Abstract On the basis of a charged BTZ black hole, we add an extra term in the metric function to describe the contribution from nonlinear electrodynamics. In this way we artificially construct a (2 + 1)-dimensional black hole in general relativity coupled with a nonlinear electrodynamics source. The thermodynamic quantities and Smarr formula are derived. It is found that this black hole has T−S criticality like that of an RN-AdS black hole. Further modifying the metric function, we obtain a (2 + 1)-dimensional black hole possessing P−V critical behaviors similar to that of van der Waals fluid. To our knowledge, this is the first example of (2 + 1)-dimensional black holes having this kind of critical behavior. Keywords：phase transition;critical phenomena;(2+1)-dimensional black hole;nonlinear electrodynamics
PDF (372KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Li-Hua Wang, Yun He, Meng-Sen Ma. Critical behaviors of a (2 + 1)-dimensional black hole with a nonlinear electrodynamics source. Communications in Theoretical Physics, 2020, 72(10): 105401- doi:10.1088/1572-9494/aba24e
1. Introduction
As thermodynamic systems, black holes have temperature and entropy and many other thermodynamic potentials. Among the great number of black holes, AdS black holes have received much interest due to their fruitful critical behaviors and phase structures. Through the Schwarzschild-AdS black hole, Hawking and Page found a first-order phase transition from the AdS black hole to the thermal AdS space, now known as the Hawking-Page phase transition [1]. Thereafter, thermodynamic properties of black holes in AdS space have been extensively studied [2-9]. In particular, when treating the cosmological constant as the thermodynamic pressure and its conjugate quantity as the thermodynamic volume, one can construct an extended phase space [10-12]. In this way, it was found that RN-AdS black hole exhibits a P−V criticality similar to that of van der Waals liquid/gas system [13]. After that, many AdS black holes have been deeply studied and many interesting critical behaviors were found [14-22, 23-31, 32].
However, almost all the previous works have focused on AdS black hole in D≥4 dimensional spacetime. As far as we know, only in [33] a P−V criticality of a (2 + 1)-dimensional black hole, which is derived from a gravity with non-minimal coupling [34, 35], was found. In fact, this is not a standard P−V criticality. As is shown in figure 2 in [33], all isotherms intersect at a point, which is called ‘thermodynamic singularity' [36, 37]. At this point, the temperature in the equation of state cancels out.
In a previous work, we proposed a procedure by which one can construct many (2 + 1)-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics [38]. Generally, a Lagrangian of the matter fields was first given and then solve the field equations to obtain metric functions and the matter fields. In our procedure, we take the opposite way. We first construct the metric function and then derive the Lagrangian of the matter fields. In the present work, we will relax the condition of regular black hole and only require Maxwell theory as the weak field limit of the nonlinear electrodynamics. In this way, we can construct many (2 + 1)-dimensional black holes possessing fruitful critical behaviors.
The paper is arranged as follows. In section 2, we construct a (2 + 1)-dimensional black hole with nonlinear electrodynamics source. In section 3, we will calculate the thermodynamic quantities and analyze its thermodynamic properties. In section 4, the critical behaviors of this black hole are analyzed. We shall give some concluding remarks in the final section.
2. (2 + 1)-dimensional black hole
Here we will simply introduce the (2 + 1)-Einstein gravity coupled with nonlinear electrodynamics. For details, one can refer to [38, 39].
The action is given by$\begin{eqnarray}S=\int {{\rm{d}}}^{3}x\sqrt{-g}\left[\displaystyle \frac{{ \mathcal R }-2{\rm{\Lambda }}}{16\pi }+L(F)\right],\end{eqnarray}$where g is the determinant of the metric tensor, Λ=−1/l2 is the cosmological constant, and L(F) is the Lagrangian of the nonlinear electrodynamics with $F={F}^{\mu \nu }{F}_{\mu \nu }$.
We are only concerned with the static, circularly symmetric spacetime and take the simplest metric ansatz$\begin{eqnarray}{\rm{d}}{s}^{2}=-f{(r){\mathrm{dt}}^{2}+f(r)}^{-1}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{\phi }^{2}.\end{eqnarray}$
Under this metric ansatz, the electromagnetic field tensor takes the form of$\begin{eqnarray}{F}_{\mu \nu }=E(r)({\delta }_{\mu }^{t}{\delta }_{\nu }^{r}-{\delta }_{\nu }^{t}{\delta }_{\mu }^{r}),\end{eqnarray}$and correspondingly F=−2E2.
The field equations under the metric ansatz have the simple forms:$\begin{eqnarray}\displaystyle \frac{f^{\prime} (r)}{2r}+{\rm{\Lambda }}=8\pi \left[L(r)-\displaystyle \frac{4q}{r}E(r)\right],\quad L^{\prime} (r)=\displaystyle \frac{4q}{r}E^{\prime} (r),\end{eqnarray}$where q is an integration constant related the the electric charge.
In the two equations there are three unknown functions: f(r),E(r),L(F)orL(r). To solve them, the general approach is to first give L(F) and then to derive metric functions. However, in this way one cannot control the properties of the derived metric functions. As mentioned above, we will first construct a metric function and then derive the Lagrangian L(r) and E(r).
With linear electrodynamics source (Maxwell field), one can obtain the charged BTZ black hole [40, 41], which has the metric function$\begin{eqnarray}f(r)=-m+{r}^{2}/{l}^{2}-{q}^{2}\mathrm{ln}({r}^{2}/{l}^{2}).\end{eqnarray}$For nonlinear electrodynamics sources, we assume an extra correction term k(r) on the basis of the charged BTZ black hole, namely $f(r)=-m+\tfrac{{r}^{2}}{{l}^{2}}-{q}^{2}\mathrm{ln}({r}^{2}/{l}^{2})+k(r)$. Therefore, all the nonlinear information about the source is contained in the correction term k(r). In general, Lagrangians of nonlinear electrodynamics source contain some parameters describing the nonlinearity. For example, the Lagrangian of Born-Infeld nonlinear electrodynamics is [42]$\begin{eqnarray}L(F)=-\displaystyle \frac{{b}^{2}}{4\pi }\left(\sqrt{1+2\displaystyle \frac{F}{{b}^{2}}}-1\right).\end{eqnarray}$In the limit ${b}^{2}\to \infty $, it will tend to the Maxwell theory. Similarly, we assume that our nonlinear model relies on a parameter a, and thus the correction term k(r) must contain the parameter a. Because the nonlinear electrodynamics should tend to Maxwell theory in the weak field limit, we require that k(r) tend to zero in the limit a→0.
In this way, we can construct many black holes satisfying the constraints. For example, we can take the metric has the form:$\begin{eqnarray}f(r)=-m+{r}^{2}/{l}^{2}-{q}^{2}\mathrm{ln}({r}^{2}/{l}^{2})+{{ar}}^{k}.\end{eqnarray}$We find that for positive a, when 0<k<1, the black hole solutions have T−S, even P−V criticality.
Below we only consider a simple example:$\begin{eqnarray}f(r)=-m+{r}^{2}/{l}^{2}-{q}^{2}\mathrm{ln}({r}^{2}/{l}^{2})+{{ar}}^{1/2},\end{eqnarray}$where a is a parameter coming from the Lagrangian of nonlinear electromagnetic fields. When a→0, the black hole solution gets back to the charged BTZ black hole.
In figure 1, we draw the graphs of f(r) for fixed values of m, q, l. Adjusting the value of a, we can obtain black holes with two horizons, one horizon and no horizon, respectively.
Figure 1.
New window|Download| PPT slide Figure 1.Behaviors of the metric function for different values of a. From top to bottom, a=1, 0.327, 0.1. Other parameters take fixed values: m=2, q=1, l=10.
Substituting the metric function into equation (2.4), we can obtain the electric field and the Lagrangian:$\begin{eqnarray}E(r)=\displaystyle \frac{16{q}^{2}-3a\sqrt{r}}{256\pi {qr}},\quad L(r)=\displaystyle \frac{8{q}^{2}-a\sqrt{r}}{64\pi {r}^{2}}.\end{eqnarray}$One can easily check that they return back to the results in Maxwell theory when a→0. However, in this method we cannot obtain an analytical expression L(F) of the nonlinear electrodynamics source generally.
3. Thermodynamics
The temperature of the black hole can be easily derived according to the metric function$\begin{eqnarray}T=\displaystyle \frac{f^{\prime} ({r}_{h})}{4\pi }=\displaystyle \frac{1}{4\pi }\left(\displaystyle \frac{a}{2\sqrt{{r}_{h}}}+\displaystyle \frac{2{r}_{h}}{{l}^{2}}-\displaystyle \frac{2{q}^{2}}{{r}_{h}}\right),\end{eqnarray}$where rh is the position of the event horizon.
Because we consider the Einstein theory of gravity, the entropy of the (2 + 1)-dimensional black hole takes the usual Bekenstein-Hawking form, which is S=A/4=π rh/2. Due to the same reason given in [38], the electric charge and mass should be Q=8πq and M=m/8, respectively.
In our solution, $m,\,q$ are both dimensionless, whereas the parameter a has dimension (length)−1/2. According to dimensional analysis and Euler theorem [10], the parameter a should be included in the Smarr formula$\begin{eqnarray}0={TS}-2{VP}-\displaystyle \frac{1}{2}{ \mathcal A }a,\end{eqnarray}$where we have taken P=1/8πl2. Besides, V is the thermodynamic volume conjugate to P and ${ \mathcal A }$ is the quantity conjugate to a. They can be derived from the first law of thermodynamics:$\begin{eqnarray}{\rm{d}}M=T{\rm{d}}S+{\rm{\Phi }}{\rm{d}}Q+V{\rm{d}}P+{ \mathcal A }{\rm{d}}a,\end{eqnarray}$here$\begin{eqnarray}\begin{array}{ccc}{\rm{\Phi }} & = & {\left.\displaystyle \frac{{\rm{\partial }}M}{{\rm{\partial }}Q}\right|}_{S,P,a}=-\displaystyle \frac{q{\rm{l}}{\rm{n}}\left(8\pi {{\Pr }}_{h}^{2}\right)}{32\pi },\\ V & = & {\left.\displaystyle \frac{{\rm{\partial }}M}{{\rm{\partial }}P}\right|}_{S,Q,a}=\pi {r}_{h}^{2}-\displaystyle \frac{{q}^{2}}{8P},\\ A & = & {\left.\displaystyle \frac{{\rm{\partial }}M}{{\rm{\partial }}a}\right|}_{S,Q,P}=\displaystyle \frac{\sqrt{{r}_{h}}}{8}.\end{array}\end{eqnarray}$With these quantities, one can check that equation (3.2) is indeed satisfied.
4. Critical behaviors of the (2 + 1)-dimensional black hole
In this part we will analyze the critical behaviors of the black hole. To demonstrate the criticality, we can employ the Gibbs free energy. In the extended phase space, the mass M is the enthalpy. Thus the Gibbs free energy is given by$\begin{eqnarray}\begin{array}{rcl}G & = & M-{TS}=\displaystyle \frac{1}{16}\left[a\sqrt{{r}_{h}}\right.\\ & & \left.-\,2{q}^{2}\mathrm{ln}\left(8\pi {{\Pr }}_{h}^{2}\right)-16\pi {{\Pr }}_{h}^{2}+4{q}^{2}\right].\end{array}\end{eqnarray}$
The behaviors of G are depicted in figure 2. It can be seen that when the pressure is less than a critical value there is a swallow tail behavior, which characterizes a first-order phase transition. At the critical point, a second-order phase transition occurs.
Now let us analyze the T−S criticality. It is shown in figure 3 that the temperature exhibits a behavior similar to that of RN-AdS black hole. Here we consider two cases. First we fix the parameters (P, q) and adjust the parameter a. We find that when a>ac the temperatures show a first-order phase transition. Then, we fix the parameters (a, P) and adjust the parameter q. We find that a first-order phase transition occurs when q<qc. This means that the effects of the parameters a, q on the criticality of the black hole are opposite. One can also fix the pair of parameter (q, a) and adjust the pressure P. In this case the critical behavior is similar to the changing q case.
Figure 3.
New window|Download| PPT slide Figure 3.Behaviors of T−S. In the left panel, we fix q=0.2, P=0.01 and take different values of a. From top to bottom, a=1, a=ac=0.889, a=0.7. In the right panel, we set a=0.5, P=0.01 and take q=0.1, q=qc=0.136, q=0.18 from top to the bottom. In both figures, the black point denotes the position of the critical point.
From equation (3.4), one can see that the thermodynamic volume is not the standard one, $\pi {r}_{h}^{2}$. This is due to the logarithmic term $\mathrm{ln}({r}^{2}/{l}^{2})$ in the metric function. Therefore, for this black hole solution there is no P−V criticality like that in the RN-AdS black hole or van der Waals fluid.
One can replace the de Sitter radius l appearing in the logarithmic term with a renormalization length scale R to give an alternative interpretation on the charged BTZ black hole [41]. After the replacement, one can obtain the standard thermodynamic volume according to the first law of black hole thermodynamics. Following the same idea, we can also modify our metric function into the form$\begin{eqnarray}f(r)=-m+{r}^{2}/{l}^{2}-{q}^{2}\mathrm{ln}({r}^{2}/{R}^{2})+{{ar}}^{1/2}.\end{eqnarray}$
This modification does not influence the electric field and Lagrangian given in equation (2.9), and also does not alter the temperature and entropy we derived above. Therefore, the T−S criticality discussed above still exists for this solution.
Now there is another dimensional quantity R which has the dimension of (length). Correspondingly, the Smarr formula should be modified$\begin{eqnarray}0={TS}-2{VP}-\displaystyle \frac{1}{2}{ \mathcal A }a+{KR},\end{eqnarray}$where K is the quantity conjugate to R and can be derived according to the new first law$\begin{eqnarray}{\rm{d}}M=T{\rm{d}}S+{\rm{\Phi }}{\rm{d}}Q+V{\rm{d}}P+{ \mathcal A }{\rm{d}}a+K{\rm{d}}R.\end{eqnarray}$Now we have K=q2/4R and $V=\pi {r}_{h}^{2}$. The electric potential Φ should also be modified correspondingly.
With the new (standard) thermodynamic volume, we can obtain the equation of state$\begin{eqnarray}P=\displaystyle \frac{8\pi {r}_{h}T-a\sqrt{{r}_{h}}+4{q}^{2}}{32\pi {r}_{h}^{2}}.\end{eqnarray}$For simplicity, we directly analyze the P−rh criticality. In figure 4, it is shown that the modified black hole solution has a typical critical behavior similar to many AdS black holes and van der Waals fluid. One can further calculate the critical point, which lies at$\begin{eqnarray}({P}_{c},{T}_{c},{r}_{{hc}})=\left(\displaystyle \frac{27{a}^{4}}{8388608\pi {q}^{6}},\,\displaystyle \frac{9{a}^{2}}{1024\pi {q}^{2}},\,\displaystyle \frac{1024{q}^{4}}{9{a}^{2}}\right).\end{eqnarray}$The ratio, ${P}_{c}{r}_{{hc}}/{T}_{c}=1/24$, is a constant and independent of the parameters a and q.
Figure 4.
New window|Download| PPT slide Figure 4.P−V criticality. We fix q=0.2, a=0.5 and take T=0.0165, T=Tc=0.0175, T=0.018.
Critical behaviors can also be reflected from critical exponents. The commonly used critical exponents are defined as [13]$\begin{eqnarray}\begin{array}{rcl}{C}_{v} & = & T{\left.\displaystyle \frac{\partial S}{\partial T}\right|}_{v}\propto | t{| }^{-\alpha },\quad {v}_{g}-{v}_{l}\propto | t{| }^{\beta },\\ {\kappa }_{T} & = & {\left.-\displaystyle \frac{1}{v}\displaystyle \frac{\partial v}{\partial P}\right|}_{T}\propto | t{| }^{-\gamma },\quad | P-{P}_{c}| \propto | v-{v}_{c}{| }^{\delta }.\end{array}\end{eqnarray}$
According to the universal analysis given in [43], the critical exponents should also be $\alpha =0,\,\beta =1/2,\,\gamma =1,\delta =3$.
5. Concluding remarks
In this paper, we studied the thermodynamic properties and critical behaviors of a (2 + 1)-dimensional black hole derived from general relativity coupled with nonlinear electrodynamics source. (2 + 1)-dimensional gravity has a simple structure. Given a nonlinear Lagrangian L(F), one can easily derive corresponding metric functions. But in this way, one cannot control the properties of the derived black hole. As far as we know, all the known black hole solutions in (2 + 1)-dimensional spacetime do not possess critical behaviors similar to that of van der Waals fluids.
We took an opposing line of thinking to study critical behaviors of (2 + 1)-dimensional black hole. We first optionally constructed a metric function which describes our black hole solution. Then substituting it into field equations, we can obtain the electric field and the corresponding Lagrangian of the nonlinear electrodynamics.
We gave a simple model in the present work. An extra parameter a describing the nonlinear electrodynamics was introduced. As a dimensional quantity, it influenced the form of the Smarr formula and the first law of black hole thermodynamics. The black hole constructed in this way can have T−S criticality. We made a further modification by replacing the de Sitter radius in the logarithmic term with a renormalization length scale R. This dimensional parameter further modified the Smarr formula and the first law. But in this way, the thermodynamic volume took the standard form, $V=\pi {r}_{h}^{2}$. We found that P−V criticality occurs in the modified black hole.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (Grant No. 11605107), the Natural Science Foundation of Shanxi Province of China (Grant No. 201701D121002), and Datong City Key Project of Research and Development of Industry of China (Grant No. 2018021).
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,1,2, Yun He
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