It is very common to set different constants or quantities to 1 in theoretical physics, such as c=1 for SR. The natural consequence is that the units of time and distance coincide. In QM one usually sets [math] \hbar = 1[/math], in other words action. This leads to the units of time and energy being reciprocal. In spectroscopy you have essentially both and that gives you [time] = [distance] = 1/[energy], only leaving 3 units: distance, mass and charge. Most other choices of natural units I know of tend to be less interesting (electron mass and charge for chemists for example), but this feels profoundly fundamental.
>>17050042The "profound" bit is the understanding that our units have always been arbitrary and sometimes the math gets easier when you define things in a way that redundant units get factored out. If a particular constant acts primarily as a linear scaling factor in the equations you're using, then it makes sense to get rid of them if what you're interested in is the relationship between variables.
>>17050052There is a difference between arbitrary (in terms of scaling) and independent. Time and energy being effectively Fourier pairs is a symmetry thing (time translation symmetry = energy conservation, same as spacial translation symmetry = momentum conservation). time and distance is much the same (Lorentz). The symmetries are the profound thing.
>>17050042Now hold on. If [math] [E]^{-1} = [x] = [t] [/math], then [eqn] [E] = [T] = [m] \cdot 1^2,[/eqn]So mass is measured in reciprocal distances as well in that unit system. Charge is not fixed unless one asserts [math] [\varepsilon_0]=[\mu_0]=1[/math], or something equivalent.
>>17050554In other words, [math] c=\hbar=\varepsilon_0=1[/math] sends pretty much every other unit to hell.
Mass Gap is Coming
>>17050556it makes a perverse sort of sense that every unit can be replaced by distance because in practice you don't measure time, you measure displacement (of the hand of a clock for example), same with force or energy
