The Axisymmetric Case for the Post-Newtonian Dedekind Ellipsoids

We consider the post-Newtonian approximation for the Dedekind ellipsoids in the case of axisymmetry. The approach taken by Chandrasekhar & Elbert (1974, 1978) excludes the possibility of finding a uniformly rotating (deformed) spheroid in the axially symmetric limit, though the solution exists at the point of axisymmetry. We consider an extension to their work that permits the possibility of such a limit.Comment: 12 pages, v1: more discussion of comparison to Chandrasekhar's and Elbert's work, modified to agree with published versio

The Parametric Transition of Strange Matter Rings to a Black Hole

It is shown numerically that strange matter rings permit a continuous transition to the extreme Kerr black hole. The multipoles as defined by Geroch and Hansen are studied and suggest a universal behaviour for bodies approaching the extreme Kerr solution parametrically. The appearance of a `throat region', a distinctive feature of the extreme Kerr spacetime, is observed. With regard to stability, we verify for a large class of rings, that a particle sitting on the surface of the ring never has enough energy to escape to infinity along a geodesic.Comment: 16 pages, 11 figures, v3: minor changes so as to coincide with published versio

Uniformly Rotating Homogeneous Rings in post-Newtonian Gravity

In this paper uniformly rotating relativistic rings are investigated analytically utilizing two different approximations simultaneously: (1) an expansion about the thin ring limit (the cross-section is small compared with the size of the whole ring) (2) post-Newtonian expansions. The analytic results for rings are compared with numerical solutions.Comment: 12 pages, 7 figures, v1: 2 tables added, agrees with published versio

Die Maclaurin-Ellipsoide in post-Newtonscher Näherung beliebig hoher Ordnung

Zusammenfassung In dieser Dissertation sind rotierende, axialsymmetrische, stationaere Flüssigkeiten konstanter Dichte im Rahmen der allgemeinen Relativitätstheorie untersucht worden. Ein iteratives Verfahren zur Bestimmung einer beliebigen Ordnung der post-Newtonschen (PN) Näherung wird vorgestellt. Anhand dieses Verfahrens wird bewiesen, dass der PN-Näherung Singularitäten im Parameterraum besitzt, die mit Bifurkationspunkten der Newtonschen Maclaurin-Sequenz zusammenhängen. Die ersten vier PN-Ordnungen werden explizit bestimmt und ihre Konvergenz durch einen Vergleich mit hoch-genauen numerischen Ergebnissen untersucht.Summary The topic of this dissertation is the study of rotating, axially symmetric, stationary fluid bodies of constant density within the theory of General Relativity. An iterative scheme to enable the determination of an arbitrary order of the post-Newtonian (PN) approximation is presented. Using this scheme, it is proved that the PN approximation becomes singular at points in a given parameter space that are related to bifurcations along the Newtonian Maclaurin sequence. The first four PN-orders are determined explicitly and their convergence is studied by comparison with highly accurate numerical results