Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect

We investigate quasi-free Hadamard states defined via characteristic initial data on null cones centred at the axis of symmetry in spherically symmetric space-times. characterize necessary singular behaviour boundary two-point functions such that one can define non-linear observables this and give formulas for calculation these observables. These results extend earlier characterizations defining bulk cone. As an application our derived formulas, we consider their implications semi-classical Einstein equations calculate vacuum polarization associated with Hawking radiation near a collapsing body.