Hawking radiation in string theory and the string phase of black holes

Abstract

The quantum string emission by black holes is computed in the framework of the ``string analog model'' (or thermodynamical approach), which is well suited to combine quantum-field theory (QFT) and string theory in curved backgrounds (particularly here, as black holes and strings possess intrinsic thermal features and temperatures). The QFT-Hawking temperature ${T}_{H}$ is upper bounded by the string temperature ${T}_{S}$ in the black hole background. The black hole emission spectrum is an incomplete gamma function of ${(T}_{H}\ensuremath{-}{T}_{S}).$ For ${T}_{H}\ensuremath{\ll}{T}_{S},$ it yields the QFT-Hawking emission. For ${T}_{H}\ensuremath{\rightarrow}{T}_{S},$ it shows that highly massive string states dominate the emission and undergo a typical string phase transition to a microscopic ``minimal'' black hole of mass ${M}_{\mathrm{min}}$ or radius ${r}_{\mathrm{min}}$ (inversely proportional to ${T}_{S})$ and string temperature ${T}_{S}.$ The string back reaction effect [self-consistent black hole solution of the semiclassical Einstein equations with mass ${M}_{+}$ (radius ${r}_{+})$ and temperature ${T}_{+}]$ is computed. Both the QFT and string black hole regimes are well defined and bounded: ${r}_{\mathrm{min}}<~{r}_{+}<~{r}_{S},{M}_{\mathrm{min}}<~{M}_{+}<~M,{T}_{H}<~{T}_{+}<~{T}_{S}.$ The string ``minimal'' black hole has a life time ${\ensuremath{\tau}}_{\mathrm{min}}\ensuremath{\simeq}{(k}_{B}c/G\ensuremath{\Elzxh}){T}_{S}^{\ensuremath{-}3}.$