SEMICLASSICAL (QFT) AND QUANTUM (STRING) ANTI-DE SITTER REGIMES: NEW RESULTS

Abstract

We compute the quantum string entropy Ss(m, H) from the microscopic string density of states ρs(m, H) of mass m in Anti-de Sitter space–time. For high m, (high Hm → c/α′), no phase transition occurs at the Anti-de Sitter string temperature Ts = (1/2πkB)L cl c2/α′, which is higher than the flat space (Hagedorn) temperature ts. (L cl = c/H, the Hubble constant H acts as producing a smaller string constant α′ and thus, a higher tension). Ts is the precise quantum dual of the semiclassical (QFT) Anti-de Sitter temperature scale T sem = ℏc/(2πkBL cl ). We compute the quantum string emission σ string by a black hole in Anti-de Sitter (or asymptotically Anti-de Sitter) space–time (bhAdS). For T sem bhAdS ≪ Ts (early evaporation stage), it shows the QFT Hawking emission with temperature T sem bhAdS (semiclassical regime). For T sem bhAdS → Ts, it exhibits a phase transition into a Anti-de Sitter string state of size [Formula: see text], and Anti-de Sitter string temperature Ts. New string bounds on the black hole emerge in the bhAdS string regime. The bhAdS string regime determines a maximal value for H : H max = 0.841c/ls. The minimal black hole radius in Anti-de Sitter space–time turns out to be rg min = 0.841ls, and is larger than the minimal black hole radius in de Sitter space–time by a numerical factor equal to 2.304. We find a new formula for the full AdS entropy S sem (H), as a function of the usual Bekenstein–Hawking entropy [Formula: see text]. For L cl ≫ ℓ Planck , i.e. for low H ≪ c/ℓ Planck , or classical regime, [Formula: see text] is the leading term with its logarithmic correction, but for high H ≥ c/ℓ Planck or quantum regime, no phase transition operates, in contrast to de Sitter space, and the entropy S sem (H) is very different from the Bekenstein–Hawking term [Formula: see text].