Journal of Physics: Conference Series PAPER OPEN ACCESS On the deformed Einstein euations and uantum black holes To cite this article: E Dil et al 016 J. Phys.: Conf. Ser. 766 01004 View the article online for updates and enhancements. Related content - Excision techniue in constrained formulations of Einstein euations: collapse scenario I Cordero-Carrión, N Vasset, J Novak et al. - Two gravitational shock waves on the AdS brane Mohamed Anber and Lorenzo Sorbo - On the vacuum Einstein euations along curves with a discrete local rotation and reflection symmetry Mikoaj Korzyski, Ian Hinder and Eloisa Bentivegna This content was downloaded from IP address 148.51..8 on 0/01/019 at 09:1
International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) 01004 doi:10.1088/174-6596/766/1/01004 On the deformed Einstein euations and uantum black holes E Dil 1, E Kolay and C C Ersanli 1 1 Department of Physics, Sinop University, Korucuk, 57000-Sinop, Turkey Department of Statistics, Sinop University, Korucuk, 57000-Sinop, Turkey Abstract. Recently -deformed Einstein euations have been studied for extremal uantum black holes which have been proposed to obey deformed statistics by Strominger. In this study, we give the solutions of deformed Einstein euations by considering these euations for the charged black holes. Also we present the implications of the solutions, such as the deformation parameters lead the charged black holes to have a smaller mass than the classical Reissner- Nordström black holes. The reduction in mass of a classical black hole can be viewed as a transition from classical to uantum black hole regime. 1. Introduction Recently, we have studied the -deformed Einstein euations [1] for describing the gravitational fields of the Strominger s extremal uantum black holes which obey the deformed statistics []. Here, Verlinde s entropic gravity approach [] have been used to obtain the -deformed Eintein euations which are the gravitational field euations of charged uantum black holes with possible minimal mass. A charged black hole loses its mass via Hawking radiation and reaches a minimum mass proportional to the charge. The resulting black hole with a minimal mass is considered to be a micro black hole. Since these black holes can be considered as deformed bosons or deformed fermions, the micro black holes defined by -deformed Einstein euations are assumed be the Ubriaco s -deformed bosons [4]. We first give a brief summary of obtaining the -deformed Einstein euations then obtain the solutions of the deformed Einstein euations for charged black holes. Since the solutions of standard Einstein euations for charged black holes are the Reissner-Nordström solutions in classical gravity, the solutions of the deformed Einstein euations for charged black holes can be considered in uantum gravity. Lastly the implications of the solutions are represented. These are that the deformation parameters lead the charged black holes to have a smaller mass than the usual Reissner-Nordström black holes. This reduction in mass of a usual black hole can be considered as a transition from classical to uantum black hole regime.. -Deformed Einstein euations We obtain the -deformed Einstein euations which describe the gravitational fields uantum black holes by using the entropy of the Ubriaco s -deformed Bose gas models in Verlinde s entropic gravity approach. The uantum algebraic structure of this -deformed Bose gas model is given by the -deformed boson algebra [4] a a a a 1, (1) Content from this work may be used under the terms of the Creative Commons Attribution.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1
International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) 01004 doi:10.1088/174-6596/766/1/01004 a 1 a1 a1 a1 ( 1) a1 a1a 1 aa, () a, () a a1 a1 a. (4) Here a and a represents the deformed annihilation and creation operators, respectively. is also a real deformation parameter with 0. The grand partition function of the -deformed boson model is [4] Z k m0 ( m 1) e k { m } z m, (5) where 1/ kt and k is the Boltzmann constant, z e is the fugacity, k is the energy of the single-particle state, m is the occupation number of the single-particle state, and {m} is the deformed occupation number and given by m 1 { m}. (6) 1 The deformed entropy of the model is also given as S / 4V (m) 5 / 5 5 E z ( ) z z ln z ( ) z h T 4 ln z, (7) where E kt is the average energy of single particle, V is the volume enclosed by the deformed bosons, m is the mass of deformed bosons, T is the temperature of the model and ( ) (1/ 4).{[/(1 ) / ] (1/ )} [4]. The deformed entropy in (7) is used to obtain the oneparameter deformed or euivalently the -deformed Einstein euations for -deformed bosons. We apply the Verlinde s proposal to the -deformed Bose gas models, in order to construct the - deformed Einstein euations from the entropy in (7). The fundamental notion needed to derive the gravity is information in the Verlinde s proposal. It is formally the amount of information associated with the matter and its location, measured in terms of entropy. When matter is displaced in space due to a reason, the result is a change in the entropy and this change causes a reaction force. This force is the gravity being an entropic force as an inertial reaction against the force causing the increase of the entropy []. The source of gravity is energy or matter and it is distributed evenly over the degrees of freedom in spacetime. The existence of energy or matter in spacetime causes a temperature in the spacetime. The product of the change of entropy during the displacement of source and the temperature is in fact the work and this work is originally led by the force which is known to be gravity []. By using the Verlinde s idea, -deformed Einstein euations have recently been derived from the deformed entropy (7) of the -deformed Bose gas model [1]. Eventually the -deformed Einstein euations is given, as [1] / 10 V (me) 1 g ( ) R g R 8 GT, (8) h where 5 5 g( ) z ( ) z z ln z ( ) z ln z. (9) 4 By rewriting (8), we get 1 R g R 8 GT, (10)
International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) 01004 doi:10.1088/174-6596/766/1/01004 / where 10 V (me) g ( ) / h. The euation (10) is the -deformed Einstein euations, and it is assumed to describe the gravitational fields generated by the extremal uantum black holes which obey the statistics of deformed particles in accordance with the Strominger s proposal. In the next section, we solve the deformed Einstein euations for a charged extremal black hole, and investigate the implications of the solutions.. Solution of the -deformed Einstein euations Deformed Einstein field euations will be used to describe the geometry of the spacetime around a charged and spherically symmetric uantum black hole. Therefore we need to solve the deformed Einstein-Maxwell euations due to the charge of the uantum black holes. Also, because of the spherical symmetry the metric has to be in the form for 4-dimension is [5] ( r, ( r, t ) ds e dt e dr r d r sin d. (11) The energy-momentum tensor T in (10) is 1 T FF g F F, (1) 4 where F is the electromagnetic field strength tensor and the trace of T is zero due to the electromagnetic strength tensor. Taking the trace of (0) leads to R 8 GT. (1) Spherical symmetry and absence of magnetic charge for our uantum black hole gives the electromagnetic field strength tensor has only radial electric field components E F F f ( r,. r tr rt By euating the non-zero components of the Ricci tensor for the metric (11) and the corresponding non-zero components of the energy-momentum tensor obtained by (1), we find ( r, ( r) ( r). The solution of the Maxwell euations g F 0 and F 0 gives [ ] f ( r, f ( r) Q / 4 r. Using this electromagnetic field strength tensor for one of the components of the Ricci tensor and the corresponding energy-momentum tensor in (1) gives the solutions of the deformed Einstein euations: 1 ds dt dr r d r sin d, (14) where Gm 1 GQ 1. (15) r r The singularities and the event horizons for these black holes are determined by the function and the radius r. There is a true curvature singularity at r 0, since the metric goes to infinity for this value. The coordinate singularity also occurs at 0 and the conditions giving this singularity occur from the solution of 0, such as The case Gm Q / r Gm hole decreases to the minimum value from the case G m GQ. (16) stands for the uantum charged black hole case, since the mass of the black Gm Q /. A minimum mass solution for the uantum black holes is reached by the Hawking radiation and remains stationary. Gm Q / case makes 0 at a single radius r Gm and refers to a single event horizon. This deformed
International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) 01004 doi:10.1088/174-6596/766/1/01004 case solution Gm Q / is the analogue of classical Reissner-Nordström solution m Q / which is often examined in the studies of uantum gravity. For Gm Q / case, the mass of the black hole is allowed to be in very large classical scales due to the ability of getting bigger values than the charge. Whereas the mass of the deformed black hole is allowed to decrease very small values which could fall into the uantum regime, because the decrease of the mass is governed by a very 6 / 7 small term 1 / being order of h. Mass reduction with respect to the classical Reissner-Nordström case can be compared for -deformed cases. Using 1 / from (10) gives the mass of the extremal uantum black hole in case of Gm Q /, such as m h 10 V (E) / 1 g ( ) 7 Q G 7 G, (17) The comparison of the deformed black hole mass and classical Reissner-Nordström black hole mass m Q / G reads as m 7 4 7 / / ) 1 10 V (E) Q / h G g ( m, (18) The euation in (18) implies that the mass of the charged extremal black hole in the deformed uantum case can decrease to a smaller value than that of the classical Reissner-Nordström case. We / 4 / 7 figure out the decrease in the mass by examining the behaviours of the factor ( h G / g( )) in / 4 the classical mass of the charged black hole in (18). We illustrate the behaviour of ( h G / g( )) with respect to z and in Figure 1 and Figure, for 1 and 1, respectively. / 7 4
International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) 01004 doi:10.1088/174-6596/766/1/01004 / 4 Figure 1. Mass reduction factor ( h G / g( )) for various values of the deformation parameter 1. / 7 / 4 Figure. Mass reduction factor ( h G / g( )) for various values of the deformation parameter 1. 4. Conclusions In this study, we review the -deformed statistics of bosons and obtained the -deformed Einstein euations, which is based on the Strominger s idea, such that the micro uantum black holes obey the deformed statistics. We then obtain the solutions of -deformed Einstein euations for the charged extremal uantum black holes. We then interpret the solutions for -deformed Einstein euations. There occur two singularities from the solutions of deformed Einstein euations, such that the true and coordinate singularities. The possible decrease in mass via Hawking radiation to the minimum value which is determined by the charge of uantum black hole is investigated and the difference between the classical black holes and uantum black holes is compared from the euations (18). We represent the decrease in uantum mass m in Figure 1 and Figure. According to the Figures 1 and, the 0 mass of the uantum black hole m in (18) is at least 10 times smaller than the classical black hole mass m, in the deformed case, with the inverse of the volume, charge and energy factors, mass 0 gets smaller than 10 m. Three independent studies, Strominger s micro black holes obeying the deformed statistics, Verlinde s entropic gravity approach and the thermostatistics of deformed Bose gas model, seem to be consistent with each other, since the theoretical possibility of concentrating a mass into its reduced Planck mass leads to a uantum black hole whose dynamics described by the deformed Einstein euations and the mass reduction to a smaller value than that of the classical black holes. / 7 m 5
International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) 01004 doi:10.1088/174-6596/766/1/01004 5. References [1] Dil E 015 Canad. J. Phys. 9 174 [] Strominger A 199 Phys. Rev. Lett. 71 97 [] Verlinde E 011 J. High Energy Phys. JHEP04(011)09 [4] Ubriaco M R 1997 Phys. Rev. E 55(1) 91 [5] Wald R M 1984 General Relativity (Chicago: The University of Chicago Press) 6