Browsing by Author "Sherif, Abbas Mohamed"

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    Computing the Arnold Tongue in the Zipoy-Voorhees Space-time
    (Stellenbosch : Stellenbosch University, 2017-03) Sherif, Abbas Mohamed; Brink, Jeandrew; Stellenbosch University. Faculty of Science. Dept. of Physics
    ENGLISH ABSTRACT : In this thesis I study the integrability of the geodesic equations of the ZipoyVoorhees metric. The Zipoy-Voorhees spacetime is a one parameter family of Stationary Axisymmetric Vacuum spacetimes (SAV’s) that is an exact solution to the vacuum Einstein Field Equations (EFE’s). It has been conjectured that the end state of any asymptotically flat black hole formed by astrophysical mechanisms, such as for example, gravitational collapse of a star, merger of two black holes etc will be a characterised by the Kerr metric. The black hole will thus be a possibly rotating, stationary axisymmetric vacuum spacetime characterised by its mass and spin and will possess no closed time-like curves. Investigating orbits in the Zipoy-Voorhees spacetime serves as a concrete example to of how the Kerr hypothesis fails. For this metric, I compute the Poincaré map and then compute the rotation curve. The Poincaré map is a tool to locate the region where chaos occurs in a dynamical system. The rotation curve is used to quantify chaos in the system. I focus my study on the 2/3 resonance for a range of the parameter values δ ∈ [1, 2]. The value δ = 1 corresponds to the Schwarzschild solution where the system is integrable. I then compute the Arnold tongue by plotting the size of the resonant regions against the parameter values to quantify the departure from integrability. I find that the shape of the tongue of instability is nonlinear and the Arnold tongue pinches off at δ = 1.6.