07 01 00 8 v 2 2 9 A ug 2 00 8 A thermodynamical model for non-extremal black p-brane

The microscopic origin of black hole entropy has been considered to be an important subject for years, since the mechanism of black hole thermodynamics is regarded to be explained by quantum gravity theory. String theory, the leading candidate of quantum gravity theory, actually explained a black hole entropy by a microscopic model [2]. A certain D-brane model exactly reproduces the black hole entropy including its coefficient. It is based on supersymmetric nature of the black hole, which ensures vanishing quantum corrections so that the string D-brane model defined at weak coupling region reproduces entropy in strong coupling region. Due to similar supersymmetric nature, AdS/CFT correspondence [27] has been providing various important results. On the other hand, when supersymmetry is broken, the relation between D-brane model in weak coupling region and supergravity (of strong coupling region) is unclear, since the quantum correction is unknown. It is difficult to relate string theoretical models to black hole thermodynamics in general. However, it is important to look for thermodynamical models which can describe black hole thermodynamics, since one might find a clue to understand non-perturbative property of quantum gravity theory. For example, [4] showed that D3-brane and open string gas model provides the microscopic description of near extremal black 3-brane thermodynamics, up to numerical factor. The reason of this agreement without supersymmetry is explained in [26] [28] [29], and the reason of the discrepancy of the coefficient is studied in [6][7]. In addition, [5] showed that the entropy of near extremal black p-brane is described as S ∝ qT b where T is the temperature and q is the charge. Surprisingly, [9] showed that D3-brane anti D3-brane model also provides the microscopic description of nonextremal (and in particular Schwarzschild) thermodynamics, including its absorption probability [37], up to numerical factor. [10] [11] extended it to general neutral p-brane and showed that the similar model reproduces the correct entropy up to numerical factor. It is also applied to non-extremal black p-branes, multi-charged branes, rotating branes [12] [13] [14] [15] [17]. Recently, [48] showed a relation between the thermodynamics of chargeless black 3-brane and D3-brane anti D3-brane system, in the supergravity framework. All those thermodynamical models for black 3-brane are T 4 model, i.e., the thermodynamics of black 3-brane is described like R massless field in finite temperature. This might indicate a correspondence between gravity and other massless theory. In addition, all those models are “extended near extremal model”, i.e., when the thermal energy of the near extremal p-brane is described by T, that of the non-extremal p-brane (far from extremality) is also described by T. In this paper, we propose an expression of a “phenomenological” thermodynamical model which provides the correct entropy and temperature and absorption probability including coefficients. This expression does not yield additional discrepancy in the coefficient, and exactly reproduces the supergravity results. Next we show a geometrical similarity between the near extremal black p-brane and non-extremal black p-brane. This geometrical similarity results in the “extended near extremal model” which we mentioned above. We also show that our thermodynamical model is in fact appropriate for blackbody radiation of non-extremal black 3-brane, by considering the action of dilaton field in the low energy region. And we show that the correct absorption probability and greybody factor for non-extremal black 3-brane are also obtained. We show that this exact agreement is due to the equality of the scalar field equation in black 3-brane background and parameter changed near extremal 3-brane background.