1 Introduction

The Last Lost Charge And Phase Transition In Schwarzschild AdS Minimally Coupled to a Cloud of Strings


In this paper we study the Schwarzschild AdS black hole with a cloud of string background in an extended phase space and investigate a new phase transition related to the topological charge. By treating the topological charge as a new charge for black hole solution we study its thermodynamics in this new extended phase space. We treat by two approaches to study the phase transition behaviour via both and criticality and we find the results confirm each other in a nice way. It is shown a cloud of strings affects the critical physical quantities and it could be observed an interesting Van der Waals-like phase transition in the extended thermodynamics. The swallow tail-like behavior is also observed in Free Energy-Temperature diagram. We observe in limit the small/large black hole phase transition reduces to the Hawking-Page phase transition as we expects. We can deduce that the impact of cloud of strings in Schwarzschild black hole can bring Van der Waals-like black hole phase transition.

The Last Lost Charge And Phase Transition In Schwarzschild AdS Minimally Coupled to a Cloud of Strings

Hossein Ghaffarnejad 1 and Mohammad Farsam 2 Faculty of Physics, Semnan University, P.C. 35131-19111, Semnan, Iran

1 Introduction

Black hole thermodynamics in Anti de-Sitter spacetime has been a hot topic in the recent past. In a remarkable work Hawking and Page discovered a phase transition between AdS black holes and a global AdS space [1]. Witten explained the Hawking-Page phase transition in terms of the AdS/CFT correspondence as a dual of the QCD confinement/deconfinement transition [2-3]. In [4-5] Chamblin et al found a Van der Waals like phase transition in Reissner-Nordstrom AdS black hole. Recently Kubiznak and Mann discovered a surprising analogy between Reissner-Nordstrom AdS black holes and Van der Waals fluid-gas system in the extended phase space of thermodynamics [6]. The extended phase space refers to a phase space in which the first law of black holes is corrected by a term and the cosmological constant is regarded as thermodynamical pressure of the black hole and its conjugate variable is a volume covered by the event horizon of the black hole [7-8]. Thermodynamical behaviors of a wide range of AdS black holes are studied in details in several works [9-27]. In the other side, Tian [28-29] introduces the spatial curvature of Reissner-Nordstrom as a topological charge that naturally arises in holography. The author called it ”the last lost charge” because it is shown that topological charge appears in an extended first law with all other known charge (mass, electric charge, angular momentum) as a new variable and satisfies the Gibbs-Duhem like relation. The phase transition related to the topological charge in Reissner-Nordstrom AdS black hole is studied in [30-31]. The thermodynamics properties of the Schwarzschild AdS black hole surrounded by a cloud of strings background in a non-extended phase space was reported in [32]. In this paper we would like to study the new extended phase space of Schwarzschild AdS black hole in a cloud of strings background related to the topological charge by two formal approaches which leads to the same result. At the first approach we seek criticality behaviour of phase transition around the critical point and try to confirm it by plotting diagrams of free energy against temperature. At the second approach we try to study these behaviours in a criticality diagram. One can see the impact of cloud of strings can bring Van der Waals like black hole phase transition in an extended phase space.

2 Topologically Schwarzschild AdS Minimally Coupled to a Cloud of Strings

The action of Einstein gravity coupled to the cloud of strings can be written as


where the last term is a Nambu-Goto action. Note is the metric of spacetime, is a non-negative constant related to the string tension, is a paramerization of the world sheet and is the determinant of the induced metric

The energy momentum tensor for a cloud of strings is given by


where the number density of a string cloud is described by and is the spacetime bivector


The black hole solution for the Einstein gravity coupled to a cloud of strings has been derived [33] as follows.




where the metric function denotes to metric function of the dimensional hyper-surface with constant scalar curvature . The constant specifies the geometric property of the hyper-surface, which takes the values , and for flat, spherical and hyperbolic respectively. In terms of the horizon , the ADM mass , the Unruh-Verlinde temperature and the Wald-Padmanabhan entropy can be written as


where is volume of the unit sphere, plane or hyperbola. Note that we assume a constant for the volume of unit space in our study [27-28].

By using an equipotential surface and by varying with respect to variables , , and , the equation (2.8) reads


The last equation can be rewritten as


for which the generalized first law become


where is topological charge, is its conjugated potential and is string cloud conjugated potential.

Figure 1: Diagram of temperature against entropy. Dot line for , dash lines for and solid line for
Figure 2: Diagram of free energy against temperature for . Dot line stands for , dash lines for:, and solid line for
Figure 3: Diagram of the free energy is plotted versus the temperature for . Dot line for , dash lines for , and solid line:

3 Critical Phenomena of Schwarzschild AdS background With a Cloud of Strings Background

In this section, we will study the phase transition of 5 dimensional Schwarzschild AdS black hole in the presence of cloud of strings by two approachs, criticality and criticality. Note that this solution for 4 dimensions reduces to Schwarzschild AdS black hole which does not have Van der Waals like phase transition.

3.1 criticality

To study phase transition and other thermodynamic behaviors of the system it must be noted that there is a critical hyper-surface in the parameter space and we can study it at a critical point by fixing some other parameters. As we can see the temperature of the black hole is a function of entropy and topological charge:


then the critical point can be calculated under the following conditions.


which leads to


The above equations show affects of the string cloud on location of the critical points. One can see in the absence of string cloud, criticality can not happened. On the other side the critical points increases with string cloud factor. We plot the diagrams of the equation of state in figure 1. In this figure we observe that for upper than critical topological charge ( is shown by solid line) we have three different solutions for the black hole’s horizon which corresponds to the identifying of them with three branches. Namely stable large black hole, stable small black hole and an unstable medium black hole. It is clear that below of critical topological charge the figure is identical to the liquid-gas transition in Van der Waals fluid. Indeed we have three cases for black hole thermodynamic in here which are small, medium and large size black holes. In above of critical topological charge, as it has investigated in figure 2, medium unstable case vanishes and small black holes transfer to large ones straightforwardly. Another way to investigate phase transition is studying the behavior of free energy which is a function of temperature and topological charges. Actually free energy has an important role in studying thermodynamic properties of a system. Any thermodynamic system always exists in a phase which has minimum value of free energy among other possible values of them and a phase transition happens when some branches of minimum free energies cross each other. The free energy for the black hole under consideration is given by:


For plotting the diagram of free energy against Unruh-Verlinde temperature we fix and in figure 2 and 3. In figure 2 for topological charge lower than critical topological charge diagram takes a swallow tail shape which indicates a first order phase transition between small and large black holes. In figure 3 we plot diagram for and we observe the small/large black hole phase transition is reduced to the Hawking-Page phase transition. In fact the effects of string cloud factor is vital to have the small/large black hole phase transition. So it is interesting to note that the impact of cloud of strings can bring Van der Waals like black hole phase transition.

3.2 P-V criticality

By treating the cosmological constant as the pressure of the black hole,


In 5 dimensional Schwarzschild black hole solution surrounded by a cloud of strings the Hawking temperature and the generalized first law can be derived respectively as follows.




where and are specific and thermodynamical volume which are defined respectively by


Using (4.1) the black hole equation of state can be written as


critical points can be calculated through the following conditions


which leads to


It is interesting to note that the effect of a cloud of strings appears in all the critical points which indicates in limit, the criticality disappears as we expects in Schwarzschild black hole. So a cloud of strings affects in criticality of the system. Van der Waals universal ratio for this black hole is which is very similar to Van der Waals fluid.

Figure 4: Pressure is plotted vs volume for and . Solid line represents , dash lines , and dot line

We plot diagram in figure 4, where one can infer that the Van der Waals-like behavior happens for temperatures lower than the critical temperature ( in contrary with diagram which criticality happens for topological charge upper than the critical topological charge).

Figure 5: Pressure is plotted versus the volume for and , which solid line indicates , dash line and dot line
Figure 6: Gibbs free energy is plotted against the temperature for and , for the pressures which are lower than the critical one.

4 Conclusion

We studied the thermodynamics of topologically Schwarzschild AdS minimally coupled in presence of a cloud of strings in an extended phase space so that topological charge behaves as new charge for black hole. The entropy of the black hole is invariant under the existence of a cloud of strings but it is interesting to note that the impact of cloud of strings can bring Van der Waals like phase transition for this black hole. It is shown that the iso-topological charges correspond to the topological charge less than the critical one which can be divided into three different branches. Two branches which are correspond to the small and the large black holes are maintained as stable while the medium black hole branch is unstable. First order phase transition is observed due to some topology charges which are upper than the critical one. In figure 5 we fixed the temperature at critical value and varying upper and lower than , we see it is in agreement with criticality.



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  1. E-mail: hghafarnejad@semnan.ac.ir
  2. E-mail: mhdfarsam@semnan.ac.ir
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