegative pressure is generated by some peculiar kind of perfect fluid, where the proportion be-tween the pressure and energy density is between 1 and 1 /
3. If the scalar field generates thisperfect fluid, it is generally called quintessence. In Rastall theory of gravitation, the variousblack hole solutions surrounded by perfect fluid has been discussed [9–14].Hawking and Bekenstein were first who proposed that black holes have thermodynamicproperties [15, 16]. There after, this subject extensively studied by many people [13, 17–24]. Inthis connection, it has been found that the thermodynamic properties of black holes in Rastallgravity investigate the nature of a non-minimal coupling between geometry and matter fields.Recently, thermodynamics of various black holes under the effect of small thermal fluctuations– 1 –ave also been studied extensively. For example, thermodynamics of Van der Waals black holes[25], static black hole in f(R) gravity [26], charged rotating AdS black holes [27], black holesin gravitys rainbow [28, 29], quasitopological black holes [30], massive gravity black hole [31],Hoˇrava-Lifshitz black hole [32] and Schwarzschild-Beltrami-de Sitter black hole [33] have beendiscussed.Meanwhile, several attempts have been made to implement differential geometric conceptsin thermodynamics of black holes. In an attempt, to formulate the concept of thermodynamiclength, Ruppeiner metric [34], which conformally equivalent to Weinhold’s metric [35], is in-troduced. Furthermore, the phase space and the metric structures suggested that Weinhold’sand Ruppeiner’s metrics are not invariant under Legendre transformations [36, 37]. Afterwards,another formalism is developed by Quevedo [38], which unifies the geometric properties of thephase space and the space of equilibrium states. Also, a new metric (HPEM metric) [39–41] wasintroduced in order to build a geometrical phase space by thermodynamical quantities. Though,several study of thermodynamics of various black holes exploiting thermodynamic geometricmethods have been made, the thermodynamics of black holes surrounded by perfect fluid inRastall theory remains unstudied yet. This provides us an opportunity to bridge this gap. Thisis the motivation of present study.In this paper, we consider a black hole solution surrounded by the (general) perfect fluid inRastall theory and describe its thermodynamics by means of two specific cases. In this regard,we first compute mass by setting the metric function representing black hole surrounded by thequintessence field to zero. Furthermore, by exploiting standard thermodynamic relations, wederive the Hawking temperature and heat capacity. The heat capacity plays a pivotal role indescribing stability of the black hole. In order to study the behavior of the resulting thermody-namic entities, we plot graphs for them in terms of horizon radius. By doing so, we find that thetemperature of black hole surrounded by the quintessence field in Rastall theory remains positiveonly in a particular range of event horizon for quintessence factor N q = 0 .
05. The larger valuesof quintessence factor decrease temperature of the system. This suggests that increasing in thevalues of quintessence the physical area of this black hole decrease. The heat capacity plot showsthat there are physical limitation points and phase transition critical points occurring at differentvalues of event horizons. For this black hole, stability occurs for some particular values horizonradius. We also study the geometric structure of the black hole surrounded by quintessencefield by calculating the Weinhold, Ruppiner, Quevedo and HPEM curvature scalars. We findthat Weinhold curvature scalar vanishes and becomes fail to describe the phase transition. Weobtain a non-zero curvature scalar for the Ruppiner and HPEM geometries and, in these cases,the singular points coincide with zero points of the heat capacity. Also for the HPEM formalism,there is another divergency of HPEM metric which coincides with the singular points of heatcapacity and therefore we can extract more information rather than other mentioned metrics.The curvature scalar of Quevedo metric is also calculated. Here, in this case we can’t find anyphysical data about the system.Moreover, thermodynamic properties of black hole surrounded by dust field in Rastall grav-ity are also studied. The mass, Hawking temperature and heat capacity of black hole in dust– 2 –eld background are calculated. In order to do comparative analysis, we plot graphs in termsof horizon radius. From the obtained plots, it is obvious that the mass of the system has oneminimum point at particular value of horizon radius. The behavior of temperature with horizonradius is similar to the quintessence case. For instance, increasing the values of dust factor( N d ) the physical area of this black hole decrease. The heat capacity plot tells that stability ofblack hole increases with the size of black hole upto a certain point. After that point a phasetransition occurs and makes the black hole more unstable. We calculate the curvature scalar todiscuss the thermodynamic geometry for this case also.The paper is presented in the following manner. In section 2, we revisit the basic setup ofblack hole surrounded by perfect fluid in Rastall theory. Thermodynamics and thermodynamicgeometry of black hole surrounded by quintessence field are discussed in section 3. Thermo-dynamics and thermodynamic geometry of black hole surrounded by dust field is presented insection 4. We summarize results and discussions in the last section. In this section, we briefly review field equations and metric components in the context of Rastalltheory. The Rastall field equations for a space-time with Ricci scalar R and an energy momentumsource of T µν can be written as G µν + kλg µν = kT µν , (2.1)where k and λ are the Rastall gravitation coupling constant and the Rastall parameter whichindicates deviation scale from the standard GR, respectively. The metric of a black hole sur-rounded by perfect fluid in Rastall theory would be deduced as follows [12] ds = − f s ( r ) dt + dr f s ( r ) + r ( dθ + sin θdφ ) , (2.2)with f s ( r ) = 1 − Mr + Q r − N s r ws − kλ (1+ ws )1 − kλ (1+ ws ) . (2.3)where subscript “s” represents the surrounding field, N s is surrounding field structure parameter, Q is charge, w s is equation of state parameter and M is mass of the black hole. By considering λ = 0 and k = 8 πG N , this metric regains the Reissner-Nordstr¨om black hole surrounded by asurrounding field as [42] ds = − (cid:18) − Mr + Q r − N s r w s +1 (cid:19) dt + dr (cid:16) − Mr + Q r − N s r ws +1 (cid:17) + r d Ω . (2.4)We use the metric (2.2) to investigate the thermodynamic geometry of a black hole surroundedby perfect fluid in Rastall theory. In particular, we consider the quintessence and dust fields assub-classes of the Rastall fields. – 3 – Black hole surrounded by the quintessence field
In this section, we are going to investigate the thermodynamic properties of a black hole sur-rounded by the quintessence field. A quintessence field plays important role for the observedaccelerated expansion of the universe [42]. By considering w s = w q = − , the metric (2.2) takesthe following form: ds = − f q ( r ) dt + dr f q ( r ) + r d Ω , (3.1) f q ( r ) = 1 − Mr + Q r − N q r − − kλ − kλ . (3.2)For the case kλ = , metric (3.1) can be written as ds = − f q ( r ) dt + dr f q ( r ) + r d Ω , (3.3) f q ( r ) = 1 − Mr + Q r − N q r . (3.4) The mass corresponding to Eq. (3.3), can be obtain using f q ( r ) = 0. Also, we can express themass of the black hole M , in terms of its entropy S , using the relation between entropy S andevent horizon radius r + ( S = πr ), as M ( S, N q , Q ) = π Q + πS − N q S π S . (3.5)First law of thermodynamics for a black hole surrounded by quintessence field can be written as dM = TdS + Ψ dN q + ϕ dQ , (3.6)where ϕ is a quantity conjugate to electric charge Q and Ψ is a quantity conjugate to quintessenceparameter N q . So, by using the Eqs. (3.5) and (3.6), we obtain thermodynamic parameters liketemperature ( T = ∂M∂S ), heat capacity ( C = T ∂S∂T ), (Ψ = ∂M∂N q ), and ( ϕ = ∂M∂Q ) as follows T = − π Q − πS + 3 N q S π S , (3.7) C = − S (cid:0) π Q − πS + 3 N q S (cid:1) π Q − πS − N q S , (3.8) ϕ = r πS Q, (3.9)Ψ = − (cid:18) Sπ (cid:19) / . (3.10)– 4 –hese thermodynamic parameters are plotted in terms of horizon radius r + , (see Fig. 1).From Fig. 1(a), we find that for N q = 0 .
05, the mass of this black hole has a minimum valueat r = 0 .
51, afterword it arrives to its maximum value at r = 2 .
53, then it becomes zero at r = 4 .
5. Also, the behaviour of the temperature versus r + is demonstrated in Fig. 1(b). Fromthis figure, we see that for low values of r + , the temperature increases to a maximum pointand then it starts decreasing with higher r + . In other words, temperature is positive only in aparticular range of event horizon ( r < r + < r ) for N q = 0 .
05, however, at r < r , and r < r , itwill be negative and leads to an unphysical solution. In addition, Fig. 1(c) show that the increasein the values of quintessence factor ( N q ) decrease the temperature of black hole surrounded byquintessence. In other words, increasing in the values of quintessence the physical area of thisblack hole decreases. Moreover, in Fig. 1(d), the behaviour of the heat capacity versus r + isrepresented. Note that, the roots of heat capacity represent a physical limitation points and thedivergences of the heat capacity demonstrate phase transition critical points of black hole [41].From Fig. 1(d), we see that the heat capacity of this black hole will be zero at r and r , whichmeans that it has two physical limitation points. Furthermore, it diverges at r + = r ∞ = 0 . r + < r , which means that the black hole system is unstable. Then, at r < r + < r ∞ , theheat capacity is positive, which means that it is in stable phase. Next, at r ∞ < r + < r , itcrashes in to a negative region (unstable phase) and, at r + > r , it becomes stable. Further, forthe case N q = 0, the physical limitation points get changed, and there will be only one physicallimitation points. Moreover, the behaviour of ϕ as a quantity conjugate to electric charge Q and Ψ as a quantity conjugate to quintessence parameter N q versus r + are shown in Fig. 1(e),(f). As can be seen, ϕ has a maximum value at r + = 0 . r + , and Ψ starts from zero and then it falls into a negative region.– 5 – a) N q = 0 .
05 (b) N q = 0 . N q = 0 .
05 (f) N q = 0 . Figure 1 : Variations of thermodynamic parameters of a black hole surrounded by quintessence field in terms of horizonradius r + for Q = √ . – 6 – .2 Thermodynamic geometry In this section, based on geometric formalism suggested by Weinhold, Ruppiner, Quevedo andHPEM, we create the geometric structure for a black hole surrounded by quintessence. TheWeinhold geometry is specified in mass representation as [35] g Wij = ∂ i ∂ j M ( S, N q , Q ) . (3.11)In this case, the line element for a black hole surrounded by quintessence is ds W = M SS dS + M N q N q dN q + M QQ dQ + 2 M SN q dSdN q + 2 M SQ dSdQ + 2 M N q Q dN q dQ, (3.12)so the matrix g W = M SS M SN q M SQ M N q S M QS M QQ . (3.13)We use the above equations to obtain the curvature scalar of the Weinhold metric ( R W ) R W = 0 . (3.14)The curvature scalar in the Weinhold formalism is equal to zero, thus we cannot describe thephase transition of this thermodynamic system. Next, we consider the Ruppiner geometry.Through the conformal property, the Ruppiner metric in the thermodynamic system is definedas [34, 43, 44] ds R = 1 T ds W , (3.15)and the relevant matrix is g R = π S Sπ − π Q − S N q ! M SS M SN q M SQ M N q S M QS M QQ . (3.16)Therefore, the curvature scalar of the Ruppiner geometry is given by R Rup = − / √ S (cid:0) π Q − S N q − π S (cid:1) ( π Q + 3 S N q − π S ) π / (cid:18) Sπ (cid:19) − / . (3.17)The resulting curvature scalar is plotted versus horizon radius to investigate thermodynamicphase transition (see Fig. 2). It can be observed from this figure that the singular points of thecurvature scalar of Ruppeiner metric coincide with zero points of the heat capacity. Moreover,variations of Ruppeiner metric and the heat capacity in terms of different values of quintessencefactor ( N q ), are shown in Figs. 2(b) and 1(d). As one can see from these Figures, the changein the values of quintessence factor causes the change in zero points of the heat capacity andalso, singular points of the curvature scalar of Ruppeiner metric. But in any case, for both low– 7 – a) N q = 0 .
05 (b)
Figure 2 : (a) Curvature scalar variation of Ruppeiner metric (orange line) and the heat capacity (green dash line) of ablack hole surrounded by quintessence field and (b) Curvature scalar variation of Ruppeiner metric for differentvalues of quintessence factor, in terms of horizon radius r + for Q = √ . and high value of the quintessence factor, again the singular points of the curvature scalar ofRuppeiner metric entirely coincide with zero points of the heat capacity.Now, we will use the Quevedo and HPEM formalism to discuss the thermodynamic proper-ties of the black hole. The general form of the metric in Quevedo formalism is given as [38, 45]: g = (cid:18) E c ∂ Φ ∂E c (cid:19) (cid:18) η ab δ bc ∂ Φ ∂E c ∂E d dE a dE d (cid:19) , (3.18)in which ∂ Φ ∂E c = δ cb I b , (3.19)where E a and I b are the extensive and intensive thermodynamic variables, respectively, and Φis the thermodynamic potential. Also, the generalized HPEM metric with n extensive variables( n ≥
2) has the following form [39–41] ds HP EM = SM S ( Q ni =2 ∂ M∂χ i ) − M SS dS + n X i =2 ( ∂ M∂χ i ) dχ i ! , (3.20)In which, χ i ( χ i = S ), M S = ∂M∂S and M SS = ∂ M∂S are extensive parameters. Moreover, TheQuevedo and HPEM metrics can be written as [39–41] ds = ( SM S + N q M N q + QM Q )( − M SS dS + M N q N q dN q + M QQ dQ ) Quevedo Case I SM S ( − M SS dS + M N q N q dN q + M QQ dQ ) Quevedo Case II .SM S ( ∂ M∂N q ∂ M∂Q ) (cid:0) − M SS dS + M N q N q dN q + M QQ dQ (cid:1) HPEM (3.21)– 8 –hese metrics have following denominator for their Ricci scalars [40, 41] denom ( R ) = M SS M N q N q M QQ ( SM S + N q M N q + QM Q ) Quevedo Case I2 S M SS M N q N q M QQ M S Quevedo Case II . S M SS M S HPEM (3.22)Solving the mentioned equations, It is evident that in case of Quevedo formalism, we can’tfind any physical data about the system. But, in case of HPEM metric, as shown in Fig. 3,divergencies of the Ricci scalar and zero points of the heat capacity will coincide. Of course,there is another divergency of HPEM metric which coincides with the singular points of heatcapacity. Actually the denominator of the Ricci scalar of HPEM metric only contains numeratorand denominator of the heat capacity. In other words, divergence points of the Ricci scalar ofHPEM metric coincide with both types of phase transitions of the heat capacity. Therefore, Itseams we can extract more information of HPEM metric compared with Ruppeiner metric. (a) (b)
Figure 3 : Curvature scalar variation of HPEM metric (orange line) and heat capacity (blue line) of a black hole surroundedby quintessence field in terms of horizon radius r + for Q = √ .
25 and N q = 0, N q = 0 .
05 for (a) and (b)respectively.
In this section, we consider a black hole surrounded by the dust field. So, by putting w s = w d = 0and kλ = , the Eq. (2.2) can be written as [12] ds = − f d ( r ) dt + dr f d ( r ) + r d Ω , (4.1) f d ( r ) = 1 − Mr + Q r − N d r. (4.2)– 9 – .1 Thermodynamics In this section, we investigate thermodynamical behaviour of a black hole surrounded by dustfield. In this case, mass of the black hole M , in terms of its entropy S , can be written as M ( S, N d , Q ) = π Q + πS − N d S π π S , (4.3)and the first law of thermodynamics for this black hole is dM = TdS + Ψ dN q + ϕ dQ , (4.4)So, by using the above equations (4.3, 4.4), the thermodynamic parameters can be obtained as T = − π Q − πS + 2 N d S π π S , (4.5) C = − S (cid:16) π Q − πS + 2 N d S π (cid:17) π Q − πS , (4.6) ϕ = r πS Q, (4.7)Ψ = − Sπ . (4.8)Plots of these thermodynamic parameters in terms of horizon radius r + , are shown in Figs. 4.Fig. 4(a) shows that mass of this black hole has one minimum point at r = 0 . r + . Moreover, similar to quintessence case, the increasein the values of dust factor ( N d ) causes decrease in the temperature of a black hole surroundedby dust field. In addition, Fig. 4(c) shows that the heat capacity is in the negative region(unstable phase), afterword at r , it becomes zero and Then for r + > r , it locate in positiveregion (stable phase). Moreover, it can be observed from Fig. 4(c) that for the case N q = 0,there are two physical limitation points, and for the case N q = 0, there will be only one physicallimitation point. Furthermore, in Fig. 4(d), (e) variation of ϕ and Ψ versus r + are demonstrated.As can be seen, similar to quintessence case, ϕ has a maximum value at r + = 0 . r + , and Ψ starts from zero and then it falls into a negative region.– 10 – a) N q = 0 .
05 (b)(c) (d) N q = 0 . N q = 0 . Figure 4 : Variations of thermodynamic parameters of a black hole surrounded by dust field in terms of horizon radius r + for Q = √ . – 11 – .2 Thermodynamic geometry In this section, again we use the geometric formalism of Weinhold, Ruppiner, Quevedo andHPEM metrics of the thermal system, and investigate the physical limitation points and phasetransition critical points of a black hole surrounded by dust field. we can use the followingWeinhold metric [35]: g Wij = ∂ i ∂ j M ( S, N d , Q ) , (4.9)and write the line element corresponding to Weinhold metric for this system in mass represen-tation as ds W = M SS dS + M N d N d dN d + M QQ dQ + 2 M SN d dSdN d + 2 M SQ dSdQ + 2 M N d Q dN d dQ. (4.10)Therefore, the relevant matrix is g W = M SS M SN d M SQ M N d S M QS M QQ , (4.11)and the curvature scalar of the Weinhold metric for this system is R W = 0 . (4.12)So, the Weinhold formalism suggests that black hole surrounded by dust field is flat and wecannot investigate the phase transition of this thermodynamic system. Moreover, we considerRuppiner formalism in which line element is conformaly transformed to the Weinhold metric as[34, 43, 44] ds R = 1 T ds W . (4.13)The matrix corresponding to this metric is g R = π S Sπ − π Q − S N d π ! M SS M SN d M SQ M N d S M QS M QQ , (4.14)and the relevant scalar curvature is R Rup = 1 / N d S / √ π − π Q + 3 π S (cid:0) N d S / √ π + π Q − π S (cid:1) S . (4.15)Plot of this scalar curvature is demonstrated in Fig. 5(a). Moreover, plot of the curvature scalarvariation of the Ruppiner metric and heat capacity, in terms of r + is shown in Fig. 5(b). It canbe seen from Fig. 5(a), (b) that changes in the value of dust factor cause change in the singularpoints of the curvature scalar of Ruppeiner metric and also in zero points of heat capacity.Moreover, similar to quintessence case, for both low and high values of dust factor, the singularpoints of the curvature scalar of Ruppeiner metric entirely coincides with zero points of heatcapacity. – 12 – a) (b) Figure 5 : (a) Curvature scalar variation of Ruppeiner metric for different values of dust factor, and (b) Curvature scalarvariation of Ruppeiner metric (green continuous line) and heat capacity (blue dash line) of a black hole sur-rounded by dust field in terms of horizon radius r + for N d = 0 . Q = √ . – 13 –ow, to extend the analysis, we consider the Quevedo and HPEM metrics as [39–41] ds = ( SM S + N d M N d + QM Q )( − M SS dS + M N d N d dN d + M QQ dQ ) Quevedo Case I SM S ( − M SS dS + M N d N d dN d + M QQ dQ ) Quevedo Case II .SM S ( ∂ M∂N d ∂ M∂Q ) (cid:0) − M SS dS + M N d N d dN d + M QQ dQ (cid:1) HPEM (4.16)These metrics have following denominator for their Ricci scalars [40, 41] denom ( R ) = M SS M N d N d M QQ ( SM S + N d M N d + QM Q ) Quevedo Case I2 S M SS M N d N d M QQ M S Quevedo Case II . S M SS M S HPEM (4.17)Solving the above equations, again similar to the quintessence section for the case of Quevedoformalism, we can’t find any physical data about the system. However, in case of HPEM metric,as shown in Fig. 6, divergencies of the Ricci scalar coincide with zero points of the heat capacity.But, In comparison with Ruppeiner metric, we can see another divergency of HPEM metricwhich coincides with the singular points of heat capacity. So, divergence points of the Ricciscalar of HPEM metric coincide with both types of phase transitions of the heat capacity. (a) (b)
Figure 6 : Curvature scalar variation of HPEM metric (orange line) and heat capacity (green line) of a black hole surroundedby dust field in terms of horizon radius r + for Q = √ . N q = 0 and Q = √ . N q = 0 .
19 for (a) and (b)respectively.
One possible explanation to the negative pressure in an expanding universe is that the universe isfilled with a peculiar kind of perfect fluid, where the proportion between the pressure and energy– 14 –ensity is between 1 and 1 /
3. If this is so, then it becomes important to study the interaction ofthe strong gravity objects such as black holes with such fluid. This idea is proposed by Kiselev[42], and has further been generalized to the Rastall model of gravity [12].In this paper, we have considered a black hole surrounded by a generic perfect fluid in Rastalltheory to emphasize its thermodynamics. To make the analysis clear, we have focused on twospecial cases of the perfect fluid. First, we have derived an expression for the mass of black holesurrounded by quintessence field. The resulting mass of this system has a minimum value forsmall horizon radius r + , afterword it arrives to its maximum value for a bit higher value of r + ,then vanishes for larger r + . By exploiting standard thermodynamic relations, we have derivedthe Hawking temperature and heat capacity. It is known that negative heat capacity correspondsto instability, however the positive value of heat capacity describe stable black holes. In orderto study the behavior of thermodynamic entities obtained for black hole surrounded by bothquintessence field and dust field, we have plotted the graphs with respect to event horizon. Forthe figure, it is clear that temperature of black hole surrounded by quintessence field in Rastalltheory remains positive only for the specific values of event horizon for quintessence factor N q = 0 .
05. It is also found that the increasing in the values of quintessence the physical areaof this black hole decrease. For heat capacity plot, we have found that there exist two physicallimitation points for the system at two different values of event horizons. Here we note that thestability do not occur corresponding to all horizon radius. Further, we have studied the geometricstructure of the black hole surrounded by quintessence field. This is done by calculating theWeinhold, Ruppiner, Quevedo and HPEM curvature scalars. We find that Weinhold curvaturescalar vanishes and in this situation one can not describe the phase transition. As well as, wehave derived the curvature scalar of Quevedo metric and haven’t found any physical data aboutthe system. The curvature scalar for the Ruppiner and the HPEM geometry does not vanishes.The curvature scalar for the Ruppiner and the HPEM geometry suggests that the singular pointscoincide with zero points of the heat capacity. Also, we have observed another divergency ofHPEM metric which coincides with the singular points of heat capacity. So, divergence pointsof the Ricci scalar of HPEM metric coincide with both types of phase transitions of the heatcapacity. Therefore, It can be extracted more information from HPEM metric compared withother mentioned metrics.Moreover, the thermodynamic properties of black hole surrounded by dust field in Rastallgravity are also studied. Following the earlier case, we have calculated the mass, Hawkingtemperature and heat capacity here also. To emphasize the behavior of these quantities, wehave plotted the graphs with respect to event horizon. From the figure, we have found that themass of the system has one minimum point at particular value of horizon radius. The behaviorof temperature with horizon radius is similar to the quintessence case, i.e., increasing in thevalues of quintessence the physical area of this black hole decrease. From the plot, we have seenthat stability of black hole increases with the size of black hole upto a certain point. After thatpoint a phase transition occurs and converts the black hole in to the more unstable state. Thegeometric properties of black hole is studied in this case also. It would be interesting to discussthe effect of thermal fluctuations on thermodynamics of the black hole surrounded by Rastall– 15 –ravity.
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