On the simplified tree graphs in gravity

TL;DR

This paper revises previous assumptions about pressure integrals in gravitational systems and derives Schwarzschild solutions using two different methods, highlighting the advantages of perturbation theory over traditional approaches.

Contribution

It introduces a simpler, more informative perturbation method for deriving Schwarzschild solutions in harmonic and isotropic coordinates, improving understanding of regional contributions.

Findings

Corrected the assumption about pressure volume integrals in gravity.

Derived Schwarzschild solutions using two different methods.

Showed the perturbation approach is simpler and more insightful.

Abstract

Firstly, I give the reason why is wrong my previously made assumption that the volume integral over the pressure may not be zero in a system where the gravitation plays no role in holding the system together. Secondly, in the first nonlinear approximation I obtain the inner and outer Schwarzschild solutions in harmonic and isotropic coordinates in two different ways. One way is to start from standard solution and make the appropriate coordinate transformation. The other way is to use the perturbation theory with elements of Schwinger and Weinberg source approach. This latter method is applicable in general case and it is useful to study all its peculiarities on known simple example such as Schwarzschild solution. It turns out that this method is simpler then S-metrics approach (previously made by Duff) and more informative as it it shows which contribution comes from what region of space.