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An analytic solution for weak-field Schwarzschild geodesics


D'Orazio, D J; Saha, P (2010). An analytic solution for weak-field Schwarzschild geodesics. Monthly Notices of the Royal Astronomical Society, 406(4):2787-2792.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute for Computational Science
Dewey Decimal Classification:530 Physics
Language:English
Date:August 2010
Deposited On:02 Mar 2011 14:32
Last Modified:21 Nov 2017 23:09
Publisher:Wiley-Blackwell
ISSN:0035-8711
Additional Information:The definitive version is available at www.blackwell-synergy.com
Publisher DOI:https://doi.org/10.1111/j.1365-2966.2010.16879.x
Related URLs:http://arxiv.org/abs/1003.5659

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Download PDF  'An analytic solution for weak-field Schwarzschild geodesics'.
Preview
Content: Accepted Version
Filetype: PDF (Accepted manuscript, Version 2)
Size: 165kB
Download PDF  'An analytic solution for weak-field Schwarzschild geodesics'.
Preview
Content: Accepted Version
Filetype: PDF (Accepted manuscript, Version 1)
Size: 165kB