Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 449study of global properties of space–times relies to a large extent on theability to integrate the geodesic equations. In the absence of exact solutionsnumerical integration is often used to get a quantitative picture.However, in the quest for a deeper understanding the exact and numericalapproaches should be viewed as complementary tools. To perform an exactinvestigation of the global properties of a given space–time, not onlymust the space–time itself be an exact solution of the Einstein equations,but in addition the geodesic equations must be integrable. Usually, in ad−dimensional space, integrability of the geodesic equations is connectedwith the existence of at least d − 1 mutually commuting Killing vectorfields which span a hypersurface in the space–time. There are exceptionshowever. The most well–known example is the Kerr space–time which hasonly two commuting Killing vectors. In that case it is the existence ofan irreducible second rank Killing tensor which makes integration possible[Walker and Penrose (1970)]. Another example is given by Ozsvath’s classIII cosmologies [Ozsvath (1970)]. In that case the geodesic system was integratedusing the existence of a non–Abelian Lie algebra of Killing vectors[Rosquist (1980)]. In general integrability can only be guaranteed if thereis a set of d constants of the motion in involution (i.e. mutually Poissoncommuting). Since the metric itself always provides one constant of themotion corresponding to the squared length of the geodesic tangent vector,the geodesic system will be integrable by Liouville’s theorem if there ared − 1 additional Poisson commuting invariants.Exact solutions of Einstein’s equations typically admit a number ofKilling vector fields. Some of these Killing vector fields may be motivatedby physical considerations. For example if one is interested in static starsthe space–time must have a time–like Killing vector. For such systems itis also very reasonable to assume spherical spatial symmetry leading to atotal of four (non–commuting) Killing vectors. In most cases, the numberof Killing vectors is limited by the physics of the problem. In a sphericallysymmetric collapse situation, for example, the space–time admits exactlythree non–commuting Killing vector fields which form an isometry groupwith 2D orbits. That structure is not sufficient for an exact integrationof the geodesic equations. However, the physics of the problem does notimpose any a priori restrictions on higher rank (≥ 2) Killing tensors. AKilling vector field, ξ, plays a double role; it is both an isometry for the metric(L ξ g = 0) and a geodesic symmetry. This last property means that itcan be interpreted as a symmetry transformation for the geodesic equations.By contrast higher rank Killing tensors are only geodesic symmetries. They