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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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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

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