Journal Article

An analytic solution for weak-field Schwarzschild geodesics

Daniel J. D'Orazio and Prasenjit Saha

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 406, issue 4, pages 2787-2792
Published in print August 2010 | ISSN: 0035-8711
Published online August 2010 | e-ISSN: 1365-2966 | DOI:
An analytic solution for weak-field Schwarzschild geodesics

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It is well known that the classical gravitational two-body problem can be transformed into a spherical harmonic oscillator by regularization. We find that a modification of the regularization transformation has a similar result to leading order in general relativity. In the resulting harmonic oscillator, the leading-order relativistic perturbation is formally a negative centrifugal force. The net centrifugal force changes sign at 3 Schwarzschild radii, which interestingly mimics the innermost stable circular orbit of the full Schwarzschild problem. Transforming the harmonic-oscillator solution back to spatial coordinates yields, for both time-like and null weak-field Schwarzschild geodesics, a solution for t, r, ϕ in terms of elementary functions of a variable that can be interpreted as a generalized eccentric anomaly. The textbook expressions for relativistic precession and light deflection are easily recovered. We suggest how this solution could be combined with additional perturbations into numerical methods suitable for applications such as relativistic accretion or dynamics of the Galactic Centre stars.

Keywords: gravitation; Galaxy: centre

Journal Article.  3295 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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