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. 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