Today we will derive the gravitational constant G, also known as Newton's constant. Here are the variables we will be working with:
Below is a crude diagram of a satellite orbiting a star or planet at velocity v, at a distance of radius r. According to its clock, the proper time is t'. The star or planet has a mass of m.
Our starting point shall be the Lorentz equation, courtesy of Einstein's theory of special relativity:
By doing some algebra we can derive equation 7 below:
Equation 7's right side expresses v^2 in terms c^2 and a time ratio. The bigger the time ratio, the faster the velocity and vice versa.
We manipulate Einstein's energy equation to get equation 8:
We make a substitution, then do some more algebra until we derive G at equation 15:
We can now see why G is the constant it is: Any change in velocity (v^2) is offset by a change of the radius-mass ratio. Any change in radius-mass ratio is offset by a change in the time ratio.