Translation in is a symmetry of Minkowski space: it can be shown that it corresponds to a boost in "x", "t" coordinate around the origin. Any time translation in quantum mechanics is generated by the Hamiltonian operator. For a detector coupled to modes with a definite frequency in , we can treat as "time" and the boost operator is then the corresponding Hamiltonian. In the Euclidean field theory, where the minus sign in front of the time in Rindler metric is changed to a plus sign by multiplying formula_4 to the Rindler time, i.e. a Wick rotation or the imaginary time, the Rindler Metric is turned into a polar-coordinate-like metric. Therefore the rotations must close itself after 2 in Euclidean metric to avoid being singular. So  A path integral with real time coordinate is dual to a thermal partition function, related by a Wick rotation. The periodicity formula_6 of imaginary time corresponds to a temperature of formula_7 in thermal quantum field theory. Now that the path integral for this Hamiltonian is closed with period 2. This means that the modes are thermally occupied with temperature . This is not an actual temperature, because is dimensionless. It is conjugate to the timelike polar angle , which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency in at position has a frequency which is determined by the square root of the (absolute value of the) metric at , the redshift factor. This can be seen by transforming the time coordinated of an Rindler observer at fixed to an inertial, co-moving observer observing a proper time. From the Rindler-line-element given above, this is just . The actual inverse temperature at this point is therefore  It can be shown that the acceleration of a trajectory at constant in Rindler coordinate is equal to , the actual inverse temperature observed is  Restoring units yields  The temperature of the vacuum, seen by an isolated observer accelerating at the Earth's gravitational acceleration of = , is only . For an experimental test of the Unruh effect it is planned to use accelerations up to , which would give a temperature of about . The Rindler derivation of the Unruh effect is unsatisfactory to some, since the detector's path is super-deterministic. Unruh later developed the Unruh–DeWitt particle detector model to circumvent this objection. The Unruh effect would also cause the decay rate of accelerating particles to differ from inertial particles.