Conserved quantities for spinning test particles in general relativity

Abstract

In 1937, Myron Mathisson initiated a research program with the aim of describing extended bodies in General Relativity, similar to what is done with rigid bodies in Classical Mechanics. In his theory, the internal structure of the bodies is described by using the so-called multipolar moments. When considering the spin of the particle beyond its translational motion, the motion of the particle is not necessarily geodesic anymore, being described by the so-called Mathisson-Papapetrou-Dixon (MPD) equations. In the early 1980s, R. Rudiger published two articles dedicated to study the most general conserved charges associated to the motion of particles with spin moving in a curved spacetime and described by the MPD equations. In particular, it was shown that, besides the well-known conserved quantity associated to Killing vectors, there is also another conserved quantity that is linear in the spin of the particle and which involves a Killing-Yano tensor. It was also shown in these papers that in order of this new scalar to be conserved, two obscure conditions involving the Killing-Yano tensor and the curvature tensor must be satisfied. In this thesis, we seek to shed light over these conditions, by trying delimitate the possible spacetimes in which these conditions are satisfied, i.e. in which this new conserved quantity is allowed to exist. In order to extract the content of the Rudiger conditions, some essential mathematical tools are introduced along the text. In this sense, we discuss about the so-called integrability conditions for the Killing-Yano tensors and also we introduce the Petrov classification, which is an important algebraic classification for the Weyl tensor valid in four-dimensional spacetimes. However, before discuss these tools, a revision concerning Mathisson’s theory and the MPD equations is made.