Temporal and spatial chaos in the Kerr-AdS black hole in an extended phase space
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Bing Tang , Department of Physics, Jishou University, Jishou 416000, China Received Date:2020-10-10 Available Online:2021-05-15 Abstract:Based on the Melnikov method, we investigate chaotic behaviors in the extended thermodynamic phase space for a slowly rotating Kerr-AdS black hole under temporal and spatial perturbations. Our results show that the temporal perturbation coming from a thermal quench of the spinodal region in the phase diagram may cause temporal chaos only when the perturbation amplitude is above a critical value, which involves the angular momentum J. Under the spatial perturbation, however, it is found that spatial chaos always occurs, independent of the perturbation amplitude.
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II.THERMODYNAMICS OF THE KERR-ADS BLACK HOLE IN AN EXTENDED PHASE SPACEIn this section, let us give an overview on the thermodynamics of a four-dimensional AdS rotating black hole. It can be described by the following Kerr-AdS metric [9, 40]
In the extended phase space, the cosmological constant $\Lambda = - \dfrac{3}{{{l^2}}}$ can be interpreted as a thermodynamic pressure P, which has the following relation
In this case, the first law of black hole thermodynamics and Smarr formula are respectively written as
$\delta M = T\delta S + {\Omega _H}\delta J + V\delta P,$
(6)
$\frac{M}{2} = TS + {\Omega _H}J - VP.$
(7)
Here, V is the thermodynamic volume conjugate to P. In the present work, we focus on a slowly rotating Kerr-AdS black hole. Thus, after neglecting all higher order terms of J, the equation of state can be written as [4, 39]
Equation (8) suggests that some P–v critical behaviors really exist. The “real” phase diagram of the slowly rotating Kerr-AdS black hole has been investigated in the extended phase space via Maxwell’s equal area law [39]. A second order transition exists, i.e., between large and small black hole phase transition, which is analogous to the gas and the fluid phase transition in the Van der Waals system. This phase transition occurs at the following critical point
In Fig. 1, we show the P?v diagram corresponding to the Kerr-AdS black hole. We can clearly see that a small-large black hole phase transition exists in the case of $T < {T_c}$. An example of this transition is specifically presented in the right panel of Fig. 1. Notice that the P?v curve contains two stable regions and one unstable region. Two stable regions are $v \in [0,\alpha ]$ and $v \in [\beta ,\infty ]$, which correspond to the small black hole region and the large black hole region, respectively. The unstable region $v \in [\alpha ,\beta ]$ is referred to as the spinodal region. In the unstable region, the small and large black hole phases can coexist. Furthermore, we note that $\dfrac{{\partial P(v,{T_0})}}{{\partial v}} < 0$ in two stable regions but $\dfrac{{\partial P(v,{T_0})}}{{\partial v}} > 0$ in the unstable region. Mathematically, the two extreme points $\alpha $ and β are determined by $\dfrac{{\partial P(v,{T_0})}}{{\partial v}}{|_{v = \alpha }} = $$ \dfrac{{\partial P(v,{T_0})}}{{\partial v}}{|_{v = \beta }} = $$ 0$, and the inflection point at $v = {v_0}$ is determined by $\dfrac{{{\partial ^2}P(v,{T_0})}}{{\partial {v^2}}} = 0$. Figure1. (color online) P?v diagram of the Kerr-AdS black hole for different temperatures. The angular momentum parameter is chosen as $J = 1$. In the right panel with the case of $T < {T_{\rm c}}$, it is obvious that the curve is divided into three regions: two stable regions (black lines) and one region (blue line). The red dotted line represents the coexisting line of the small black hole (with specific volume $v_m^ - $) and the large black hole (with specific volume $v_m^ + $) with transition pressure ${P_0}$, which satisfies Maxwell’s equal area law.
Some works have suggested that the Melnikov method is well suited for studying chaotic behavior of the black hole within the extended thermodynamic phase space [34, 36-38]. Hence, we shall investigate the chaos phenomenon of the Kerr-AdS black hole under periodic thermal perturbations with the help of the Melnikov method.
III.TEMPORAL CHAOS IN A SPINODAL REGIONHere, the effect of a weak temporally periodic perturbation shall be considered, when the Kerr-AdS black hole is quenched by the spinodal region. According to the standard procedure, we need to make use of the Kerr-AdS black hole equation of state to construct the Hamiltonian for the fluid flow and derive the Melnikov function containing information about the occurrence of temporal chaos. With respect to the temporal perturbation, we first consider a specific volume ${v_0}$ (the inflection point) in the spinodal region, which corresponds to an isotherm ${T_0}$(${T_0} < {T_{\rm c}}$). The weak time-periodic fluctuation of the absolute temperature near ${T_0}$ can be written as [42]
with $\varepsilon < < 1$. According to Ref. [34], one can suppose that the black hole flow takes place along the x axis in a finite tube with a unit cross section, which includes a total mass of ${{2\pi } / q}$ of the black hole in a volume of $\left( {{{2\pi } / q}} \right){v_0}$. Here, q is a positive constant. From Ref. [34], ${x_0}$ can be denoted as the Eulerian coordinate of a reference system. Thus, the mass M of a column of the Kerr-AdS black hole with the unit cross section between the reference ${x_0}$ and a general Eulerian coordinate x can be expressed as
$M = \int_{{x_0}}^x {\rho (\xi ,t){\rm d}\xi } ,$
(12)
where $\rho (x,t)$ denotes the black hole density at the position x and the time t. It should be pointed out that $\rho {(x(M,t),t)^{ - 1}} = {x_M}(M,t) = v$, where v represents the specific volume. Now, let us consider the thermodynamic phase transition displayed in the one dimensional thermal compressible Kerr-AdS black hole flow, which can be depicted in Lagrangian coordinates via the following system:
Here, $\varepsilon $ and ${\mu _0}$ are positive constants. For convenience, "~" has been omitted in Eq. (16). Then, the mass-volume constraint can be written as
$\int_0^{2\pi } {v(M,t){\rm d}M = 2\pi {v_0}} .$
(17)
It is logical to construct the Hamiltonian of the system, which has the following form
Notice that $\bar P(\zeta ,T) = P(\zeta ,T){{{\rm d}V} / {{\rm d}\zeta }}$ is an effective equation of state obtained by replacing $\zeta $ in terms of the thermodynamic volume $V = {{\pi {\zeta ^3}} / 6}$ before performing the integral. Considering that $v = {v_0}$ and is the infection point in the spinodal region, we have ${P_{vv}}({v_0},{T_0}) = 0$, ${P_v}({v_0},{T_0}) > $$ 0$, and ${P_{vvv}}({v_0},{T_0}) < 0$. At the equilibrium point $({v_0},{T_0})$ with $v = {v_0}$ and $u = 0$, v and u can be expanded in Fourier cosine and sine series on $[0,2\pi ]$, respectively. By using Eq. (17), we can obtain that
For the sake of carrying out the perturbation analysis, we need to expand $\bar P(v,T)$ near the equilibrium point $({v_0},{T_0})$ in a Taylor series and keep terms to third order. Thus, one has [36]
Considering that ${\bar P_{TT}}({v_0},{T_0})$, ${\bar P_{vTT}}({v_0},{T_0})$, and ${\bar P_{TTT}}({v_0},{T_0})$ are to be zero, they have been ignored in the above expression. By substituting Eqs. (19)-(21) into Eq. (18), the Hamiltonian can be rewritten as
where $({x_1},{x_2})$ and $({u_1},{u_2})$ stand for the positions and velocities of the first two modes. Consequently, it is not difficult to drive the corresponding equations of motion, which are as follows:
and $\varepsilon $($ > 0$) is small enough, then the first mode is unstable with ${\lambda _1} > 0$ and ${\lambda _2} < 0$ but the second and higher modes are stable. Next, let us analyze the unperturbed ($\varepsilon = 0$) system $\dot z(t) = {g_0}(z)$ existing in a two-dimensional invariant symplectic manifold, which contains a homoclinic orbit connecting the origin to itself. The corresponding analytical form can be written as [35]
In Fig. 2, we show the phase portrait corresponding to the homoclinic orbit of the unperturbed system, which agrees with the analytical expression (30). From this figure, we can see that the homoclinic orbit possesses two branches, which are the two wings of the butterfly-like orbit, respectively. According to Eq. (30), it is easy to deduce that ${z_0}$ tends to a saddle point (i.e., the origin point) as $t \to \pm \infty $. Figure2. (color online) Homoclinic orbit of the unperturbed system with $T = {\rm{0}}{\rm{.02894}} < {T_{\rm c}}$. The parameters are set to $J = 1$, $q = 1$, and $A = 0.01$. These arrows stand for the time flow direction.
After introducing the time-periodic perturbation (10) (i.e., $\varepsilon \ne 0$), the above homoclinic orbit may be destroyed, which means that the chaotic behavior of the system becomes possible. In light of the Melnikov method, the Melnikov function for the present perturbed system can be computed by using the following formula [42, 43]
Hence, it is logical to conclude that the sufficiently small temporal perturbation may cause the occurrence of the chaos behavior due to the temporal thermal fluctuation. What is more, Eq. (38) can be translated into a critical value for the perturbation parameter $\gamma $, which reads
We note that the small temporal perturbation with $\gamma > {\gamma _{\rm c}}$ guarantees the transversal intersection between unstable and stable manifolds, which may give rise to the emergence of the Smale horseshoe chaotic motion [40]. In order to check this analytical condition on the chaotic threshold, numerical results for different values of $\gamma $ are shown in Fig. 3. For convenience, both ${x_2}$ and ${u_2}$ have been fixed at zero. Fig. 3(a) displays normal trajectories of the system in the presence of the small temporal perturbation (for $\gamma < {\gamma _{\rm c}}$). Figure 3(b) shows the occurrence of chaotic motion for $\gamma > {\gamma _c}$. Figure3. (color online) Temporal evolution in the phase space of velocity vs displacement for the perturbed system in the specific temperature $T = {\rm{0}}{\rm{.02894}} < {T_{\rm c}}$: (a) $\gamma = 0.0188 < {\gamma _{\rm c}}$: (b) $\gamma = 5 > {\gamma _{\rm c}}$. Parameters are set to $J = 1$, $q = 1$, $A = 0.01$, $\varepsilon = 0.001$, $\omega = 0.01$, and ${\mu _0} = 0.1$. The initial conditions are chosen as a fixed point ($\sqrt {{{{\beta ^2}{v_0}^7}/ {(180{J^2})}}} ,0$) in the right branch of the homoclinic orbit.
From Eq. (39), one can conclude that the critical value ${\gamma _c}$ depends on the value of the angular momentum J. In Fig. 4, one can clearly see that ${\gamma _{\rm c}}$ decreases as the value of the angular momentum J increases. This means that a larger J value makes the occurrence of the chaos behavior easier under the time-periodic thermal perturbation. Figure4. (color online) Dependence of the critical value ${\gamma _{\rm c}}$ fon the angular momentum J for the Kerr-AdS black hole. Other parameters are set as in Fig. 2.
IV.SPATIAL CHAOS IN THE EQUILIBRIUM STATEIn this section, let us consider the effect of the small spatially periodic perturbation in the equilibrium configuration with an absolute temperature (${T_0} < {T_{\rm c}}$) of the form [42]
$T = {T_0} + \varepsilon \cos (qx).$
(40)
On the basis of the Korteweig theory, the stress tensor can be written as
$\tau = - p(v,T) - Av'',$
(41)
where $p(v,T)$ satisfies the Kerr-AdS black hole equation of state in Eq. (8), $A > 0$ is a constant, and the symbol $^\prime $ stands for $\dfrac{\rm d}{{{\rm d}x}}$. It should be pointed out that, for a static equilibrium without body forces, one can obtain $\dfrac{{{\rm d}\tau }}{{{\rm d}x}} = 0$ so as to have $\tau = {\rm con}st = - B$. Furthermore, B is the ambient pressure as $\left| x \right| \to \infty $. Based on this fact, Eq. (41) can be changed into the following form
$v'' = B - p(v,T).$
(42)
Here, the constant A has been set to $A = 1$ as in Refs. [34, 36]. First, we discuss the unperturbed system ($T = {T_0}$). For arbitrary temperature $T = {T_0} < {T_c}$, the nonlinear systems in Eq. (42) exist at three fixed points, ${v_1}$, ${v_2}$, and ${v_3}$, respectively. In light of the magnitude of the ambient pressure B, Eq. (42) can generate three different types of portraits in the $v - v'$ phase plane: Case 1. The ambient pressure B lies in the range ${P_0} < B < P(\beta ,{T_0})$. ${P_0}$ is the phase transition pressure. Then, the values ${v_1}$, ${v_2}$, and ${v_3}$ , where $P({v_1},{T_0}) = $$ P({v_2},{T_0}) = P({v_3},{T_0}) = B$ , are displayed in Fig. 5(a); meanwhile the portrait of Eq. (42) in the $v - v'$ phase plane is displayed in Fig. 5(b). Figure5. (color online) Case 1: (a) B and ${v_1}$, ${v_2}$, ${v_3}$ in $P - v$ diagram; (b) $v - v'$ phase portrait. One can see a homoclinic orbit (read dash dot line) connecting ${v_3}$ to itself.
Case 2. The ambient pressure B lies in the range $P(\alpha ,{T_0}) < B < {P_0}$. Then, the values ${v_1}$, ${v_2}$, and ${v_3}$ , where $P({v_1},{T_0}) = P({v_2},{T_0}) = P({v_3},{T_0}) = B$ , are presented in Fig. 6(a), meanwhile, the corresponding $v - v'$ phase plane is presented in Fig. 6(b). Figure6. (color online) Case 2: (a) B and ${v_1}$, ${v_2}$, ${v_3}$ in $P - v$ diagram; (b) $v - v'$ phase portrait. One can see a homoclinic orbit (blue dash dot line) connecting ${v_1}$ to itself.
Case 3. The ambient pressure B is equal to the phase transition pressure ${P_0}$, i.e., $B = {P_0}$. In this case, the values ${v_1}$, ${v_2}$, and ${v_3}$ , where $P({v_1},{T_0}) = P({v_2},{T_0}) = P({v_3},{T_0}) = $$ {P_0}$ , are exhibited in Fig. 7(a); meanwhile the corresponding $v - v'$ phase plane is exhibited in Fig. 7(b). Figure7. (color online) Case 3: (a) B and ${v_1}$, ${v_2}$, ${v_3}$ in $P - v$ diagram; (b) $v - v'$ phase portrait. One can see a heteroclinic orbit (read dash dot line) connecting ${v_1}$ to ${v_3}$.
From these figures, one can conclude that the present unperturbed system exists with a homoclinic orbit for both case 1 and case 2 while it has a heteroclinic orbit for the final case. It is obvious that the above characteristics allow us to calculate the Melnikov function for these orbits. After adding a small spatial perturbation, expressed in Eq. (40), we can rewrite the Eq. (43) for a perturbed system as follows:
$v'' = B - p(v,{T_0}) - \frac{{\varepsilon \cos (qx)}}{v}.$
(43)
Then, the Melnikov function can be expressed as [42]
From Eq. (50), it can be deducted that $M({x_0})$ always contains simple zeros for any given value of L and K, which signals that spatial chaos can occur in the thermodynamic system of the Kerr-AdS black hole under spatially periodic thermal perturbation. This result is in agreement with other AdS black holes [34, 36-38]. In Figs. 8-10, the numerical solutions of the perturbed dynamical Eq. (46) for these three cases are plotted in the $v - v'$ plane, respectively. The corresponding initial configurations are chosen to be the homoclinic orbit or heteroclinic orbit. These figures clearly indicate that spatial chaos under spatially periodic thermal perturbation indeed exists. Figure8. (color online) Portrait of the perturbed equation in $v - v'$ phase plane for case 1.
Figure9. (color online) Portrait of the perturbed equation in $v - v'$ phase plane for case 2.
Figure10. (color online) Portrait of the perturbed equation in $v - v'$ phase plane for case 3.