  Thanks to a local interpetation of the KMS condition, the mapping from
(unbounded) wedge regions of Minkowski space-time to (bounded) double-cone
regions is extended to the Unruh temperature associated to relevant observers
in both regions. A previous result, the diamond's temperature, is shown to be
proportional to the inverse of the conformal factor (Weyl rescaling of the
metric) of this map. 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.
