For a pair of interacting masses, Newton’s law of gravitation is typically expressed as:
where is the distance between the two centres of mass, define the masses of each body and is the gravitational constant.
Note that if , we approximately have:
Given that and , we may linearise using logarithms as follows:
so we have a linear equation in terms of the logarithm of the mass-distance ratio, a new variable, with the logarithm of the Gravitational constant as the y-intercept. This is quite interesting as it shows that when transformed by the logarithm, a non-linear operator, a non-linear interaction between objects may be transformed into a linear interaction.
It must be noted that these objects are now mathematical however and no longer have a physical interpretation.
is useful as it leads us to a couple important insights.
First, we may infer the Gravitational constant :
Second, we may deduce the existence of singularities:
which may theoretically allow the gravitational force to impede the movement of any particle possessing mass, anticipating the existence of black holes.