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Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in general relativity. (English) Zbl 1213.83039

Summary: The description of many dynamical problems like the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of a holomorphic hyperelliptic integral. The result of the inversion is defined only locally, and is done using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the \(\theta \)-divisor. For a representation of the hyperelliptic functions the Klein-Weierstraß multivariable sigma function is introduced. It is shown that all parameters needed for the calculations like period matrices and Abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and theta-constants. The cases of genus 2 and genus 3 are considered in detail. The method is exemplified by particle motion associated with a genus 3 hyperelliptic curve.

MSC:

83C10 Equations of motion in general relativity and gravitational theory
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
83C57 Black holes
83C22 Einstein-Maxwell equations
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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