Fermat Principle and weak deflection angle from Lindstedt-Poincare method

TL;DR

The paper addresses light propagation in curved spacetime within the weak-deflection regime by formulating the problem via Fermat's principle and solving the resulting nonlinear oscillator equations with the Lindstedt-Poincaré method. It derives a perturbative, algebraic expansion in the inverse invariant impact parameter $1/b$ to third order for Schwarzschild, Reissner–Nordström, and Kerr (equatorial) geometries, avoiding elliptic integrals. The approach preserves the solutions' periodic structure and yields uniformly bounded approximations, with Kerr and RN corrections captured through a frequency–amplitude relation. The work provides a systematic, scalable framework for gravitational lensing calculations and insights into nonlinear frequency shifts in curved spacetime.

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--Poincaré method of solving perturbatively the nonlinear oscillation equations. The expansion in terms of inverse invariant impact parameter for Schwarzschild, Reissner--Nordström 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--Poincaré 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.

Figures (14)

• Figure 1: The geometry of deflection angle.
• Figure 2: The potential corresponding to the Schwarzschield metric. Note that the value $u=\frac{1}{3}$ corresponds to the radius of circular light rays.
• Figure 3: The potential corresponding to the Reissner--Nordström black-hole; $u_+$ and $u_-$ correspond to the outer and inner horizons, respectively.
• Figure 4: Trajectory in the Schwarzschield metric for $\frac{m}{b} = 0.135$; solid line represents exact solution and the dashed line -- the approximate one.
• Figure 5: Trajectory in the Schwarzschield metric for $\frac{m}{b} = 0.1$; solid line represents exact solution and the dashed line -- the approximate one.