text: In special relativity, an observer moving with uniform proper acceleration  through Minkowski spacetime is conveniently described with Rindler coordinates, which are related to the standard (Cartesian) Minkowski coordinates by
  The line element in Rindler coordinates, i.e. Rindler space is
  where , and where  is related to the observer's proper time  by  (here ).
  An observer moving with fixed  traces out a hyperbola in Minkowski space, therefore this type of motion is called hyperbolic motion. The coordinate  is related to the Schwarzschild spherical coordinate  by the relation 
  An observer moving along a path of constant  is uniformly accelerating, and is coupled to field modes which have a definite steady frequency as a function of . These modes are constantly Doppler shifted relative to ordinary Minkowski time as the detector accelerates, and they change in frequency by enormous factors, even after only a short proper time.
  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 Euclidean field theory, where the minus sign in front of the time in the Rindler metric is changed to a plus sign by multiplying  to the Rindler time, i.e. a Wick rotation or imaginary time, the Rindler metric is turned into a polar-coordinate-like metric. Therefore any rotations must close themselves after 2 in a 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  of imaginary time corresponds to a temperature of  in thermal quantum field theory. Note 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 coordinate of a 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 coordinates is equal to , so 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.