# Diamonds's Temperature: Unruh effect for bounded trajectories and thermal time hypothesis

Pierre Martinetti, Carlo Rovelli
December 17, 2002
We study the Unruh effect for an observer with a finite lifetime, using the thermal time hypothesis. The thermal time hypothesis maintains that: (i) time is the physical quantity determined by the flow defined by a state over an observable algebra, and (ii) when this flow is proportional to a geometric flow in spacetime, temperature is the ratio between flow parameter and proper time. An eternal accelerated Unruh observer has access to the local algebra associated to a Rindler wedge. The flow defined by the Minkowski vacuum of a field theory over this algebra is proportional to a flow in spacetime and the associated temperature is the Unruh temperature. An observer with a finite lifetime has access to the local observable algebra associated to a finite spacetime region called a "diamond". The flow defined by the Minkowski vacuum of a (four dimensional, conformally invariant) quantum field theory over this algebra is also proportional to a flow in spacetime. The associated temperature generalizes the Unruh temperature to finite lifetime observers. Furthermore, this temperature does not vanish even in the limit in which the acceleration is zero. The temperature associated to an inertial observer with lifetime $T$, which we denote as "diamond's temperature", is $2\hbar/(\pi k_b T)$.This temperature is related to the fact that a finite lifetime observer does not have access to all the degrees of freedom of the quantum field theory.

Keywords:
thermal time hypothesis, Unruh temperature, diamond regions