Conformal mapping of unruh temperature

In the framework of conformal field theory, the mapping from (unbounded) wedge regions of Minkowski space time to (bounded) double-cone regions is extended to the Unruh temperature associated to a uniformly accelerated observer. The link between a previous result, the diamond's temperature, and the conformal factor (Weyl rescaling of the metric) is worked out. One thus explains from a mathematical point of view why an observer with finite lifetime experiences the vacuum as a thermal state whatever his acceleration, even vanishing.