Hi, let me introduce myself, My name is Terell Randolph and I am a freshman at University of Washington majoring in physics and mathematics. Anyways I have been working on a equation to describe the motion on moving bodies caused by gravity. After two or three weeks I developed one to describe the force of gravity:

((mc^2)/r)*Sin(x)

where m=mass Force gravity is acting upon, c=Speed of light, x=motion of wave, and r=Distance from sun.

My equation and Newton's equation answers match for and mass or radius, but when it comes to black holes for example Schwarzschild radius with my equation you get 2.5km where as Newton equation you get 3km (even though Schwarzschild radius was derived from GR you can use Newton equation). I need help seeing why my equation only works for long distances. First I'll show how I derived c*Sin(x) which is not important so you can skip over it but my equation is based of this. Then I'll show how I derived my force equation.

This derivation is based on Special relativity and Mass-energy equivalence

E(1)/k=E(2)

Where E(1) is energy of mass at rest, E(2) is mass new energy from observer perspective, and k=sqrt(1-v^2/c^2) c=speed of light v=velocity of moving body (throughout this derivation the velocity is 0)

(hc)/(λk)=mc^2

h=Planck constant c=speed of light λ=wavelength m=mass

h/(λck)=m

h^2/(λ^2k^2c^2)=m^2

h^2/(λ^2(1-v^2/c^2)c^2)=m^2

h^2/(λ^2(c^2-v^2))=m^2

h/(λ*sqrt(c^2-v^2))=m **Equation (1)**

The following derivation sets the condition for λ

f=c/λ

f=Frequency, c=speed of light,λ=wave length

f=c/1 λ=1

Then hf=hc/λ equals hf=hc

Using the conditions above

mc^2=hf

hc^2/(λ*sqrt(c^2-v^2))=hf **Using Equation (1)**

hc^2/sqrt(c^2-v^2)=hf when λ=1

h^2c^4/(c^2-v^2)=h^2f^2

h^2c^4 =h^2f^2(c^2-v^2)

h^2c^4 =(h^2f^2c^2)-(h^2f^2v^2)

h^2f^2c^2 =(h^2c^4)+(h^2f^2v^2) add h^2f^2v^2 to both sides

(h^2f^2c^2)-(h^2c^4) =(h^2f^2v^2) subtract h^2c^4 from both sides

h^2f^2c^2(1-(c^2/f^2)) =h^2f^2v^2 Factor out h^2f^2c^2

c^2(1-(c^2/f^2)) =v^2 Divide both sides by h^2f^2

if f=c/λ then λ=c/f

c^2(1-λ^2)=v^2

c*sqrt(1-λ^2)=v **Equation (2) **

This equation satisfy the conditions above because if λ=1 then v=0

solving for λ

λ=\sqrt(1-(v^2/c^2))

Since (v^2/c^2)is a ratio, I can place them on a unit circle, y component is v and the radius is c. Now I can replace (v^2/c^2) with Sin^2(x)

λ=sqrt(1-Sin^2(x))

Cos(x)=sqrt(1-Sin^2x) Replacing λ with Cos(x)

Going back to Equation(2)

c*sqrt(1-λ^2)=v

c*sqrt(1-Cos^2x)=v replacing λ with Cos(x)

Using trig-identities

**c*Sin(x) =v Equation(3)**

I'll resume in a second post