Dimensional Analysis

As we have already seen, length, mass, and time are three fundamentally different entities that are measured in terms of three completely independent units. It, therefore, makes no sense for a prospective law of physics to express an equality between (say) a length and a mass. In other words, the prospective physical law,

$$m = l,$$ (1.3)

where $m$ is a mass and $l$ is a length, cannot possibly be correct. One easy way of seeing that Equation (1.3) is invalid (as a law of physics) is to note that this equation is dependent on the adopted system of units. That is, if $m=l$ in mks units then $m\neq l$ in fps units, because the conversion factors which must be applied to the left- and right-hand sides of the equation differ. Physicists hold very strongly to the maxim that the laws of physics possess objective reality. In other words, the laws of physics are equivalent for all observers. One immediate consequence of this maxim is that a law of physics must take the same form in all possible systems of units that a prospective observer might choose to employ (because the choice of units is arbitrary, and has nothing to do with physical reality). The only way in which this can be the case is if all laws of physics are dimensionally consistent. In other words, the quantities on the left- and right-hand sides of the equality sign in any given law of physics must have the same dimensions (i.e., the same combinations of length, mass, and time). A dimensionally consistent equation naturally takes the same form in all possible systems of units, because the same conversion factors are applied to both sides of the equation when transforming from one system to another.

As an example, let us consider what is probably the most famous equation in physics; that is, Einstein's mass-energy relation,

$$E = m\,c^{2}.$$ (1.4)

(See Section 3.3.4.) Here, $E$ is the energy of a body, $m$ is its mass, and $c$ is the speed of light in vacuum. The dimensions of energy are $[M]\,[L]^2/[T]^2$, and the dimensions of speed are $[L]/[T]$. Hence, the dimensions of the left-hand side are $[M]\,[L]^2/[T]^2$, whereas the dimensions of the right-hand side are $[M]\,([L]/[T])^2= [M]\,[L]^2/[T]^2$. It follows that Equation (1.4) is indeed dimensionally consistent. Thus, $E=m\,c^{2}$ holds good in mks units, in cgs units, in fps units, and in any other sensible set of units. Had Einstein proposed $E=m\,c$, or $E=m\,c^{3}$ then his error would have been immediately apparent to other physicists, because these prospective laws are not dimensionally consistent. In fact, $E=m\,c^{2}$ represents the only simple, dimensionally consistent way of combining an energy, a mass, and the velocity of light in a law of physics.

The last comment leads naturally to the subject of dimensional analysis. That is, the use of the idea of dimensional consistency to guess the forms of simple laws of physics.