## Introduction:

For a pair of interacting masses, Newton’s law of gravitation is typically expressed as:

where $r$ is the distance between the two centres of mass, $m_i$ define the masses of each body and $G$ is the gravitational constant.

## Linearization:

Note that if $m = m_1 \approx m_2$, we approximately have:

Given that $G > 0$ and $\frac{m}{r} > 0$, we may linearise $(2)$ 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.

## Analysis:

$(3)$ is useful as it leads us to a couple important insights.

First, we may infer the Gravitational constant $G$:

Second, we may deduce the existence of singularities:

which may theoretically allow the gravitational force $F$ to impede the movement of any particle possessing mass, anticipating the existence of black holes.