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In this section, we review the derivation of the shadow radius for the Hayward-AdS BH. The metric of a static spherically symmetric BH can be written as
$ \begin{array}{*{20}{l}} {\rm d} s^{2}=-f(r) {\rm d} t^{2}+h(r)^{-1} {\rm d} r^{2}+r^{2} ({\rm d} \theta^{2}+ \sin ^{2} \theta {\rm d} \phi^{2}), \end{array} $
(1) where
$ f(r) $ and$ h(r) $ represent the lapse functions associated with the radius parameter r. For a photon moving in such spacetime background of the metric (1), the corresponding Hamiltonian satisfies$ \begin{array}{*{20}{l}} 2\mathcal{H}= g^{\mu \nu} p_{\mu} p_{\nu}=0, \end{array} $
(2) where
$ p_\mu=\dfrac{\mathrm{d} x_\mu}{\mathrm{d} \lambda} $ is the proper four-momentum of the photon and λ denotes the affine parameter. For convenience, we assume that the photon is moving on the equatorial plane and therefore we can fix the coordinate$ \theta=\pi/2 $ and$ \dot{\theta}=0 $ . Then, with Eq. (2) and the metric (1), one can obtain$ -\frac{p_{t}^{2}}{2f(r)}+\frac{h(r) p_{r}^{2}}{2}+\frac{p_{\phi}^{2}}{2r^{2}}=0, $
(3) where
$ p_r $ is the radial momentum of the photon. For the geodesic motion in the context of such BH background, there are two conserved quantities corresponding to the Killing vector fields$ \dfrac{\partial}{\partial t} $ and$ \dfrac{\partial}{\partial \phi} $ , i.e.,$-E =p_{t}=\frac{\partial \mathcal{H}}{\partial \dot{t}}, \quad L =p_{\phi} = \frac{\partial \mathcal{H}}{\partial \dot{\phi}}, $
(4) where
$ E=-p_{t} $ represents the energy of the photon and$ L=p_{\phi} $ represents the angular momentum of the photon. Note that the dot indicates the derivative with respect to the affine parameter λ. Then, one can obtain the equation of motion of the photon from Eq. (3),$ \begin{eqnarray} \dot{t}=\frac{\partial \mathcal{H}}{\partial p_{t}}=-\frac{p_{t}}{f(r)}, \quad \dot{\phi}=\frac{\partial \mathcal{H}}{\partial p_{\phi}}=\frac{p_{\phi}}{r^{2}}, \quad \dot{r}=\frac{\partial \mathcal{H}}{\partial p_{r}}=p_{r} h(r). \end{eqnarray} $
(5) From Eq. (2), the effective potential of the photon can be defined as
$ \begin{array}{*{20}{l}} \dot{r}^{2} + V_{\rm eff}(r) = 0. \end{array} $
(6) With Eqs. (3) and (5), the effective potential
$V_{\rm eff}$ can be written as$ V_{\rm eff}(r) = h(r)\Bigg(\frac{L^2}{r^2}-\frac{E^{2}}{f(r)}\Bigg). $
(7) The circular orbit radius
$ r_p $ of the photon needs to satisfy the following conditions:$ V_{\rm eff}\left(r_p\right)=0,\left.\quad \frac{\partial V_{\rm eff}(r)}{\partial r}\right|_{r=r_p}=0. $
(8) The impact parameter
$ \mu_{p} $ of the photon can be deduced from Eqs. (7) and (8), which is given by$ \mu_{p} = \frac{L}{E}=\left.\frac{r}{\sqrt{f(r)}}\right|_{r=r_{p}}. $
(9) Using Eq. (5), the orbit equation of the photon can be expressed as
$ \frac{{\rm d} r}{{\rm d}\phi}=\frac{\dot{r}}{\dot{\phi}}=\frac{r^{2} h(r) p_{r}}{L}. $
(10) Combining with Eqs. (3) and (10), one can obtain
$ \frac{{\rm d} r}{{\rm d} \phi} = \pm r \sqrt{h(r)\left(\frac{r^{2}E^{2}}{f(r)L^{2}}-1\right)}. $
(11) For the turning point (
$ r=\chi $ ) of the photon orbit, one has the mathematical constraint$\dfrac{{\rm d} r}{{\rm d} \phi}{\big |}_{r=\chi}=0$ , which means that$ \dfrac{E^2}{L^2}=\dfrac{f(\chi)}{\chi^2} $ . Then, Eq. (11) can be rewritten as$ \frac{{\rm d} r}{{\rm d} \phi} = \pm r \sqrt{h(r)\left(\frac{r^{2}f(\chi)}{f(r)\chi^{2}}-1\right)}. $
(12) Now, we study the BH shadow observed by a static observer located at position
$ r_o $ . For a light ray emitting from the observer and transmitting into the past with an angular ψ with respect to the radial direction, we have [70–73],$ \cot \psi = \frac{\sqrt{g_{r r}}}{\sqrt{g_{\phi \phi}}} \cdot \frac{{\rm d} r}{{\rm d} \phi}{\Bigg |}_{r=r_{o}} = \left.\frac{1}{r \sqrt{h(r)}} \cdot \frac{{\rm d} r}{{\rm d} \phi}\right|_{r=r_o}. $
(13) Combining with Eqs. (12) and (13), we obtain
$ \cot ^2 \psi=\frac{r_o^2 f(\chi)}{f\left(r_o\right) \chi^2}-1 \quad {\text{or}} \quad \sin ^2 \psi=\frac{f\left(r_o\right) \chi^2}{r_o^2 f(\chi)}. $
(14) For the static observer located at position
$ r_o $ , one can set$ \chi = r_p $ to obtain the shadow radius$ r_{s} $ ,$ r_{s}= r_{o} \sin \psi = \chi\sqrt{\frac{f(r_{o})}{f(\chi)}}{\Bigg |}_{\chi = r_{p}}. $
(15) For the Hayward-AdS BH, the coefficients in the metric (1) are given as [81, 82]
$ f(r)=h(r)^{-1}=1-\frac{2 M r^{2}}{r^{3}+g^{3}}+\frac{8 \pi P r^{2}}{3}, $
(16) where M represents the BH mass and g represents the magnetic charge. The thermodynamic pressure P is associated with the cosmological constant, namely,
$P =-\dfrac{\Lambda}{8\pi} = \dfrac{3}{8 \pi l^{2}}$ . One can use Eqs. (8) and (16) to obtain the circular orbit radius of the photon, which is given as$ \begin{aligned}[b] r_{p}=&M+\frac{M^{2}}{\left(M^{3}-g^{3}+\sqrt{g^{6}-2 g^{3} M^{3}}\right)^{1 / 3}}\\&+\left(M^{3}-g^{3}+\sqrt{g^{6}-2 g^{3} M^{3}}\right)^{1 / 3}. \end{aligned} $
(17) Finally, with Eqs. (15) and (16), the expression for the shadow radius of the Hayward-AdS BH for a static observer located at position
$ r_o $ can be directly written as$ r_{s} = r_{p} \sqrt{\frac{f(r_{o})}{f(r_{ p})}}, $
(18) where
$ r_p $ is given by Eq. (17). -
For the Hayward-AdS BH, the mass M and temperature T are given by [82]
$ \begin{aligned}[b] M=&\frac{g^{3}}{2 r_{h}^{2}}+\frac{r_{h}}{2}+\frac{4}{3} P \pi\left(r_{h}^{3}+g^{3}\right), \\\ T=&\frac{r_{h}^{3}-2 g^{3}}{4 \pi r_{h}\left(g^{3}+r_{h}^{3}\right)}+\frac{2 P r_{h}^{4}}{g^{3}+r_{h}^{3}}, \end{aligned} $
(19) where
$ r_{h} $ is the horizon radius of the Hayward-AdS BH. As the expression of$ r_{h} $ is extremely verbose, we do not display the exact form here. The equation of state of the Hayward-AdS BH is given as$ P=\frac{g^{3}}{4 \pi r_{h}^{5}}-\frac{1}{8 \pi r_{h}^{2}}+\frac{g^{3} T}{2 r_{h}^{4}}+\frac{T}{2 r_{h}}. $
(20) To obtain the critical point of the thermodynamic PT for the Hayward-AdS BH, one can take Eq. (20) into the critical condition
$ ({\partial P}/{\partial r_{ h}})=0=({{\partial}^{2}P}/{\partial r_{ h}^{2}}) $ , which yields$ P_{ c} =\frac{3(57-23 \sqrt{6})(14+6 \sqrt{6})^{1 / 3}}{800 g^{2} \pi} \simeq 0.002418 g^{-2}, $
(21) $ r_{ c}=(14+6 \sqrt{6})^{1 / 3} g \simeq 3.061577 g, $
(22) $ T_{ c}=\frac{(5-2 \sqrt{6})(7+3 \sqrt{6})^{2 / 3}}{4 \times 2^{1 / 3} g \pi}\simeq 0.037675 g^{-1}. $
(23) Substituting Eq. (19) into Eq. (17), one can rewrite the circular orbit radius of the photon as
$ \begin{aligned}[b] r_{ p}=&\frac{g^3}{2 r_{ h}^2}+\frac{r_{ h}}{2}+\frac{4}{3} \left(g^3+r_{ h}^3\right)P \pi\\&+\frac{\left(3+8 P \pi r_{ h}^2\right)^2\left(g^3+r_{ h}^3\right)^2}{36 r_{ h}^4 \xi^{1 / 3}}+\xi^{1 / 3}, \end{aligned} $
(24) where
$ \begin{aligned}[b] \xi=&\frac{\left(3+8 P \pi r_{ h}^2\right)^3\left(g^3+r_{ h}^3\right)^3}{216 r_{ h}^6}\\&+\sqrt{g^6-\frac{g^3\left(3+8 P \pi r_{ h}^2\right)^3\left(g^3+r_{ h}^3\right)^3}{108 r_{ h}^6}}-g^3. \end{aligned} $
(25) Combining with Eqs. (18) and (24), we plot the
$ r_{ s}-r_{ h} $ diagrams of the Hayward-AdS BH in Fig. 1, where we have set$ f(r_{o}=100)=1 $ for the static observer [70]. For different values of magnetic charge g and pressure P, the shadow radius$ r_{ s} $ invariably increases monotonically with the horizon radius$ r_{h} $ . This result implies that the horizon radius$ r_{h} $ can be replaced with the shadow radius$ r_{ s} $ to study the thermodynamics of the Hayward-AdS BH.Figure 1. Shadow radius
$ r_{ s} $ versus horizon radius$ r_{ h} $ . The solid, dashed, and dotted lines correspond to$ P=0.6P_{c} $ ,$ P=P_{c} $ , and$ P=1.4P_{c} $ , respectively.With Eq. (19), we plot the
$ T-r_{ h} $ diagrams with three types of pressures based on the critical pressure$ P_c $ (see Figs. 2(a), 2(b), and 2(c) with the magnetic charges$ g= $ 0.3, 0.6, 0.9). It is found that when the pressure is less than the critical pressure, such as$ P=0.6P_{ c} $ , the corresponding temperature curve is not monotonous and there exists a vdW-like PT. The slopes of the small radius region ($ r_{h}<r_{h1} $ ) and the large radius region ($ r_{h}>r_{h2} $ ) are both positive, which correspond to the small and large BH phases, respectively. The intermediate region ($ r_{h1}<r_{h}<r_{h2} $ ) represents a coexistence region of the small and large BH. The intermediate region is thermodynamically unstable and the corresponding temperature$ T_{co} $ is known as coexistence temperature. For the pressure equal to the critical pressure ($ P=P_{ c} $ ), there will be an unstable PT, whereas for the pressure larger than the critical pressure ($ P=1.4P_{ c} $ ), PT will not appear.Figure 2. (color online) Temperature T versus
$ r_h $ (2(a), 2(b), 2(c)) and$ r_{s} $ (2(d), 2(e), 2(f)) for the magnetic charges$ g = 0.3,\; 0.6,\; 0.9 $ .$ T_{co} $ (black lines) is the coexistence temperature. The solid, dashed, and dotted lines correspond to$ P=0.6P_{c} $ ,$ P=P_{c} $ , and$ P=1.4P_{c} $ , respectively.We can also study the thermodynamic PT of the Hayward-AdS BH by replacing the horizon radius
$ r_{h} $ with the shadow radius$ r_{s} $ . In Figs. 2(d), 2(e), and 2(f), we plot the$ T-r_{s} $ diagrams with the parameters adopted in the$ T-r_{h} $ diagrams. To observe the difference between the two types of diagrams, we also consider three sets of values of the magnetic charge g and three sets of values of the pressure P. The results show that the PT process denoted by the shadow radius$ r_s $ is similar to the PT process denoted by the horizon radius$ r_h $ . The critical point of the PT process is still dependent on the pressure P. For$ P<P_c $ , there exists a vdW-like PT. The small ($ r_{ s}<r_{ s1} $ ) and large ($ r_{ s}>r_{ s2} $ ) radius regions correspond to the stable small and large BH, respectively. The radius region$ r_{ s1}< r_{s}<r_{ s2} $ corresponds to the unstable intermediate BH. For$ P=P_c $ , there will be an unstable PT. When$ P>P_c $ , there is no PT. Moreover, from the entire Fig. 2, it can be observed that for the pressure$ P=0.6P_c $ , as the magnetic charge g increases, the shadow radius$ r_{s} $ becomes larger while the coexistence temperature$ T_{co} $ becomes lower.To further analyze the thermodynamic stability of the Hayward-AdS BH, we investigate its heat capacity. The positive (negative) heat capacity corresponds to the thermodynamically stable (unstable) BH phase. The heat capacity
$ C_P $ of the Hayward-AdS BH with a fixed pressure P and magnetic charge g is given by$\begin{aligned}[b] C_{ P} =& T\Bigg(\frac{{\rm d}S}{{\rm d}T}\Bigg)_{ P,g} \\ = & \frac{2 \pi r_{h}^{2}\left(g^{3}+r_{h}^{3}\right)\left(-2 g^{3}+r_{h}^{3}+8 P \pi r_{h}^{5}\right)}{2 g^{6}+r_{h}^{6}\left(-1+8 P \pi r_{h}^{2}\right)+2 g^{3} r_{h}^{3}\left(5+16 P \pi r_{h}^{2}\right)}.\end{aligned} $
(26) The
$ C_{P}-r_{h} $ diagrams of the Hayward-AdS BH for different magnetic charge g and pressure P values are shown in Figs. 3(a), 3(b), and 3(c). For the pressure$ P=0.6P_c $ ,$ \left(\partial_{r_{h}} T\right)_P=0 $ has two roots$ r_{h1} $ and$ r_{h2} $ as shown in Fig. 2. These two roots indicate two divergences of the heat capacity$ C_P $ . The sign of$ C_P $ is changed at these two divergences. When the pressure is increased to$ P=P_c $ , these two roots coincide with each other, so there is only one point of divergence in the heat capacity$ C_P $ . For the pressure$ P=1.4P_c $ , it can be observed that the heat capacity$ C_P $ is a positive and continuous function of the horizon radius$ r_{h} $ .Combining with Eqs. (18) and (26), we can also study the heat capacity of the Hayward-AdS BH by replacing the horizon radius
$ r_{h} $ with the shadow radius$ r_{s} $ . The$ C_{P}-r_{s} $ diagrams of the Hayward-AdS BH are shown in Figs. 3(d), 3(e), and 3(f). When the pressure is less than the critical pressure, the heat capacity$ C_P $ also diverges at two points. When the pressure is equal to the critical pressure, there is only one divergent behavior in the heat capacity$ C_P $ . For the pressure larger than the critical pressure, the heat capacity$ C_P $ is also a positive and continuous function of the shadow radius$ r_{s} $ .To summarize, the behaviors of the heat capacity
$ C_P $ in the$ C_{P}-r_{s} $ diagrams are similar to those in the$ C_{P}-r_{h} $ diagrams. In addition, Fig. 2 shows that for$ P<P_c $ , both the small ($ r_{ s}<r_{ s1} $ ) and large ($ r_{ s}>r_{ s2} $ ) BH branches have positive slopes (i.e., positive heat capacities), which means that these two branches are thermodynamically stable. The slope of the intermediate BH branch is negative, which represents a negative heat capacity, and thus it is thermodynamically unstable. These results are consistent with the phenomena depicted in Fig. 3. -
To show the phase structure of the Hayward-AdS BH more intuitively, we calculate the boundary curve of the shadow of the Hayward-AdS BH. For such a spherically symmetric BH, the shape of the BH shadow is circular for an arbitrary observer [83]. Therefore, one can use a thermal profile in a two-dimensional plane to study the relation between the thermodynamic PT and shadow of the Hayward-AdS BH. In this case, for the stereographic projection of the Hayward-AdS BH located in the celestial coordinate onto the two-dimensional plane, the boundary curve of the shadow is given as follows [20],
$ x=\lim \limits_{r \rightarrow \infty}\left(-r^2 \sin \theta_0 \frac{{\rm d} \phi}{{\rm d} r}\right)_{\theta_0 \rightarrow \frac{\pi}{2}}, \quad y=\lim \limits_{r \rightarrow \infty}\left(r^2 \frac{{\rm d} \theta}{{\rm d} r}\right)_{\theta_0 \rightarrow \frac{\pi}{2}}, $
(27) where x and y are the Cartesian coordinates with the inclination angle
$ \theta_0 \rightarrow \dfrac{\pi}{2} $ .The shadow contours of the Hayward-AdS BH for different parameters are shown in Fig. 4, where we have set
$ M=60 $ and$ f(r_{o}=100)=1 $ for the static observer. The results show that the shadow radius consistently increases with the pressure. The solid lines denote$ P<P_{c} $ , which correspond to the largest shadow radius region. As we clarified earlier, in this case, there exits a vdW-like PT between the small and large BH. The dashed lines represent$ P=P_{c} $ , which is a critical point because there is an unstable PT. The dotted lines correspond to$ P>P_{c} $ and the shadow represents a supercritical phase, for which no PT occurs. Moreover, for the same pressure, comparing the radii of the shadow contours with different magnetic charges, one can find that the shadow radius of the Hayward-AdS BH increases with the magnetic charge g.Figure 4. Boundary curves of the shadow of the Hayward-AdS BH for the magnetic charges
$ g = 0.3,\; 0.6,\; 0.9 $ . The solid, dashed, and dotted lines correspond to$ P=0.6P_{c} $ ,$ P=P_{c} $ , and$ P=1.4P_{c} $ , respectively. We have set$ M=60 $ and$ r_{o}=100 $ .To analyze further, from multiple aspects, the relation between the phase structure and the shadow of the Hayward-AdS BH, one can study the superposition of the temperature diagram (see Fig. 2) and the shadow cast diagram (see Fig. 4), which is called the thermal profile of the Hayward-AdS BH (see Fig. 5). To do that, we need to fix the pressure of the Hayward-AdS BH and then the shadow contour can be considered as a function of temperature for a static observer located at position
$ r_o = 100 $ . Figure 5 illustrates the thermal profiles of the Hayward-AdS BH for three sets of magnetic charges and three sets of pressures. Figures 5(a), 5(d), and 5(g) represent$ P<P_{c} $ , which correspond to the solid lines in Fig. 4. It can be found that when the shadow radius is smaller than a certain value, the temperature increases from the center to the boundary of the shadow. Then, the temperature oscillates as the shadow radius increases, i.e., the N-type change trend [74]. Figures 5(b), 5(e), and 5(h) denote$ P=P_{c} $ , which correspond to the dashed lines in Fig. 4. It exits a critical region of thermodynamic instability, where the temperature remains constant. Figures 5(c), 5(f), and 5(i) represent$ P>P_{c} $ , which correspond to the dotted lines in Fig. 4. One can find that the temperature keeps increasing from the center to the boundary of the shadow, which is consistent with the result depicted by the dotted lines in Fig. 2. In addition, comparing these diagrams, one can find that when the magnetic charge g is fixed, the temperature increases with the pressure while the region of the thermal profile decreases with the pressure.Figure 5. (color online) Thermal profiles of the Hayward-AdS BH for the magnetic charges
$ g = 0.3,\; 0.6,\; 0.9 $ and pressures$ P=0.6P_c $ ,$ P_c $ ,$ 1.4P_c $ . We have set$ M=60 $ and$ r_{o}=100 $ .Finally, for the Hayward-AdS BH, we study the region where the N-type change trend occurs. For the case of
$ P<P_{c} $ , the thermal profiles of the Hayward-AdS BH with different magnetic charges from$ r_{s1} $ to$ r_{s2} $ are shown in Fig. 6. It can be found that the temperature increases first, then decreases, and finally increases again from the center to the boundary of the shadow, which is consistent with the result depicted by the solid lines in Fig. 2. The results imply that when$ P<P_{c} $ , the thermal profile can be used to investigate the phase structure of the Hayward-AdS BH.
Shadow thermodynamics of the Hayward-AdS black hole
- Received Date: 2023-02-08
- Available Online: 2023-06-15
Abstract: In this paper, the phase structure of the Hayward-anti-de Sitter (AdS) black hole (BH) is studied using shadow formalism. It has been found that the shadow radius is a monotonic function of the horizon radius and can therefore play an equivalent role to the horizon radius in characterizing the thermodynamics of the Hayward-AdS BH. The thermodynamic phase transition (PT) of the Hayward-AdS BH is investigated with the shadow radius. It is shown that as the magnetic charge increases, the shadow radius becomes larger, while the coexistence temperature becomes lower. The thermal profile of the Hayward-AdS BH is established by combining the temperature diagram and the shadow cast diagram, which shows that for a fixed magnetic charge, the temperature of the Hayward-AdS BH increases with the pressure whereas the region of the thermal profile decreases with the pressure. In particular, the temperature of the Hayward-AdS BH follows an N-type change trend when it is smaller than the critical temperature. It implies that the BH shadow may be used to investigate the thermodynamics of the Hayward-AdS BH.