*** First, I introduce the full idea I came up with. Below are actual formulae to play with, so if you want to skip down to that, scroll down ***
I recently played around with the following thought:
Newton's law of gravity states:
F_m1=-F_m2= m1*m2*G* 1/(r.r) * r/norm(r)
Now, the universe is said to be edge-free and the geometry it's estimated macroscopically, usually is a 3-sphere.
This means, that the universe has an (expanding) Radius R and if you go in a straight line, you'll eventually end up at the position you started (edge-freeness)
Since in Newtonian Physics, Gravitational pull is both infinite and unconstrained by speed of light (in relativistic physics it would be limited by speed of light but considering String theory, Gravitons could take shortcuts that might mean, it actually can appear to go faster.), you gotta sum up for infinite distances in both directions.
If you now do the math on one of the grand-circles (straight paths) in that universe, you end up with the following as a sum of the net-force across the entire universe:
|F|=m1*m2*G/(2 R Sin(r/(2 R))²
Now simplifying a bit, the masses and the gravitational constant can be normalized to 1. Furthermore, you can change the Universe's Radius to its Diameter D=2R instead:
1/(D Sin(r/D)²
Inverting gives
(D Sin(r/D)²
And from there you simply introduce the typical constant c.
r = (D Sin(r/D)²+c
This is roughly like an MSet with constrained "universe-Radius" R as additional parameter.
In case you consider all three parameters complex-valued, you end up with the following formulae for the real and imaginary parts of r:
r=x+yi // standard MSet variable, representing distance between mass-points
D=c+di // Diameter of the "Universe"
C=a+bi // standard MSet constant
//variables to simplify the actual formulae:
sqD = c²+d²
Dpos = 2*(c*x+d*y)/sqD
Dneg = 2*(d*y-c*x)/sqD
reD = c²-d²
imD = 2*c*d
rer = 1/2 * (1- Cos(Dpos)*Cosh(Dneg))
imr = -1/2 * Sin(Dpos)*Sinh(Dneg)
// iterated Functions:
x = reD*rer-imD*imr+a
y = reD*imr+imD*rer+b
(The actual Newtonian law of gravity might end up being more complicated, since you have to integrate over all directions. This is essentially the law applied to a single direction.
Though it behaves correctly in at least the following ways:
- It repeats every 2 pi R, as expected
- In the limit where R goes to infinity, the usual Force is recovered
)