Entropy of 2+1 dimensional de Sitter space in terms of brick wall method

We calculate the statistical entropy of a scalar field on the background of three-dimensional De Sitter space in terms of the brick wall method and finally derive the perimeter law of the entropy. ∗Electronic address: [email protected] 1 Entropy of black holes has an universal area law [1] and the entropy of the well-known Schwarzschild black hole satisfies the area law by means of thermal radiation based on the quantum field theory [2]. On the other hand, ’t Hooft has argued that when one calculates the black hole entropy, the modes of a quantum field in the vicinity of a black hole horizon should be cut off due to gravitational effects rather than infinitely piling up by suitably choosing a brick-wall cutoff just beyond the horizon [3]. Most of brick-wall calculations have been done for the asymptotically flat cases. For 2+1 dimensional anti-de Sitter space, Bañados, Teitelboim, and Zanelli (BTZ) have obtained a black hole solution which is asymptotically anti-de Sitter spacetime rather than asymptotically flat [4]. The theormodynamic properties has been extensively studied in this black hole [5,6]. Recently the statistical entropy of the de Sitter(DS) space is studied in terms of Chern-Simons formulation [7]. The DS space has a cosmological horizon and asymptotically non-flat spacetime, furthermore the spacetime is bounded by the horizon as the two-dimensional cavity. In this Brief Report, we shall calculate the entropy of a scalar field on the DS space background by using the brick wall method. As a result, the divergent entropy is obtained in the vicinity of horizon and by properly choosing the brick-wall cutoff we finally obtain the expected perimeter law. Let us start with the following action I = 1 2π ∫