Quantization of Open strings in time dependent Black Holes

In this letter, the open string is quantized in a time dependent black hole background. The geometry defined through an adiabatic approximation of Vaydia metric. worldsheet two-point function derived and it shown to have same type singularity as flat space one. However, equal times depends on particular Cauchy surface where fields are defined. Finite temperature effects incorporated Liouville-von Neumann approach non equilibrium thermodynamics.