Fermat Principle and weak deflection angle from Lindstedt-Poincare method
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| Format: | Preprint |
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2024
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| _version_ | 1866912304429268992 |
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| author | Piwnik, Joanna Gonera, Joanna Kosinski, Piotr |
| author_facet | Piwnik, Joanna Gonera, Joanna Kosinski, Piotr |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fermat Principle and weak deflection angle from Lindstedt-Poincare method Piwnik, Joanna Gonera, Joanna Kosinski, Piotr General Relativity and Quantum Cosmology Mathematical Physics 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. |
| title | Fermat Principle and weak deflection angle from Lindstedt-Poincare method |
| topic | General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2405.14462 |