Research Article | Open Access
M. Sharif, Rabia Saleem, "Cardy-Verlinde Formula and Its Self-Gravitational Corrections for Regular Black Holes", Advances in High Energy Physics, vol. 2014, Article ID 926589, 7 pages, 2014. https://doi.org/10.1155/2014/926589
Cardy-Verlinde Formula and Its Self-Gravitational Corrections for Regular Black Holes
We check the consistency of the entropy of Bardeen and Ayón Beato-García-Bronnikov black holes with the entropy of particular conformal field theory via Cardy-Verlinde formula. We also compute the first-order semiclassical corrections of this formula due to self-gravitational effects by modifying pure extensive and Casimir energy in the context of Keski-Vakkuri, Kraus and Wilczek analysis. It is concluded that the correction term remains positive for both black holes, which leads to the violation of the holographic bound.
The idea of massive bodies from which nothing can escape due to strong gravity was firstly proposed in 1783 by Michell  and named as “dark stars.” In 1967, Wheeler  used the term “black hole” for such objects. Black hole (BH) is the most important prediction of general relativity which is detected by its effect on the nearby matter. A BH can be defined as a spacetime singularity which is covered by an event horizon. Hawking  found that BHs also radiate particles from their boundary with finite thermal spectrum dubbed after his name as Hawking radiations. Several efforts have been made to visualize this spectrum by examining the quantum effects on a scalar particle . Taking quantum effects into account, BH evaporation converts pure quantum states into mixed states known as “information loss paradox” . Much work has been done to resolve this paradox using fixed geometrical background during emission process.
In 1995, Kraus and Wilczek [6, 7] put forward an analysis in which dynamical BH background was used by treating the Hawking radiations as a tunneling process. They found that Hawking radiations are modified and their spectrum becomes nonthermal after considering self-gravitational interaction. In this procedure, total mass (Arnowitt-Deser-Misner (ADM)) remains constant, while the mass of the Schwarzschild BH decreases due to emitted radiations. A specific Painlevé coordinate transformation  is used in order to avoid the existence of singularity at the horizon. Initially, this analysis was implemented on spherically symmetric geometry (Schwarzschild type BHs ) to evaluate their corrected temperature and entropy. An extra nonthermal term appears in thermodynamical quantities due to this modification. The modified BH temperature also depends upon the emitted shell energy along with BH characteristics, while entropy is different from the Bekenstein-Hawking entropy for the corresponding Schwarzschild BH.
Vagenas  applied Keski-Vakkuri, Kraus and Wilczek (KKW) analysis on non-Schwarzschild type BH using more generalized coordinate transformation. This transformation allows studying the across horizon BH physics whose generalization leads to exact expressions of thermodynamical quantities. The temperature and entropy of non-Schwarzschild type BH are not any more the Hawking temperature and Bekenstein-Hawking entropy, respectively. Parikh and Wilczek  were the first to utilize this technique to four-dimensional Schwarzschild BH. Later, this analysis was used for several BHs such as -dimensional anti-de Sitter (AdS) BH, -dimensional AdS BH, two-dimensional charged and uncharged dilatonic BH, -dimensional charged BTZ BH, spinning BTZ BH, and magnetic stringy BH [11–17].
Holographic principle proposed by Susskind  is one of the fundamental principles of quantum gravity. It states that Bekenstein-Hawking entropy should be larger than the entropy associated with volume , where . Generally, radiations can be described by using interacting conformal field theory (CFT) and the AdS/CFT correspondence is the most common example of the holography. Verlinde  proposed a universal formula (Cardy-Verlinde formula (CV)) which relates the entropy of a certain CFT to its total energy and Casimir energy valid for all dimensions. The generalization of this formula is the major outcome of AdS/CFT correspondence. This formula remains valid for topological dS, Schwarzschild-dS, Reissner-Nordström-dS, Kerr-dS, Kerr-Newman-dS BHs, and so forth [20–26].
The issue of quantum corrections in entropy attracted many researchers [27–29]. Carlip  deduced the leading order quantum correction to the classical CV formula. Setare and Vagenas [31, 32] proved that this formula holds for Achúcarro-Ortiz BH and calculated the self-gravitational corrections in in the framework of KKW analysis. Setare and Jamil  showed that the entropy of the charged rotating BTZ BH can be expressed in terms of . Darabi et al.  found the self-gravitational corrections in this formula for the charged BTZ BH. Recently, Abbas  showed that the entropy of noncommutative Schwarzschild BH can be expressed in terms of CV formula.
In this paper, we study entropy of the Bardeen and Ayón Beato-García-Bronnikov (ABGB) BHs in terms of CV formula and also find the first-order leading self-gravitational corrections. The paper is organized as follows. In Section 2, we calculate quantities like mass, potential, temperature, and entropy of both BHs. We prove that the entropies of these BHs coincide with that of CFT via CV formula. Section 3 is devoted to the computation of the self-gravitational corrections in the CFT entropy of these BHs using KKW analysis. Finally, we summarize the results in the last section.
2. Entropy of Regular Black Holes via Cardy-Verlinde Formula
In this section, we evaluate the entropy of the Bardeen and ABGB BHs through the Cardy-Verlinde formula.
2.1. Entropy of Bardeen Black Hole
The four-dimensional spherically symmetric BH is represented by the following line element: where For different choices of , this reduces to some well-known BHs. Bardeen  proposed a model obeying weak energy condition that can be interpreted as the solution of magnetic monopole. The mass function has the following specific choice for Bardeen BH: with mass and charge . For zero charges, this mass function leads to the Schwarzschild BH. The lapse function provides a unique root corresponding to the event horizon (outer horizon). Substituting this value of , the event horizon takes the form leading to the value of as The corresponding electric potential and the Hawking temperature can be calculated as The corresponding area and entropy are given as The generalized Cardy-Verlinde formula is defined as  where and are arbitrary positive constants and is the radius of the sphere. The Casimir energy is generated by quantum fluctuations of CFT at finite volume while it disappears when volume becomes infinite. Casimir effects are usually significant at zero temperatures but can also be discussed at its finite values of the temperature. Thus, an extensive (in thermodynamical system, the energy is called extensive when it satisfies the relation ) part is added in the total energy expressed as where is the pure extensive part of energy and the factor is used for convenience .
The violation of Euler identity is given as which provides -dimensional Casimir energy in the following form: where pressure is defined as . The product of energy and radius (ER) is independent of volume due to conformal invariance which is satisfied by both and . Hence, both energies can be expressed in terms of and in arbitrary dimensions as follows : Taking and using the above expressions of in (11), the Casimir energy takes the form Using mass energy relation , this reduces to The extensive energy is found through (9) and (13) as Using (7), the generalized formulas of Casimir and extensive energy in one dimension can be written as Comparing the Casimir as well as extensive energy given in (14)–(16), we obtain two different spherical radii as Multiplication of the above two equations provides Substituting (14), (15), and (18) in CV formula, the final result is This shows that the entropy of Bardeen BH can be expressed in terms of CV formula in the context of CFT.
2.2. Entropy of Ayón Beato-García-Bronnikov Black Hole
Ayón Beato and García  and Bronnikov  proposed a nonsingular BH by solving the system of equations coupled with nonlinear electrodynamics and gravity. For this BH, the mass function is given as which also reduces to the Schwarzschild BH for . The event horizon has the following form: The corresponding electric potential, Hawking temperature, and entropy turn out to be Using (11), the Casimir energy can be written as For , this reduces to The extensive energy is given as Comparison of (24) and (25) with (16) leads to yielding the unique radius Inserting (24)–(27) in (8), we find that the entropy of CFT is equivalent to the entropy of ABGB BH as
3. Self-Gravitational Corrections to the Cardy-Verlinde Formula
In this section, we evaluate the self-gravitational corrections in the CV formula for both BHs using KKW analysis. This leads to modifying all the quantities in the above section under self-gravitational effects.
3.1. Self-Gravitational Corrections for Bardeen Black Hole
Here, we evaluate the self-gravitational corrections in CV formula for Bardeen BH. The modified Casimir energy is defined as where and stand for modified temperature and entropy, respectively. The third term of (11) is modified to , whereas the total energy, charge, radius of the sphere, and electric potential of BH remain invariant under this effect. In order to evaluate the corrections in BH temperature and entropy, we use the KKW analysis. In this analysis, the total mass is kept fixed, while BH mass is assumed to fluctuate because a shell of energy radiates massless particles and hence BH mass reduces to . This shell energy does not contain charge due to which electric potential remains the same; that is, . Applying this analysis, we obtain a relationship between BH entropy and Bekenstein-Hawking entropy as The corresponding self-gravitational corrected temperature is given by which reduces to the Hawking temperature by considering corrections up to first-order in .
Thus, the self-gravitational corrections in take the form Substituting this value in (29), the modified Casimir energy turns out to be where the second term is the correction term. The modification in extensive part of the energy can be obtained by adding the term in the above expression as The extensive energy (16) can also be modified in the context of KKW analysis as follows: Inserting (18), (33), and (34) in modified CV formula, that is, it follows that where . This gives the corrections in CV formula due to the effects of self-gravitation.
The total energy is the combination of extensive and Casimir energies, so , which implies that the term is positive. As the correction term is positive, so the modified term is greater in magnitude than the original one. The self-gravitational correction of CV formula for Bardeen BH remains positive as the term inside square root is real and positive. The above term can be expanded up to the first-order in as follows: which relates the modified self-gravitational energy with the original energy. It is seen that the self-gravitational modified temperature and entropy derived in the context of KKW analysis of Bardeen BH are different from the Bekenstein-Hawking temperature and entropy, respectively. The BH entropy described by CV formula violates the holographic bound as [39, 40].
3.2. Self-Gravitational Corrections for Ayón Beato-García-Bronnikov Black Hole
Now, we consider ABGB BH to calculate the self-gravitational corrections in the CV formula using KKW analysis. The corresponding self-gravitational corrections in are Equation (29) gives the modified Casimir energy as where the second term is the correction term. Additionally, the modified extensive energy can be written as Inserting (27), (40), and (41) in modified CV formula, we obtain the self-gravitational corrected CV formula for ABGB BH as follows: As the correction term , this implies that . The corresponding first-order expansion of the correction term is This shows that the ABGB BH entropy described by CV formula also violates the holographic bound.
4. Concluding Remarks
The description of BH entropy by CV formula and its quantum corrections have attained much attention. There have been a number of papers to find the semiclassical corrections in this formula for different BHs using various techniques such as self-gravitational effects , generalized uncertainty principle , space noncommutativity , and quantum corrections . The self-gravitational correction allows one to study across horizon physics and provides the solution of information loss paradox.
This paper is devoted to study this analysis for the Bardeen and ABGB BHs which correspond to the Schwarzschild BH for zero charges. Firstly, we have evaluated the thermodynamical quantities such as electric potential, Hawking temperature, and Bekenstein-Hawking entropy of both of the BHs. It is found that there exists unique event horizon whose entropy can be expressed in the form of CV formula indicating that this formula holds on both event horizons.
Secondly, we have used the self-gravitational effect to evaluate the temperature and entropy of these BHs in the context of KKW analysis. The corrections in the CV formula due to self-gravitational effects are evaluated by taking dynamical background of the BH. The corrections are restricted up to the linear order in ; for the zeroth order, the modified temperature corresponds to the Hawking temperature. We have also found the corrections in all the quantities which can be modified under self-gravitational effect, that is, pure extensive energy and the Casimir energy. The electric potential, charge, spherical radius, and total energy remain invariant under this effect. It turns out that the self-gravitational correction term in modified CV formula is positive for both BHs as the modified term is greater than the original. It is interesting to mention here that the positive self-gravitational corrections for regular BHs do not satisfy the inequality which proves that the holographic bound is not universal.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the Higher Education Commission, Islamabad, Pakistan, for its financial support through the Indigenous Ph.D. Fellowship for 5K Scholars, Phase-II, Batch-I.
- J. Michell, “On the means of discovering the distance, magnitude, &c. of the fixed stars, in consequence of the diminution of the velocity of their light, in case such a diminution should be found to take place in any of them, and such other data should be procured from observations, as would be farther necessary for that purpose,” Philosophical Transactions of the Royal Society, vol. 74, pp. 35–57, 1784.
- J. A. Wheeler, “Our universe: the known and the unknown,” The American Scientist, vol. 56, p. 1, 1968.
- S. W. Hawking, “Black hole explosions?” Nature, vol. 248, no. 5443, pp. 30–31, 1974.
- J. B. Hartle and S. W. Hawking, “Path-integral derivation of black-hole radiance,” Physical Review D, vol. 13, no. 8, pp. 2188–2203, 1976.
- Q. Q. Jiang and X. Cai, “Remarks on self-interaction correction to black hole radiation,” Journal of High Energy Physics, vol. 2003, no. 11, article 110, 2009.
- P. Kraus and F. Wilczek, “Self-interaction correction to black hole radiance,” Nuclear Physics B, vol. 433, no. 2, pp. 403–420, 1995.
- P. Kraus and F. Wilczek, “Effect of self-interaction on charged black hole radiance,” Nuclear Physics B, vol. 437, no. 1, pp. 231–242, 1995.
- P. Painleve, “La mécanique classique et la théorie de la relativité,” Comptes Rendus de l'Académie des Sciences, vol. 173, pp. 677–680, 1921.
- E. C. Vagenas, “Generalization of the KKW analysis for black hole radiation,” Physics Letters B, vol. 559, no. 1-2, pp. 65–73, 2003.
- M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” Physical Review Letters, vol. 85, no. 24, pp. 5042–5045, 2000.
- S. Hemming and E. Keski-Vakkuri, “Hawking radiation from AdS black holes,” Physical Review D, vol. 64, no. 4, Article ID 044006, 8 pages, 2001.
- Y. Kwon, “Hawking radiation in AdS2 black hole,” Il Nuovo Cimento della Società Italiana di Fisica B. Serie 12, vol. 115, no. 4, pp. 469–471, 2000.
- E. C. Vagenas, “Are extremal 2D black holes really frozen?” Physics Letters B, vol. 503, no. 3-4, pp. 399–403, 2001.
- E. C. Vagenas, “Two-dimensional dilatonic black holes and Hawking radiation,” Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, vol. 17, no. 10, pp. 609–618, 2002.
- E. C. Vagenas, “Semiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitation,” Physics Letters B, vol. 533, no. 3-4, pp. 302–306, 2002.
- A. J. M. Medved, “Radiation via tunnelling in the charged BTZ black hole,” Classical and Quantum Gravity, vol. 19, no. 3, pp. 589–598, 2002.
- A. J. M. Medved, “Radiation via tunneling from a de Sitter cosmological horizon,” Physical Review D, vol. 66, no. 12, Article ID 124009, 7 pages, 2002.
- L. Susskind, “The world as a hologram,” Journal of Mathematical Physics, vol. 36, no. 11, pp. 6377–6396, 1995.
- E. Verlinde, “On the holographic principle in a radiation dominated universe,” http://arxiv.org/abs/hepth/0008140.
- D. Birmingham and S. Mokhtari, “The Cardy-Verlinde formula and Taub-bolt-AdS spacetimes,” Physics Letters B, vol. 508, no. 3-4, pp. 365–368, 2001.
- D. Klemm, A. C. Petkou, and G. Siopsis, “Entropy bounds, monotonicity properties and scaling in CFTs,” Nuclear Physics B, vol. 601, no. 1-2, pp. 380–394, 2001.
- B. Wang, E. Abdalla, and R.-K. Su, “Relating Friedmann equation to Cardy formula in universes with cosmological constant,” Physics Letters B, vol. 503, no. 3-4, pp. 394–398, 2001.
- M. R. Setare, “The Cardy-Verlinde formula and entropy of topological Reissner-Nordström black holes in de Sitter spaces,” Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, vol. 17, no. 32, pp. 2089–2094, 2002.
- M. R. Setare and M. B. Altaie, “The Cardy-Verlinde formula and entropy of topological Kerr-Newman black holes in de Sitter spaces,” The European Physical Journal C. Particles and Fields, vol. 30, no. 2, pp. 273–277, 2003.
- M. R. Setare and R. Mansouri, “Holographic thermodynamics on the brane in topological Reissner-Nordström de Sitter space,” International Journal of Modern Physics A, vol. 18, no. 24, pp. 4443–4450, 2003.
- C. O. Lee, “Cardy-Verlinde formula in Taub-NUT/bolt-(A)dS space,” Physics Letters B, vol. 670, no. 2, pp. 146–149, 2008.
- A. J. M. Medved, “Quantum-corrected Cardy entropy for generic -dimensional gravity,” Classical and Quantum Gravity, vol. 19, no. 9, pp. 2503–2513, 2002.
- S. Mukherji and S. S. Pal, “Logarithmic corrections to black hole entropy and AdS/CFT correspondence,” Journal of High Energy Physics, vol. 2002, no. 5, article 026, 2002.
- J. E. Lidsey, S. Nojiri, S. D. Odintsov, and S. Ogushi, “The AdS/CFT correspondence and logarithmic corrections to braneworld cosmology and the Cardy-Verlinde formula,” Physics Letters B, vol. 544, no. 3-4, pp. 337–345, 2002.
- S. Carlip, “Logarithmic corrections to black hole entropy, from the Cardy formula,” Classical and Quantum Gravity, vol. 17, no. 20, pp. 4175–4186, 2000.
- M. R. Setare and E. C. Vagenas, “Cardy-Verlinde formula and Achúcarro-Ortiz black hole,” Physical Review D, vol. 68, no. 6, Article ID 064014, 5 pages, 2003.
- M. R. Setare and E. C. Vagenas, “Self-gravitational corrections to the Cardy-Verlinde formula of the Achúcarro-Ortiz black hole,” Physics Letters B, vol. 584, no. 1-2, pp. 127–132, 2004.
- M. R. Setare and M. Jamil, “The Cardy-Verlinde formula and entropy of the charged rotating BTZ black hole,” Physics Letters B, vol. 681, no. 5, pp. 469–471, 2009.
- F. Darabi, M. Jamil, and M. R. Setare, “Self-gravitational corrections to the Cardy-Verlinde formula of charged BTZ black hole,” Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, vol. 26, no. 14, pp. 1047–1057, 2011.
- G. Abbas, “Cardy-Verlinde formula of noncommutative Schwarzschild black hole,” Advances in High Energy Physics, vol. 2014, Article ID 306256, 4 pages, 2014.
- J. Bardeen, “Non-singular general relativistic gravitational collapse,” in Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity, Tbilisi Unversty Press, Tbilisi, Georgia, September 1968.
- E. Ayón-Beato and A. García, “New regular black hole solution from nonlinear electrodynamics,” Physics Letters B, vol. 464, no. 1-2, pp. 25–29, 1999.
- K. A. Bronnikov, “Comment on ‘regular black hole in general relativity coupled to nonlinear electrodynamics’,” Physical Review Letters, vol. 85, p. 4641, 2000.
- R. Bousso, “A covariant entropy conjecture,” Journal of High Energy Physics, vol. 1999, no. 7, article 004, 1999.
- G. 't Hooft, “Dimensional reduction in quantum gravity,” http://arxiv.org/abs/gr-qc/9310026.
- M. R. Setare and E. C. Vagenas, “Self-gravitational corrections to the Cardy-Verlinde formula and the FRW brane cosmology in SdS5 bulk,” International Journal of Modern Physics A, vol. 20, no. 30, pp. 7219–7232, 2005.
- M. R. Setare, “The generalized uncertainty principle and corrections to the Cardy-Verlinde formula in SAdS5 black holes,” International Journal of Modern Physics A, vol. 21, no. 6, p. 1325, 2006.
- M. R. Setare, “Space noncommutativity corrections to the Cardy-Verlinde formula,” International Journal of Modern Physics A, vol. 21, no. 13-14, p. 3007, 2006.
- M. Sharif and W. Javed, “Quantum corrections for ABGB black hole,” Astrophysics and Space Science, vol. 337, no. 1, pp. 335–341, 2012.
Copyright © 2014 M. Sharif and Rabia Saleem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.