Please explain these Equations in GR, Modified Universal law of Gravity.

[SBC]

Space time Geometry and GR, taken from lecture notes

http://www.mth.kcl.ac.uk/courses/cm334.html

http://www.mth.kcl.ac.uk/courses/cm334/Lectures0910.pdf
from pages 3 and 4

Newton's universal law of gravity is getting modified.

it says using the result, from where this result comes from?

and as follows with this page

after equation 4, it says that using the fact that... where this equation is coming from and how can I understand this, can anyone explain please?

 

[atyy]

Try http://www.math.okstate.edu/~binegar/5233-S96/5233-l20.pdf

The alternative way is to recognize Coulomb's law and the Principle of superposition as being the solution to Gauss's law, and from there with the assumption of time independence get to Poisson's equation.

I don't know which came first historically. Undergraduate textbooks usually take the second route nowadays.