Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum

W. G. Dixon


Definitions are proposed for the total momentum vector p$^\alpha$ and spin tensor S$^{\alpha\beta}$ 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$_\beta$S$^{\alpha\beta}$ = 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 $\Phi$ 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 $M = m + \frac{1}{2}S^\kappa\Omega_\kappa + \Phi$, where $\Omega_\kappa$ is the angular velocity of the body and S$^\kappa$ its spin vector.

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