He chose to define and by second moments. A similar and perhaps more basic concept than the product of time and frequency spreads is the relationship between spectral bandwidth and rise time of a system response function. The rise time of a system response is also defined somewhat arbitrarily, often as the time span between the time of excitation and the time at which the system response is half its ultimate value. In principle, a broad frequency response can result from a rapid decay time as well as from a rapid rise time. Slackness could also result from other combinations of rises and falls such as random combinations.
Many systems respond very rapidly compared to the rate at which they subsequently decay.
The first step is to choose a definition for rise time. The choice is determined not only for clarity and usefulness but also by the need to ensure tractability of the subsequent analysis. The numerator is just a normalizing factor. The denominator says we have defined by the first negative moment. Presumably this disadvantage is not fundamental since we can always go to a limiting process in which the discretized time domain tends to a continuum.