Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions

It is known that all spatially homogeneous solutions of the vacuum Einstein equations in four dimensions which exist for an infinite proper time towards the future are future geodesically complete. This paper investigates whether the analogous statement holds in higher dimensions. A positive answer to this question is obtained for a large class of models which can be studied with the help of Kaluza-Klein reduction to solutions of the Einstein-scalar field equations in four dimensions. The proof of this result makes use of a criterion for geodesic completeness which is applicable to more general spatially homogeneous models.Comment: 18 page

Die lineare Instabilität der Uniform Black String Lösungen

Black strings are black hole solutions of Einstein’s equations in more than four dimensions. In spacetime dimensions n + 1 they have horizon topology S^(n−1) × S^1. Using numerical simulations, it was discovered in the 1990s that black strings are linearly unstable. Since then, most research surrounding this result was focused on understanding physical aspects of the instability. In this thesis we will return to the original problem of linear stability and prove an existence result for a special class of mode solutions. This constitutes an important step towards proving linear instability.Black strings sind schwarze Löcher und Lösungen der Einsteinschen Feldgleichungen in mehr als vier Dimensionen. In Raumzeitdimension n + 1 hat ihr Ereignishorizont die Topologie S^(n−1) × S^1. Durch numerische Simulationen wurde in den 1990ern entdeckt, dass black strings linear instabil sind. Ausgehend von diesem Resultat lag der Schwerpunkt der Forschung darauf, die physikalischen Aspekte dieser Instabilität zu verstehen. In dieser Arbeit werden wir zu dem Problem der linearen Instabilität zurückkehren und die Existenz einer speziellen Familie von Lösungen beweisen. Dies ist ein wichtiger Schritt für den Beweis der linearen Instabilität