# Linearizing Newton's Law of Gravitation

## Introduction:

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.

## Linearization:

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.

## Analysis:

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.