Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2011-06-13
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
34 pages, 3 figures, 1 table; We extended the sections 'Introduction' and 'Conclusions and Outlook' and made some cosmetic cha
Scientific paper
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 hyperelliptic integrals of all three kinds. The result of the inversion is defined locally, 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--Weierstra{\ss} 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 two, three and four are considered in detail. The method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves. Applications are for instance solutions to the geodesic equations in the space-times of static, spherically symmetric Ho\v{r}ava-Lifshitz black holes.
Enolski Victor
Hartmann Betti
Kagramanova Valeria
Kunz Jutta
L{ä}mmerzahl Claus
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