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

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The discussion focuses on the modifications to Newton's universal law of gravity as explained through the lens of General Relativity (GR) and spacetime geometry. Participants seek clarification on the origins and implications of specific equations presented in lecture notes, particularly regarding the transition from Newtonian gravity to GR. The conversation highlights the relationship between Coulomb's law, Gauss's law, and Poisson's equation in understanding gravitational forces. Historical context regarding the development of these concepts is also mentioned, noting that modern undergraduate textbooks often favor the latter approach. Overall, the thread emphasizes the need for a clearer understanding of the mathematical foundations of these gravitational theories.
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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?

Screenshot1.png


and as follows with this page

Screenshot2.png


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?
 
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