We compute the quantum string entropy S s (m, H) from the microscopic string density of states ρ s (m, H) of mass m in de Sitter space–time. We find for high m (high Hm → c/α') a new phase transition at the critical string temperature T s = (1/2πk B )L cl c2/α', higher than the flat space (Hagedorn) temperature t s (L cl = c/H, the Hubble constant H acts at the transition, producing a smaller string constant α' and thus, a higher tension). T s is the precise quantum dual of the semiclassical (QFT Hawking–Gibbons) de Sitter temperature T sem = ħ c/(2πk B L cl ). By precisely identifying the semiclassical and quantum (string) de Sitter regimes, we find a new formula for the full de Sitter 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 [Formula: see text] is the leading term, but for high H near c/ℓ Planck , a new phase transition operates and the whole entropy S sem (H) is drastically different from the Bekenstein–Hawking entropy [Formula: see text]. We compute the string quantum emission cross-section σ string by a black hole in de Sitter (or asymptotically de Sitter) space–time (bhdS). For T sem bhdS ℓ T s (early evaporation stage), it shows the QFT Hawking emission with temperature T sem bhdS (semiclassical regime). For T sem bhdS → T s , σ string exhibits a phase transition into a string de Sitter state of size [Formula: see text], [Formula: see text], and string de Sitter temperature T s . Instead of featuring a single pole singularity in the temperature (Carlitz transition), it features a square root branch point (de Vega–Sanchez transition). New bounds on the black hole radius r g emerge in the bhdS string regime: it can become r g = L s /2, or it can reach a more quantum value, r g = 0.365 ℓ s .
Hawking Non-thermal and Purely Thermal Radiations of Rotating Black Hole in Anti-de Sitter Space Time by Hamilton-Jacobi Method
