  The inverse relationship between energy and time is as familiar as Planck's
constant. From the point of view of a system with many states, perhaps a better
representation of the system is a vector of characteristic times (one per
state) for example, a canonically distributed system. In the vector case the
inverse relationship persists, this time as a relation between the $L_2$ norms.
That relationship is derived herein. An unexpected benefit of the vectorized
time viewpoint is the determination of surfaces of constant temperature in
terms of the time coordinates. The results apply to all empirically accessible
systems, that is situations where details of the dynamics are recorded at the
microscopic level of detail. This includes all manner of simulation data of
statistical mechanical systems as well as experimental data from actual systems
(e.g. the internet, financial market data) where statistical physical methods
have been applied.
