A path integral with real time coordinate is dual to a thermal partition function, related by a Wick rotation. The periodicity β β of imaginary time corresponds to a temperature of β β = 1 / T in thermal quantum field theory. Note that the path integral for this Hamiltonian is closed with period 2 π. This means that the H modes are thermally occupied with temperature ⁠ 1 / 2 π ⁠. This is not an actual temperature, because H is dimensionless. It is conjugate to the timelike polar angle σ, which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency f in σ at position ρ has a frequency which is determined by the square root of the (absolute value of the) metric at ρ, the redshift factor. This can be seen by transforming the time coordinate of a Rindler observer at fixed ρ to an inertial, co-moving observer observing a proper time. From the Rindler-line-element given above, this is just ρ. The actual inverse temperature at this point is therefore