Killing Tensors in Koutras-McIntosh Spacetimes

Abstract

This thesis is concerned with the (non)existence of Killing Tensors in Koutras-McIntosh spacetimes. Killing tensors are of particular interest in general relativity, because these correspond to conserved quantities for the geodesic motion. For instance, Carter found such a conserved quantity in the Kerr metric which he used to explicitly integrate the geodesic equations.

The equation defining a Killing tensor is actually an overdetermined linear first order partial differential equation. We shall study the Killing equation using methods from the geometric theory of PDEs. More precisely, we use Cartan's prolongation method to prove the (non)existence of Killing tensors in several Koutras-McIntosh spacetimes. A subclass of the Koutras--McIntosh spacetimes are the conformally flat pp-waves. We show that a generic conf. flat pp-wave has an irreducible Killing 2-tensor, which reproves a result obtained by Keane and Tupper using a different method. Moreover, we prove in particular examples of pp-waves that all Killing tensors of degree 3 and 4 are reducible.

We then study the Wils metric, another subclass of the Koutras-McIntosh spacetimes. This metric has a univariate function as its parameter. By using Cartan's prolongation method we deduce the explicit form of the function for which the Wils metric admits a Killing vector, and for which a Killing 2-tensor. This existence result for a Killing vector makes a statement by Koutras and McIntosh more precise. Finally, we show in particular examples of a Wils metric that all Killing 3- and 4-tensors are reducible.