arXiv:1108.6149v1 [gr-qc] 31 Aug 2011 Bose-Einstein Condensation on Holographic Screens Behrouz Mirza a,b , ∗ Hosein Mohammadzadeh a , † and Zahra Raissi a ‡ a Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran b Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran, P. O. Box: 55134-441 We consider a boson gas on holographic screens of the Rindler and Schwartzschild spacetimes. It is shown that the gas on the stretched horizon is in a Bose-Einstein condensed state with the Hawking temperature T c = T H if the particle number of the system be equal to the number of quantum bits of spacetime N ≃ A/l p 2 . A boson gas on a holographic screen (r > 2M) with the same number of particles and at Unruh temperature is also in a condensed state. Far from the horizon, the Unruh temperature is much lower than the condensation temperature (T c = T Unruh +  f (r)T Planck ). This analysis implies a possible physical model for quantum bits of spacetime on a holographic screen. We propose a unique and physical interpretation for equipartition theorem on holographic screens. Also, we will argue that this gas is a fast scrambler. PACS numbers: 04.70.-s, 05.30.-d, 04.60.-m Many efforts have been made to find a statistical interpre- tation for the entropy of black holes. Starting from a theorem proved by Hawking [1, 2], Bekenstein conjectured that the entropy of black holes is proportional to the area of its event horizon , S BH = A/4l p 2 , where, A is the area of event horizon and l p is the Planck length [3]. Further evidence for this was given in studies by Hawking [4]. Until 1995, no one was able to make a precise calculation of black hole entropy based on a fundamental theory. The situation changed when A. Stro- minger and C. Vafa calculated the right Bekenstein-Hawking entropy of a supersymmetric black hole in the string theory using a method based on D-branes [5]. Their calculation was followed by many similar computations of entropy of large classes of other extremal and near-extremal black holes. How- ever, these methods cannot be applied to the more general case of nonsupersymmetric neutral black holes. Also, loop quan- tum gravity has yielded a detailed prescription for identifying microscopic quantum states corresponding to an isolated hori- zon [6]. Despite strong evidence for Bekenstein’s conjecture, the physical nature of quantum mechanically distinct internal states has remained unknown. There is another different per- spective for microstates of a Schwarzschild black hole based on earlier ideas of ’t Hooft [7], Susskind [8] and some other works [9, 10]. Gerlach tried to interpret the Hawking radia- tion as one produced by zero-point fluctuations on the surface of a star that collapsed to form black holes. He concluded that the number W zp of zero-point fluctuation modes gives rise to the Hawking radiation of freely evaporating Schwarzschild black holes is satisfied at ln W zp ∼ = 280S BH [11]. Also, accord- ing to York’s proposal, Hawking radiation is produced by the black hole’s quantum ergosphere of thermally excited gravi- tational quasinormal modes. He concluded that the number of ways W qe this quantum ergosphere can be excited and re- excited, during the evaporation of Schwarzschild hole into a surrounding radiation bath is satisfied at ln W qe ∼ = 1.106S BH ∗ Electronic address: b.mirza@cc.iut.ac.ir † Electronic address: h.mohammadzadeh@ph.iut.ac.ir ‡ Electronic address: z.raissi@ph.iut.ac.ir [12]. It has been shown that one can consider the number of quantum mechanically distinct ways that the black holes could have been made by infalling quanta particles [9]. The number of ways a Schwarzschild black hole of mass M can be made by accretion of quanta from infinity (r ≫ 2M) will be a precise statistical explanation of W . A great deal of effort has gone into resolving the puzzles of black hole thermodynamics and information loss. In this context, the idea of a stretched horizon arose as a useful tool for thinking about black holes. It originated as a classical de- scription of black holes seen by an outside observers, but the concept was later borrowed to help give a consistent quan- tum mechanical interpretation of black hole physics [13, 14]. It was also shown that the degrees of freedom of a stretched horizon can be viewed as a gas of quasiparticles. This simple picture makes manifest several universal properties of hori- zons, including the universal relationship between entropy and horizon area [15]. Although a lot of research work has been focused on explaining the physics of horizon, our knowledge of holographic screens and their thermodynamics is still far from perfect. In this paper, we introduce a new perspective for investigating the condensation of an ideal boson gas in the background of Rindler and Schwartzschild spacetimes. For earlier studies on condensation on a curved spacetime see Ref. [16]. This analysis could be considered as a probe into the quantum structure of spacetime. Our calculations indicates that the equipartition theorem on a holographic screen should run as follows, U u ≃ N e k B T u (1) where, U u and T u denote the internal energy and the Un- ruh temperature on a holographic screen, respectively. N e = N u (T u /T c ) 2 is the number of excited quantum states of space- time where, T c and N u ≃ A/l 2 p are the local condensation tem- perature and number of quantum bits of spacetime (particle number), respectively. First, we will evaluate the density of states in the back- ground of a curved spacetime. We consider a system with N non-interacting particles described by a Hamiltonian H ( p i , q i ). The system is confined to a specified volume and has an energy E . Therefore, H ( p i , q i )= E designates the