The Kepler Problem and Its Relation to Extremal Black Holes

Anyszka, Wojciech (2023) The Kepler Problem and Its Relation to Extremal Black Holes. Bachelor's Thesis, Physics.

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Abstract

This thesis explores the unique properties of the classical Kepler problem, including Bertrand's theorem, the connection between Kepler and harmonic oscillator potentials, and the existence of an additional conserved quantity --- the Laplace-Runge-Lenz (LRL) vector. The role of symmetries in this context is explored. Then a comprehensive proof of Moser's construction, establishing the correspondence between nonconstant geodesics on an n-dimensional sphere and Kepler orbits with negative energies in n-dimensions, is presented. This construction demonstrates that the Kepler problem has a larger symmetry group compared to an arbitrary central potential. The relativistic corrections to two-body problems, which generically induce perihelion precession, are then investigated. Notably, a specific relativistic system involving an extremal test particle near an oppositely charged extremal Einstein-Maxwell-dilaton black hole does not exhibit perihelion precession. However, this phenomenon is limited to a specific value of the dilaton coupling constant, specifically a=\sqrt{3}. A generalized theorem based on this construction is established, followed by an examination of cases where a\neq \sqrt{3}. It was shown that away from a=\sqrt{3}, the test-particle orbits correspond to the orbits of a perturbed Kepler problem.

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Seri, M. and Roest, D.
Degree programme:	Physics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	08 Nov 2023 10:12
Last Modified:	08 Nov 2023 10:12
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/31614

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