Un extrait du premier papier. J'abuserais en te disant que je comprend correctement, mais en gros, l'expression de l'entropie en dimension usuelle s'exprime également correctement en dimension plus élevée selon une formule dite de Cardy - Verlinde.
A black hole hologram in de Sitter space
Published by Institute of Physics Publishing for SISSA/ISAS
Accepted: March 08, 2002
Ulf H. Danielsson
Institutionen for teoretisk fysik
Box 803, SE-751 08 Uppsala, Sweden
In this paper we show that the entropy of de Sitter space with a black hole in arbitrary dimension can be understood using a modified Cardy-Verlinde entropy formula.
We also comment on the observer dependence of the de Sitter entropy.
Keywords: Black Holes in String Theory, AdS/CFT Correspondence, Cosmology of Theories beyond the SM.
1. Introduction 1
2. Constructing the hologram 2
2.1 The metric 2
2.1.1 Cylindrical coordinates 3
2.1.2 Planar coordinates 3
2.2 The Brown-York tensor 4
2.2.1 Planar coordinates 4
2.2.2 Cylindrical coordinates 5
3. Interpreting the hologram 6
3.1 Black holes in AdS 6
3.2 Black holes in de Sitter 7
3.2.1 The Nariai black hole 9
Physics is very different depending on the presence and sign of the cosmological constant. With a vanishing cosmological constant, space time is asymptotically flat and physics can be conveniently described by an S-matrix. This is a consequence of the presence of a light like infinity; if we wait long enough particles may be separated by an arbitrary spatial distance and interactions will be suppressed. In the case of a negative cosmological constant the universe is anti de Sitter (AdS) on large scales and there is no light like infinity. There is, however, at an infinite spatial distance a time like in¯nity which can be used as a holographic screen of lorentzian signature, [1, 2, 3]. This discovery was the ¯rst rigorous implementation of the idea of holography, [4, 5], and has lead to many subsequent studies. But it is very likely that our universe is neither flat nor AdS but instead is a de Sitter space (dS) on large scales with a positive cosmological constant, [6, 7]. While being the most realistic possibility, de Sitter space is at the same time theoretically the most challenging. To this date there is, in fact, no successful implementation of de Sitter space in string theory. One of the intriguing features of de Sitter space is the presence of a horizon and an associated temperature and entropy, . While string theory successfully has addressed the problem of entropy for black holes, dS entropy remains a mystery. One reason is that the ¯nite entropy seems to suggest that the Hilbert space of quantum gravity for asymptotically de Sitter space is finite dimensional, . A recent discussion of this and other connected issues can be found in . Another, related, reason is that the horizon and entropy in de Sitter space have an obvious observer dependence. For a black hole in flat space (or even in AdS) we can take the point of view of an outside observer who can assign a unique entropy to the black hole. The problem of what an observer venturing inside the black hole experiences, is much more tricky and has not been given a satisfactory answer within string theory. While the idea of black hole complementarity provides useful clues, , rigorous calculations are still limited to the perspective of the outside observer. In de Sitter space there is no way to escape the problem of the observer dependent entropy. This contributes to the difficulty of de Sitter space.
Recently there has been some progress towards a holographic understanding of de Sitter space, [10, 12]. See also [13, 14, 15]. A review with references may be found in . The main observation is that there is a possibility of introducing a holographic screen at either time like past infinity, I¡, or time like future infinity, I+. The theory on the screen will, just as in the case of AdS, be a conformal ¯eld theory with a scale that encodes the dimension transverse to the screen. But in contrast to the AdS case the holographic theory of de Sitter space will be euclidean and it is time that is given a description through the scale. Large scales on I+ will correspond to early times, while late times will correspond to small scales. The central charge of the theory is given by the area of the horizon. Very recently it was suggested, , (see also ) that our universe is described by a RG °ow in the euclidean theory on I+. In the IR the theory has a ¯xed point of relatively low central charge corresponding to an early phase of in°ation with a horizon a few orders of magnitude larger than the Planck scale. In the UV, on the other hand, there is a ¯xed point of large central charge corresponding to the universe we now are approaching where a cosmological constant again will be dominating. In this paper we will continue the study of a holographic description of de Sitter space. In particular we will investigate the properties of black hole holograms in de Sitter space. We will show that the entropy of the cosmological horizon in the presence of a black hole can be understood using a Cardy-Verlinde formula, , in a way very similar to the entropy of black holes in AdS. We will also make some speculations on how to address the problem