Fermat Principle and weak deflection angle from Lindstedt-Poincare method

TL;DR

This paper applies the Lindstedt-Poincaré method to analyze light deflection in curved spacetime, providing a systematic perturbative approach that accurately computes deflection angles up to third order without complex integrations.

Contribution

It introduces a novel application of the Lindstedt-Poincaré method to gravitational lensing, improving perturbative calculations of light trajectories in various metrics.

Findings

Deflection angles computed to third order.

Method preserves periodicity and boundedness of solutions.

No integrations or elliptic function expansions needed.

Abstract

The Fermat principle is advocated to be a convenient tool to analyze the light propagation in a curved space time. It is shown that in the weak deflection regime the light ray trajectories can be systematically described by applying the Lindstedt--Poincar\'e method of solving perturbatively the nonlinear oscillation equations. The expansion in terms of inverse invariant impact parameter for Schwarzschild, Reissner--Nordstr\"om and Kerr (equatorial motion) metrics is described. The corresponding deflection angles are computed to the third order. Only algebraic operations are involved in the derivation; no integrations or Fourier expansion of elliptic functions are necessary. It is argued, that contrary to the naive perturbative expansion, the Lindstedt--Poincar\'e approach correctly represents the main properties of light propagation in asymptotic regime. At each step it preserves the periodicity of the relevant nonlinear oscillations of the inverse radial coordinate which allows to group the trajectories with the same invariant impact parameter into disjoint sets of ones generated by particular oscillations. Moreover, it allows for partial summation of perturbative expansion leading to the uniformly bounded approximations.