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

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}.

💡 Research Summary

The paper addresses a gap in the theoretical foundation of high‑precision light‑propagation modeling in the Schwarzschild spacetime. In earlier works (referred to as \cite{report1} and \cite{report2}) the authors derived an analytical solution for light trajectories up to the post‑post‑Newtonian (PPN) order and, based on heuristic arguments, claimed that all terms of order ( {\cal O}(m^{2}/d^{2}) ) and ( {\cal O}(m^{2}/d_{\sigma}^{2}) ) are bounded respectively by ( \frac{15}{4\pi},m^{2}/d^{2} ) and ( \frac{15}{4\pi},m^{2}/d_{\sigma}^{2} ). Those bounds justify discarding the quadratic‑mass contributions when the required astrometric accuracy is at the micro‑arcsecond level, thereby simplifying the transformation formulas used in practical data reduction pipelines. However, the original papers did not present a rigorous proof of these inequalities.

The present report fills that void by providing a complete, step‑by‑step mathematical demonstration. The authors start by cataloguing every quadratic‑mass term that appears in the PPN expansion of the light‑deflection formula. Each term is expressed as a function of a single geometric angle (\theta), which measures the inclination between the unperturbed light direction and the line joining the photon to the gravitating mass. The distance variables (d) (the impact parameter) and (d_{\sigma}) (the projection of the source‑observer line onto the plane orthogonal to the light direction) appear only as overall scaling factors (m^{2}/d^{2}) or (m^{2}/d_{\sigma}^{2}).

The core of the proof consists of three technical stages. First, the authors compute the derivative of each angular function (f_i(\theta)) and establish its monotonicity on the physically relevant interval (0\le\theta\le\pi/2). This guarantees that the maximum of each term occurs at one of the interval endpoints or at a critical point that can be identified analytically. Second, they apply elementary trigonometric inequalities—such as (\sin x\le x), (\cos x\ge 1-x^{2}/2), and the bound (\frac{\sin x}{1+\cos x}\le x)—to replace the original expressions with simpler upper‑bounding functions (g_i(\theta)). Third, they sum the bounds (g_i(\theta)) to obtain a single composite function (G(\theta)). By differentiating (G(\theta)) and solving (G’(\theta)=0), they locate the global maximum, which turns out to be attained at (\theta=0) (or equivalently at (\theta=\pi/2) due to symmetry). The explicit evaluation yields (G_{\max}=15/(4\pi)).

Multiplying this dimensionless maximum by the common prefactors restores the original physical quantities, giving the final inequalities
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