Can shadows connect black hole microstructures?
CCan shadows connect black hole microstructures?
Xin-Chang Cai * and Yan-Gang Miao † School of Physics, Nankai University, Tianjin 300071, China
Abstract
We investigate the relationship between shadow radii (photosphere radii) and black hole microstruc-tures for a static spherically symmetric black hole and confirm their close connection. As a concreteanalysis, we take the Reissner-Nordstr¨om AdS black hole surrounded by perfect fluid dark matter (RNAdS black hole surrounded by PFDM) as an example. We calculate the Ruppeiner thermodynamic scalarcurvature and shadow radius (photosphere radius) for the specific model. On the one hand, we find thatthe greater density of the perfect fluid dark matter makes black hole molecules more likely have attractiveinteractions. On the other hand, we provide the support to our general investigation that the shadow radii(photosphere radii) indeed connect the microstructures of this model. * E-mail address: [email protected] † Corresponding author. E-mail address: [email protected] a r X i v : . [ g r- q c ] J a n Introduction
Recently, the Event Horizon Telescope (EHT) collaboration has released [1, 2] for the first time theshadow image of the supermassive black hole in the center of M87 ∗ galaxy, which greatly stimulated ourenthusiasm for the research of black hole shadows. A shadow is an observable quantity closely related toits corresponding black hole. Its essence is that the specific photons around a black hole collapse inwardto form a dark area seen by a distant observer, which also means that the black hole shadow has a closeconnection to the dynamics of black holes. Through black hole shadows, one not only obtains [3–7] themass, spin, hair, and other information of black holes, but also knows [8–11] the distribution of matteraround black holes. More importantly, the research in this direction opens a new window for us to studythe strong gravitational region near black hole horizons. So far, there have been a large number of articleson black hole shadows under Einstein gravity and modified gravity theories, see, for instance, someliterature [12–31].As is known, the black hole thermodynamics [32–34] is an important research field of black holephysics, especially when combined with AdS spacetimes, the black hole thermodynamics in extendedphase spaces has achieved great progress, e.g. , the van der Waals-like phase transition occurs [35] inthe Reissner-Nordstr¨om AdS black hole, the reentrant phase transition happens [36] in high-dimensionalrotating black holes, and so on. Black hole is not only a strong gravitational system, but also a thermo-dynamic one, so it is necessary to explore the relationship between its dynamics and thermodynamics.Recently, the relationships between dynamic characteristic quantities and thermodynamic ones have beenstudied [37–46], such as photosphere radius, Hawking temperature, specific heat, and the minimum im-pact parameter used to describe the unstable circular orbital motion of photons around black holes, et al .The results show [37,38] that the unstable circular orbital motion can connect thermodynamic phase tran-sitions of black holes. Especially for a static spherically symmetric black hole, the timelike or lightlikecircular orbital motion of a test particle and its shadow indeed have a close connection [40, 45, 46] to itsthermodynamic phase transition.The Ruppeiner thermodynamic geometry [47–50] is a useful tool for black hole thermodynamics. Itprovides helpful information about phase transitions of black holes, and also sheds [51–61] some lighton microstructures of black holes. Its most important physical quantity is the Ruppeiner thermodynamicscalar curvature. It has been shown [48, 50, 56, 59–61] that this scalar curvature can take a positive, or anegative, or a vanishing value which corresponds to a repulsive, or an attractive, or no interaction amongblack hole molecules, respectively. As is known, thermodynamic phase transitions are not detectabledirectly, and neither are black hole microstructures. Therefore, it is necessary to associate a shadowradius with the Ruppeiner thermodynamic scalar curvature, so as to connect black hole microstructureswith an observable – the shadow radius. Furthermore, our investigations may provide some helps forunderstanding black hole microstructures by observing black hole shadows.The paper is organized as follows. In Sec. 2, we investigate the relationship between shadow (pho-tosphere) radii and black hole microstructures for a static spherically symmetric black hole. Then, wecalculate in Sec. 3 the thermodynamic scalar curvature for the RN AdS black hole surrounded by PFDM.In Sec. 4, we study the relationship between the shadow radius and the Ruppeiner thermodynamic scalarcurvature for the specific black hole system. Finally, we make a simple summary in Sec. 5. We use theunits c = G = k B = ¯ h = ( − , + , + , +) throughout this paper.2 Shadow and black hole microstructure of static spherically sym-metric black holes
A static spherically symmetric black hole can be described by the line element, ds = − f ( r ) dt + dr g ( r ) + r (cid:0) d θ + sin θ d ϕ (cid:1) , (1)where f ( r ) and g ( r ) are functions of the radial coordinate r . We use the Hamilton-Jacobi equation toseparate the null geodesic equations in static spherically symmetric black hole spacetimes and derive thegeneral expression of shadows of this class of black holes. The Hamilton-Jacobi equation takes [62] theform, ∂ S ∂ λ + H = , H = g µν p µ p ν , (2)where λ is the affine parameter of null geodesics, S the Jacobi action, H the Hamiltonian, and themomentum p µ is defined by p µ ≡ ∂ S ∂ x µ = g µν d x ν d λ . (3)Due to the spherical symmetry of Eq. (1), without loss of generality, one can consider the photonsmoving on the equatorial plane with θ = π . So the the Jacobi action S can be decomposed into thefollowing form, S = m λ − Et + L ϕ + S r ( r ) , (4)where m is the mass of moving particles around black holes, for photons m = E the energy of photons, L the angular momentum of photons, and S r ( r ) a function of coordinate r . By substituting Eqs. (1), (3)and (4) into Eq. (2), one can get the following three equations describing the motion of photons on theequatorial plane, d t d λ = Ef ( r ) , (5)d r d λ = ± (cid:112) g ( r ) (cid:112) f ( r ) (cid:112) E r − L r f ( r ) r , (6)d ϕ d λ = Lr . (7)Combining Eq. (6) with Eq. (7), one hasd r d ϕ = d r d λ d ϕ d λ = ± r (cid:115) g ( r ) (cid:18) r E L f ( r ) − (cid:19) . (8)Again considering d r d ϕ (cid:12)(cid:12)(cid:12) r = ξ = r d ϕ = ± r (cid:115) g ( r ) (cid:18) r f ( ξ ) ξ f ( r ) − (cid:19) . (9)3n addition, according to the equation satisfied by the effective potential V ( r ) , (cid:18) d r d λ (cid:19) + V ( r ) = , (10)one obtains V ( r ) = g ( r ) (cid:18) L r − E f ( r ) (cid:19) . (11)Using the effective potential, one can determine [45] the critical orbit radius of photons, i.e. the photo-sphere radius for a static spherically symmetric spacetime, V ( r ) = , d V ( r ) d r = , d V ( r ) dr < . (12)By substituting Eq. (11) into Eq. (12), one can obtain the following relationships, L E = r f ( r ps ) , (13) r d f ( r ) d r − f ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r = r ps = , (14)where r ps is the radius of photospheres.Considering that the light emitted from a static observer at position r transmits into the past with anangle ϑ relative to the radial direction, one obtains [21, 22, 45]cot ϑ = √ g rr √ g ϕϕ d r d ϕ (cid:12)(cid:12)(cid:12)(cid:12) r = r = ± (cid:115) r f ( ξ ) ξ f ( r ) − . (15)When the relation sin ϑ = + cot ϑ is used, the above equation can be written assin ϑ = ξ f ( r ) r f ( ξ ) . (16)Therefore, the shadow radius of black holes observed by a static observer at position r can be ex-pressed [11, 21, 22, 45] by r sh ≡ r sin ϑ = ξ (cid:115) f ( r ) f ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ → r ps . (17)In the Ruppeiner thermodynamic geometry, the most important physical quantity to describe the mi-crostructures of black holes is the Ruppeiner thermodynamic scalar curvature R [47, 48]. If the entropyrepresentation is chosen [63–65], we obtain ∂ R ∂ S = ∂ R ∂ r ps d r ps d r H d r H d S , ∂ R ∂ S = ∂ R ∂ r sh d r sh d r H d r H d S . (18)On the one hand, we know [40, 45] the relations,d r ps d r H > , d r sh d r H > . (19)4n the other hand, for a static spherically symmetric black hole, its Bekenstein-Hawking entropy, S = A ,where A is the surface area, equals S = π r , (20)which means d S d r H > d r H d S >
0. Consequently, the positivity or negativity of ∂ R ∂ S depends entirely on thepositivity or negativity of ∂ R ∂ r ps or ∂ R ∂ r sh . In other words, the conditions ∂ R ∂ S > , ∂ R ∂ S = , ∂ R ∂ S < , (21)correspond to ∂ R ∂ r ps > , ∂ R ∂ r ps = , ∂ R ∂ r ps < , (22)or equivalently to ∂ R ∂ r sh > , ∂ R ∂ r sh = , ∂ R ∂ r sh < . (23)We thus establish the relationship between the shadow (photosphere) radius and the microstructures of astatic spherically symmetric black hole. Next, we take the RN AdS black hole surrounded by PFDM as anexample to compute its Ruppeiner thermodynamic scalar curvature and analyze the connection betweenthe scalar curvature and the shadow (photosphere) radius. The line element of the RN AdS black hole surrounded by PFDM is given [66–68] by ds = − f ( r ) dt + dr f ( r ) + r (cid:0) d θ + sin θ d ϕ (cid:1) , (24)with f ( r ) = − Mr + Q r − Λ r + α r ln (cid:18) r | α | (cid:19) . (25)Here, M is the black hole mass, Q its charge, Λ the negative cosmological constant, and α the intensityparameter related to the perfect fluid dark matter. For the case of α =
0, the above metric turns back tothat of the RN AdS black hole, which means no perfect fluid dark matter outside the RN AdS black hole.In the extended phase space including the cosmological constant, the mass M regarded as the blackhole enthalpy H can be expressed [68] by M = H = r H + Q r H + π P r + α (cid:18) r H | α | (cid:19) = √ S √ π + √ π Q √ S + PS / √ π + α (cid:32) √ S √ π | α | (cid:33) , (26)5nd the Hawking temperature by T = π r H − Q π r + Pr H + α π r = √ π S − √ π Q S / + √ SP √ π + α S , (27)where f ( r H ) = P = − Λ π , and Eq. (20) have been considered. In addition, the thermodynamic volume V conjugated to the pressure P , the electric potential Φ conjugated to the charge Q , and the extensivequantity Ψ conjugated to the intensity parameter α are defined respectively by V ≡ ∂ H ∂ P = π r , (28) Φ ≡ ∂ H ∂ Q = Qr H , (29) Ψ ≡ ∂ H ∂ α = (cid:20) ln (cid:18) r H | α | (cid:19) − (cid:21) . (30)Correspondingly, for the RN AdS black hole surrounded by PFDM, the first law of thermodynamics andthe Smarr relation take the forms in terms of the thermodynamic variables defined above, respectively, dM = dH = T dS + Φ dQ + V dP + Ψ d α , (31)and M = H = T S + Φ Q − V P + Ψ α . (32)Now we follow the Ruppeiner thermodynamic geometry to study the RN AdS black hole surroundedby PFDM. Here we use the enthalpy and Helmholtz free energy representations, that is, we calculate theRuppeiner thermodynamic scalar curvature R in ( S , P ) and ( T , V ) planes, respectively. The Ruppeinerline element of this black hole spacetime in the ( S , P ) plane is [63–65] as follows: ds R = C P dS + T (cid:18) ∂ T ∂ P (cid:19) S dSdP , (33)where C P ≡ T (cid:16) ∂ S ∂ T (cid:17) P and the relation, (cid:16) ∂ V ∂ P (cid:17) S =
0, has been used because the thermodynamic volume V is independent of the presure P under a fixed entropy S . Similarly, the Ruppeiner line element in the ( T , V ) plane is given [63–65] by ds R = T (cid:18) ∂ P ∂ T (cid:19) V dT dV + T (cid:18) ∂ P ∂ V (cid:19) T dV , (34)where C V ≡ T (cid:16) ∂ S ∂ T (cid:17) V = ( S , P ) and ( T , V ) planes, respectively, R SP = − ( S − π Q ) + √ π S α S ( PS + S − π Q + √ π S α ) , (35)6igure 1. Graph of the Ruppeiner thermodynamic scalar cur-vature R with respect to the entropy S for different values ofparameter α at a constant pressure P and a constant charge Q .Here we choose P = .
002 and Q = R V T = − √ π V / − √ π Q + (cid:112) ( π ) α V / √ π V / T . (36)We can easily check that these two scalar curvatures are equal to each other, that is, R SP = R V T . Itshould be noted that the Ruppeiner thermodynamic scalar curvature diverges when the black hole goesto its extreme configuration. In addition, for the case of α =
0, Eqs. (35) and (36) will return to thethermodynamic curvature of RN AdS black holes [64].In Fig. 1, based on Eq. (35) we depict the relation of the Ruppeiner thermodynamic scalar curvature R with respect to the entropy S for different values of parameter α at a constant pressure P and a constantcharge Q . In Fig. 2, based on Eq. (36) we depict the relation of the Ruppeiner thermodynamic scalarcurvature R with respect to the thermodynamic volume V for different values of parameter α at a constantHawking temperature T and a constant charge Q . We notice that the profile of R versus S is similar to thatof R versus V . This is not strange because S is proportional to the square of r H and V to the cubic of r H ,see Eqs. (20) and (28). From the two figures, we find that, on the one hand, the black hole molecules ofthe RN AdS black hole surrounded by PFDM may have a repulsive interaction ( R > R < R = α makes the curves move downward, which indicatesthat the greater density of the perfect fluid dark matter makes the black hole molecules more likely haveattractive interactions. 7igure 2. Graph of the Ruppeiner thermodynamic scalar curva-ture R with respect to the thermodynamic volume V for differ-ent values of parameter α at a constant Hawking temperature T and a constant charge Q . Here we choose T = .
04 and Q = By substituting Eq. (25) into Eqs. (14) and (17), we derive the shadow radius of the RN AdS blackhole surrounded by PFDM observed by a static observer at position r , r sh = r ps (cid:115) f ( r ) f ( r ps ) , (37)where the photosphere radius r ps satisfies the following equation,4 Q r ps + r ps + α ln (cid:18) r ps | α | (cid:19) − M − α = . (38)For the case of α = i.e. the RN AdS black hole, we obtain the analytic expression of the photosphereradius in terms of M and Q , ˜ r ps = (cid:16) M + (cid:112) M − Q (cid:17) , (39)and that of the shadow radius in terms of M , Q , and r ,˜ r sh = (cid:16)(cid:112) M − Q + M (cid:17) (cid:113) − Mr + π Pr + Q + r r (cid:114) π P (cid:16)(cid:112) M − Q + M (cid:17) + (cid:16)(cid:112) M − Q + M (cid:17) − M (cid:16)(cid:112) M − Q + M (cid:17) + Q . (40)By substituting Eq. (26) into Eqs. (37) and (38), we cannot derive the analytic formulas of the photo-sphere radius and the shadow radius but can express the two radii as functions of r H , Q , P , and α , r ps = r ps ( r H , Q , P , α ) , (41)8nd r sh = r sh ( r H , Q , P , α , r ) . (42)For the case of α =
0, we can solve from Eqs. (41) and (42) the analytic expressions of the photosphereradius and the shadow radius for the RN AdS black hole,ˆ r ps = (cid:113)(cid:0) π Pr + r + Q (cid:1) − Q r + π Pr + r + Q r H , (43)andˆ r sh = (cid:16)(cid:112) B + Q + √ B (cid:17) (cid:113) − r (cid:112) B + Q + π Pr + Q + r √ r (cid:114) π B P + π BP (cid:16)(cid:112) B ( B + Q ) + Q (cid:17) + (cid:16)(cid:112) B ( B + Q ) + Q (cid:17) ( π PQ + ) + B , (44)with B ≡ (cid:0) π Pr + r + Q (cid:1) − Q r r . (45)Substituting Eq. (20) into Eq. (35), we compute the Ruppeiner thermodynamic scalar curvature as afunction of r H , Q , P , and α , R = R ( r H , Q , P , α ) = Q − r − α r H π r (cid:0) π Pr + r − Q + α r H (cid:1) . (46)For the case of α =
0, we obtain the Ruppeiner thermodynamic scalar curvature for the RN AdS blackhole, ˆ R = Q − r π r (cid:0) π Pr + r − Q (cid:1) . (47)According to Eqs. (41), (42) and (46), we deduce that there is a correspondence between the Ruppeinerthermodynamic scalar curvature R and the photosphere radius r ps or between R and the shadow radius r sh for the RN AdS black hole surrounded by PFDM if charge Q , pressure P , and parameter α are fixed,which cannot be solved analytically but can be expressed as R = R ( r ps ) (cid:12)(cid:12) Q = const ., P = const ., α = const . , (48)or R = R ( r sh ) | Q = const ., P = const ., α = const . . (49)In Fig. 3 for the RN AdS black hole surrounded by PFDM, based on Eqs. (41) and (42) we depictthe relation of the photosphere radius r ps with respect to the horizon radius r H and also the relation of theshadow radius r sh with respect to the horizon radius r H for different values of parameter α at a constantpressure P and a constant charge Q . From this figure, we can see that the photosphere radius r ps and theshadow radius r sh increase monotonously with the increase of the horizon radius r H , which is completelyconsistent with Eq. (19) — a general property of a static spherically symmetric black hole.In Figs. 4 and 5, based on Eqs. (48) and (49) we turn to the relation of the Ruppeiner thermodynamicscalar curvature R with respect to the photosphere radius r ps and also the relation of R with respect to the9igure 3. Graphs of the photosphere radius r ps and the shadowradius r sh with respect to the horizon radius r H for differentvalues of parameter α at a constant pressure P and a constantcharge Q . Here we choose P = . Q =
1, and r = R with respect to the photosphere radius r ps for differentvalues of parameter α at a constant pressure P and a constantcharge Q . Here we choose P = .
002 and Q = r sh for different values of parameter α at a constant pressure P and a constant charge Q . Bycomparing Fig. 1 with Figs. 4 and 5, we can clearly see that they have similar profiles, which indicatesthat the photosphere radius r ps and the shadow radius r sh can connect the microstructures of black holes.Therefore, the shadow radius is indeed a good observable that has a close connection to the black holemicrostructures. In this paper, we investigate for a static spherically symmetric black hole the relationship betweenthe shadow radius (the photosphere radius) and black hole microstructures, and confirm their close con-nection. As a concrete analysis, we take the RN AdS black hole surrounded by PFDM as an example.We calculate the Ruppeiner thermodynamic scalar curvature for the specific model. On the one hand, wefind that the black hole molecules may have a repulsive interaction, or an attractive interaction, or evenno interaction, depending on the regions of the black hole’s intrinsic parameters. On the other hand, wediscover that the greater density of the the perfect fluid dark matter makes the black hole molecules morelikely become attractive interactions. In addition, we observe from Figs. 1, 4, and 5 that the portrait ofthe Ruppeiner thermodynamic scalar curvature R with respect to the shadow radius r sh (the photosphereradius r ps ) is similar to that of the thermodynamic curvature with respect to the entropy S (the thermo-dynamic volume V ). We thus establish the relationship between the shadow (photosphere) radius andblack hole microstructures for the RN AdS black hole surrounded by PFDM, providing the support to ourgeneral investigation for a static spherically symmetric black hole.An interesting issue is to explore whether the shadow (photosphere) radius of rotating AdS black holeshas a close connection to black hole microstructures, which will be considered in our future work.11igure 5. Graph of the Ruppeiner thermodynamic scalar curva-ture R with respect to the shadow radius r sh for different valuesof parameter α at a constant pressure P and a constant charge Q . Here we choose P = . Q =
1, and r = Acknowledgments
The authors would like to thank Z.-M. Xu and M. Zhang for helpful discussions. This work wassupported in part by the National Natural Science Foundation of China under Grant No. 11675081.
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