Computing the Arnold Tongue in the Zipoy-Voorhees Space-time

[摘要] ENGLISH ABSTRACT : In this thesis I study the integrability of the geodesic equations of the ZipoyVoorheesmetric. The Zipoy-Voorhees spacetime is a one parameter family ofStationary Axisymmetric Vacuum spacetimes (SAV's) that is an exact solution tothe vacuum Einstein Field Equations (EFE's). It has been conjectured that the endstate of any asymptotically flat black hole formed by astrophysical mechanisms,such as for example, gravitational collapse of a star, merger of two black holesetc will be a characterised by the Kerr metric. The black hole will thus be apossibly rotating, stationary axisymmetric vacuum spacetime characterised by itsmass and spin and will possess no closed time-like curves. Investigating orbits inthe Zipoy-Voorhees spacetime serves as a concrete example to of how the Kerrhypothesis fails. For this metric, I compute the Poincaré map and then computethe rotation curve. The Poincaré map is a tool to locate the region where chaosoccurs in a dynamical system. The rotation curve is used to quantify chaos in thesystem. I focus my study on the 2/3 resonance for a range of the parameter valuesδ ∈ [1, 2]. The value δ = 1 corresponds to the Schwarzschild solution where thesystem is integrable. I then compute the Arnold tongue by plotting the size ofthe resonant regions against the parameter values to quantify the departure fromintegrability. I find that the shape of the tongue of instability is nonlinear and theArnold tongue pinches off at δ = 1.6.

[发布机构] Stellenbosch University