Formal proof of some inequalities used in the analysis of the post-post-Newtonian light propagation theory

TL;DR

This paper provides rigorous mathematical proofs for inequalities used in simplifying post-post-Newtonian light propagation equations in Schwarzschild spacetime, ensuring their validity at microarcsecond accuracy levels.

Contribution

It offers formal proofs for key inequalities that justify neglecting certain terms in light propagation analysis at high precision.

Findings

Validated inequalities used in light propagation approximations

Confirmed neglecting higher-order terms at microarcsecond accuracy

Strengthened the mathematical foundation of post-post-Newtonian light theory

Abstract

A rigorous analytical solution of light propagation in Schwarzschild metric in post-post Newtonian approximation has been presented in \cite{report1}. In \cite{report2} it has been asserted that the sum of all those terms which are of order ${{\cal O} (\frac{m^2}{d^2})}$ and ${{\cal O}(\frac{m^2}{d_\sigma^2})}$ is not greater than $15/4 \pi \frac{m^2}{d^2}}$ and $15/4 \pi \frac{m^2}{d_\sigma^2}}$, respectively. All these terms can be neglected on microarcsecond level of accuracy, leading to considerably simplified analytical transformations of light propagation. In this report, we give formal mathematical proofs for the inequalities used in the appendices of \cite{report2}.