Dynamics of extended bodies in general relativity. I. Momentum and angular momentum

Abstract

Definitions are proposed for the total momentum vector p α and spin tensor S αβ of an extended body in arbitrary gravitational and electromagnetic fields. These are based on the requirement that a symmetry of the external fields should imply conservation of a corresponding component of momentum and spin. The particular case of a test body in a de Sitter universe is considered in detail, and used to support the definition p β S αβ = 0 for the centre of mass. The total rest energy M is defined as the length of the momentum vector. Using equations of motion to be derived in subsequent papers on the basis of these definitions, the time dependence of M is studied, and shown to be expressible as the sum of two contributions, the change in a potential energy function ϕ and a term representing energy inductively absorbed, as in Bondi’s illustration of Tweedledum and Tweedledee. For a body satisfying certain conditions described as ‘dynamical rigidity’, there exists, for motion in arbitrary external fields, a mass constant m such that M = m + ½ S κ Ω κ + ϕ , where Ω k is the angular velocity of the body and S κ its spin vector.