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

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)Lclc2/α′, which is higher than the flat space (Hagedorn) temperature ts. (Lcl = 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 Tsem = ℏc/(2πkBLcl). We compute the quantum string emission σstring by a black hole in Anti-de Sitter (or asymptotically Anti-de Sitter) space–time (bhAdS). For Tsem bhAdS ≪ Ts (early evaporation stage), it shows the QFT Hawking emission with temperature Tsem bhAdS (semiclassical regime). For Tsem bhAdS → Ts, it exhibits a phase transition into a Anti-de Sitter string state of size , 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 : Hmax = 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 Ssem(H), as a function of the usual Bekenstein–Hawking entropy . For Lcl ≫ lPlanck, i.e. for low H ≪ c/lPlanck, or classical regime, is the leading term with its logarithmic correction, but for high H ≥ c/lPlanck or quantum regime, no phase transition operates, in contrast to de Sitter space, and the entropy Ssem(H) is very different from the Bekenstein–Hawking term .