Topics in the dynamics of general relativity
Fischer, A. E. and J. E. Marsden
Isolated Gravitating Systems in General Relativity, Italian Physical Society, (J. Ehlers, ed.), 322-395
In this chapter we discuss some of the inter-relationships between the initial value problem, the canonical formalism, linearization stability and the space of gravitational degrees of freedom. In the last decade, these topics have experienced a resurgence of interest as more advanced mathematical methods and viewpoints have begun to show the intimate relationships among these topics. At present, the literature regarding these areas of general relativity is a rapidly expanding body of knowledge.
Our purpose here is to present the current state of affairs from our own point of view. We shall use geometric methods developed by the authors to establish various connections between the above-mentioned topics. The main tools we shall use to develop this material are nonlinear functional analysis, an adjoint formalism for Hamiltonian field theories, and infinite-dimwensional symplectic geometry. As we shall see, these tools and the topics we shall consider are naturally related.