The energy-momentum tensor of the massless scalar field is calculated from the transformation :
if and (Vendrell (1997a)). The black hole emits thus spontaneously a thermal radiation (Hawking (1975)) of temperature
To the first loop approximation, the back-reaction of the massless radiation on space-time is analysed by adding to eq.1 the trace anomaly . One obtains
where . The mass and temperature of the black hole are renormalised and become smaller
The radius of the black hole becomes larger.