The Entropic Force in Reissoner-Nordström-De Sitter

Physics Letters B 797 (2019) 134798

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Physics Letters B

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The entropic force in Reissoner-Nordström-de Sitter spacetime ∗ Li-Chun Zhang a,b, Ren Zhao a,b, a Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China b Department of Physics, Shanxi Datong University, Datong 037009, China a r t i c l e i n f o a b s t r a c t

Article history: We compare the entropic force between the black hole horizon and the cosmological horizon in Received 2 March 2019 Reissoner-Nordström-de Sitter (RN-dS) spacetime with the Lennard-Jones force between two particles. Received in revised form 21 July 2019 It is found that the former has an extraordinary similarity to the latter. We conclude that the black hole Accepted 22 July 2019 horizon and the cosmological horizon are not pure geometric surface, but have a definite “thickness”, Available online 24 July 2019 which is proportional to the radius of the horizons. In the case of no external force, according to different Editor: M. Cveticˇ initial condition, the relative position of the two horizons will experience first accelerated expansion, then decelerated expansion, or first accelerated contraction, then decelerated contraction. Therefore the whole Universe is oscillating. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction nuity is of the first order; and the one with continuous entropy is of the second order. Exactly, for the first-order phase transition The accelerating Universe manifests that it is an asymptotically Gibbs free energy is continuous and its first derivative is not; for de Sitter one. It is a tough issue to study the thermodynamic prop- the second-order phase transition and its first derivative are both erties of black holes in dS space, because the cosmological horizon continuous while the second derivative is discontinuous, such as and the black hole horizon generally have different temperatures. heat capacity, isothermal compressibility and expansion coefficient. The cosmological constant may be a candidate of the dark energy, VdW system, which has no latent heat at the critical point, has the which arouses extensive attention on the thermodynamics of dS second-order phase transition there. Similarly, we found that the space [1–12]. phase transition of RN-dS black hole is of second order, just like Although multiple horizons exist for the black holes in dS space, its RN-AdS counterpart. the thermodynamic quantities of these horizons are all dependent Eight years ago, Verlinde [17]proposed to link gravity with on the black hole mass, electric charge, the cosmological constant. an entropic force. The ensuing conjecture was proved recently From this point of view, the thermodynamic quantities for the [18–20], in a purely classical environment. According to our pre- respective horizons are not independent, but are related to each vious results [15], we calculate the entropic force between the other. On the basis of this consideration, we can introduce effective two horizons, especially the relation between the entropic force thermodynamic quantities, by which we studied the phase transi- and the horizon’s position. Comparing the entropic force with the tion and critical phenomena of four and higher dimensional RN-dS Lennard-Jones force between two particles [21], we find that they black holes. We also derived the entropy of the RN-dS black holes have nearly the same behavior. According to the Lennard-Jones po- by considering the correlations between the black hole horizon and tential, when the distance between the two particles is the diam- the cosmological horizon [13–16]. Treating the cosmological con- eter of the particle, the potential is zero. Similarly, we guess that stant as a variable and relating it to the thermodynamic pressure, the black hole horizon and the cosmological horizon both have a we found that dS black holes have the similar critical behaviors finite thickness. As is well known, a cutoff should be introduced to to that of Van der Waals system. Ehrenfest had ever classified the obtain the black hole entropy [22–24]. However the exact mean- phase transitions: the one corresponding to the entropy disconti- ing of the cutoff is still unknown. As we stated in this paper, if the horizons have a finite thickness, the lower limit of the integral for the black hole entropy cannot take to be at the position of the Corresponding author at: Department of Physics, Shanxi Datong University, Da- * horizon. This can explain the cutoff issue. tong 037009, China. E-mail addresses: [email protected] (L.-C. Zhang), [email protected] The paper is arranged as follows: in Section 2, we first simply (R. Zhao). review the results in Ref. [15] and derive the entropic force be- https://doi.org/10.1016/j.physletb.2019.134798 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 L.-C. Zhang, R. Zhao / Physics Letters B 797 (2019) 134798 tween the two horizons. In Section 3, we compare the entropic force with the Lennard-Jones force between two particles. We derive the thickness of the horizons by analogy in Section 4. In the last section, we analyze the oscillation of the two horizon under the entropic force and obtain an oscillating Universe. (We use the units G = h¯ = kB = c = 1.)

2. Entropic force between the two horizons

The line element of the RN-dS black hole is given by

− ds2 =−h(r)dt2 + h(r) 1dr2 + r2d2, (1) where Fig. 1. f (x) − x curve. 2 2M Q 2 h(r) = 1 − + − r . (2) RNdS black hole is also a thermodynamic system. We can de- r r2 3 rive its entropic force. According to Eq. (3), the entropy from the For RN-dS black hole, the black hole horizon and the cosmo- interaction between the two horizons is logical horizon both depend on the same family of parameters: = 2 M, Q , . All other geometric and thermodynamic quantities can S πrc f (x). (8) be expressed by these parameters. So we think that the two hori- By analogy with Eq. (7), we can define the entropic force of the zons are not independent. There should be some correlations be- RN-dS black hole as tween them. The evolution of the black hole horizon will lead to ∂ S the evolution of the cosmological horizon and vice versa. On the F =−Tef f , (9) ∂r basis of this consideration, we proposed that the total entropy of Tef f the RN-dS black hole should include an correction term describing where Tef f is the effective temperature, and r = rc −r+ = rc(1 − x). the correlations between the two horizons besides their respective In our opinion, the entropic force between the two horizons origi- horizon entropy. Requiring the temperature of the lukewarm black nate from the interactions between them. Thus in the definition of hole equal to the effective temperature, we can derive the correc- the entropic force we are only concerned with the extra correction tion term. The result has been given in [15], which is entropy. In detail, the entropic force is1 ∂ S ∂ Tef f ∂ S ∂ Tef f = 2[ + 2 + ]= + + − Stotal πrc 1 x f (x) Sc, S, (3) ∂rc ∂x ∂x rc ∂rc x rc x F (x) = Tef f ∂ Tef f ∂ Tef f where rc and r+ are the radius of the cosmological horizon and (x − 1) − rc ∂x r ∂rc x the black hole horizon respectively, and we define x = r+/rc , thus c 2[ − 2 + 3 − 5]− 2[ − 3 + 4 − 7] 0 ≤ x ≤ 1. From Eq. (3), the total entropy includes two parts. = x 1 3x 3x x φ 1 3x 3x x = 2 + 2 3 + 4 Sc,+ πrc (1 x ) is the entropy without considering the correla- 4x (1 x ) tion between the two horizons. It is just the sum of the respective + + 4 × 2Af(x) x(1 x )Bf (x) entropy of the two horizons. In fact, the two horizons are not com- , (10) A(x − 1) + x(1 + x4)B = 2 pletely independent. This gives the correction term S πrc f (x), where where = 2[− − 2 + 3 − 4 − 5 + 6 − 7] 8 2(4 − 5x3 − x5) A x 1 3x 6x 5x 4x 9x 6x f (x) = (1 − x3)2/3 − . (4) 5 5(1 − x3) + φ2[3 + 4x4 − 8x7 + 9x8], 2 2 3 5 2 3 4 7 The behavior of f (x) is shown in Fig. 1. B = x [1 − 3x + 3x − x ]−φ [1 − 3x + 3x − x ]. The volume of the system is [25–27] For different values of φ, we depict the F − x curves in the range 0 ≤ x ≤ 1in Fig. 2 (see also Fig. 1). Clearly, the three curves 4π = − = 3 − 3 nearly coincide. The values of horizontal coordinate correspond- V V c V + rc (1 x ). (5) 3 ing to F = 0are x = 0.9748057, x = 0.9748059, and x = 0.9748064 The effective temperature is [15] with the increase of φ. Obviously, the values of φ has almost no effect on the F − x x2[1 − 3x2 + 3x3 − x5]−φ2[1 − 3x3 + 3x4 − x7] curves. In particular, F →+∞ when x → 1. T = , (6) ef f 3 4 4π x rc(1 + x ) 3. Comparison between the entropic force and the = where φ Q /rc is the electric potential at the cosmological hori- Lennard-Jones force zon. Entropic force in a thermodynamic system can be given as [17, Interactions between neutral molecules or atoms with a cen- 18,20,28–32] ter of mass separation r are often approximated by the so-called Lennard-Jones potential energy φ , given by [21,33] ∂ S L, J F =−T , (7) ∂r u 1 To derive this result, we have used the following formulae: ∂ = where T is the temperature of the system, r is the radius location ∂x y ∂(u,y) ∂(u,v) = ∂(u,v) ∂(x,y) of the boundary surface. ∂(x,y) and ∂(x,y) ∂(t,s) ∂(t,s) . L.-C. Zhang, R. Zhao / Physics Letters B 797 (2019) 134798 3

× 10 Fig. 4. The F (y) − y curves with 3 2 φmin = 1. r0

Fig. 2. The F (x) − x curves for different φ.

× 10 Fig. 5. The F (y) − y curves (solid) with 3 2 φmin = 275 and F (x) − x curve (dot r0 8 = Fig. 3. The φL, J (y) − y curves with 2 φmin = 1. dash line) with φ 0.001. 12 6 r 1 r0 r0 0 = F (y) = namely ∞ F (r)dr 0or 2 dy 0, because the potential φL, J = 4φmin − , (11) 0 y r r should be zero when no external force acts on the two-particle = where the first term is a short-range repulsive interaction and system. Thus y 1should be the starting point of the integral 1 F (y) dy = 0, but not the true diameter of a particle. From Fig. 4, the second term is a longer-range attractive interaction. A plot of 0 y2 →∞ φL, J /φmin versus r/r0 is shown in Fig. 3. The value r = r0 corre- F when y > 1, which indicated that infinite large force is sponds to φL, J = 0, and the minimum value of φL, J is φL, J /φmin = needed to compress the particle and the diameter of the particle −1at should be invariant without external force. A comparison of Fig. 2 and Fig. 4 indicates that the entropic 1/6 rmin/r0 = 2 ≈ 1.122. (12) force of the two horizons in de Sitter space is similar to the Lennard-Jones force of two particles. For further comparison, we By the definition of potential energy, the force between a molecule put them in a single figure with the same coordinates, Fig. 5. and a neighbor in the radial direction from the first molecule is Obviously, the two curves have nearly the same behavior. It =− Fr dφL, J /dr, which is positive (repulsive) for r < rmin and neg- should be emphasized that F (x) − x relation is derived theoreti- ative (attractive) for r > rmin. cally in the general relativity, while F (y) − y is a semi-empirical Let the center of the first particle at the origin, and set the result according to the experimental data. radius of the particle is r0/2. Assume that the center of the second = + particle sit at r and the boundary at r2. Then r r2 r0/2. We 4. The thickness of horizon take the change of the coordinate, y = r0/(2r2),0 < y ≤ 1. Eq. (11) − →+∞ → becomes From F (x) x, we know F when x 1, which means 12 6 r0 r0 that infinite large force is needed to make the black hole horizon φL, J (y) = 4φmin − r r and the cosmological horizon coincide. When there is no external force between the two horizons, energy is conserved. Thus the 12 6 6 6 y y entropic force between the two horizons is similar to the force be- = 4φmin2 2 − . (13) 1 + y 1 + y tween two particles. In analogy with the two-particle system, the potential in the interval (0, x0] between the two horizon should The force between the two particles is also be zero. The value of x is determined by the thickness of the 0 12 6 horizon. According to the entropic force, the potential in the inter- =−dφL, J = 6 r0 − r0 F (y) 4φmin 2 [ − ] dr r r r val x, 1 x should be 10 13 7 1−x 3φmin2 y y = 27 − . (14) F r 1 + y 1 + y A = dx. (15) 0 x2 x As is shown in Fig. 4, the force is positive at the interval 7/6 7/6 −10 [1/(2 − 1), 1] and is negative at the interval (0, 1/(2 − 1)). From Fig. 6, when x0 = 8.15 × 10 the potential between According to Eq. (11), the potential is zero at r = r0 and r →∞, the two horizons is zero in the interval (0, 1 − x0]. When x < 4 L.-C. Zhang, R. Zhao / Physics Letters B 797 (2019) 134798

Fig. 7. F (x¯) − x¯ curve (dotted) and F (y) − y curve (solid).

Fig. 6. A − x curves for different x.Hereφ = 0.001. The horizontal coordinates of the minimal values of the two ¯ curves are ymin = 0.6716 and xmin = 0.904, which are obviously in the range (0, 1]. When x¯ lies in the range (0, 0.904], there is an at- x0 the potential in the interval [0, 1 − x] is greater than zero tractive force between the two horizons under the entropic force; and when x > x0 the potential in the interval [0, 1 − x] is when x¯ lies in the range (0.904, 1], there is an repulsive force be- smaller than zero. Without external force the value of x0 de- tween the two horizons. When the ratio of the boundaries of the pends on the thickness of the horizon, similar to the diameter of two horizons is x¯ = 1, under the influence of the entropic force the the particle. The thickness of the black hole horizon and the cos- two horizons accelerate relatively until x¯ = 0.904, when the two mological horizon are assumed to be r+ = r+z and r = r z, c c horizons decelerate relatively. At last when x¯ = 0, the relative ve- respectively. Between the ratio of the two horizons and the ratio locity of the two horizons tends to zero. In this case the black hole of their boundaries has the following relation horizon and the cosmological horizon are at rest relatively. Once ¯ r+(1 + z) r+ certain perturbation makes x > 0, the attractive entropic force will x¯ = ≈ (1 + 2z) = x(1 + 2z), (16) accelerate the two horizons to shrink until x¯ = 0.904, after which r (1 − z) r c c the two horizons begin to shrink slowly. When x¯ = 1, the relative where z is an infinitesimal dimensionless quantity. When the velocity becomes zero. After this epoch, the two horizons begin the boundaries of the two horizon touches, namely x¯ = 1, we find accelerated expansion under the entropic force again. The dS black x = 1 − x0. Correspondingly, hole enters another cycle. Therefore, under the influence of the entropic force two horizons suffer periodic oscillation. Our Universe x0 2z = . (17) will evolve into a new de Sitter phase due to the entropic force 1 − x0 and perturbations. In the framework of GR, the theoretically derived entropic force Therefore, we can obtain the thickness of the black hole horizon x0 between the black hole horizon and the cosmological horizon is and the cosmological horizon are r+ = − r+ and rc = 2(1 x0) extraordinary similar to the Lennard-Jones force between two par- x0 rc , respectively. 2(1−x0) ticles. Our results reflect the intrinsic connections among general relativity, quantum mechanics and thermodynamics. Besides, this 5. Conclusions work provide a new route for studying the interaction the micro- scopic states in the interior of black hole. The extraordinary similarity between the entropic force between the two horizons and the Lennard-Jones force between the Acknowledgements two particles indicates that horizons are not pure surfaces but have certain thickness. The result derived from general relativity and This work is supported in part by the National Natural Science the result from experiments are consistent, which should not be Foundation of China (Grant No. 11475108). a coincidence. This thickness can be used to explain the cutoff in the calculation of black hole entropy. Because of the thickness, one References cannot calculate black hole entropy from the position of the black [1] Y. Sekiwa, Thermodynamics of de Sitter black holes: thermal cosmological con- hole horizon. In this way, the thickness of the black hole horizon stant, Phys. Rev. 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