Minimum size of charged particles in general relativity

Spherical charged matter distributions are examined in a coordinate-free manner within the framework of general relativity. Irrespective of models chosen to describe the interior structure of a charged particle, it is found that the latter's total gravitational mass is positive definite, being finite only when there exists a lower bound for its invariant extension. For a simple choice of matter and charge distributions it is then shown that there is a minimum invariant size for the particle, below which no solution of the field equations exists, the matter density becoming negative and the spacetime developing an intrinsic singularity in the exterior of the particle for radii less than this minimum. A mass renormalization is derived, valid at the moment of time symmetry, which relates the particle's total mass to its charge, bare mass and invariant extension. Our results are compared with those obtained previously by Arnowitt, Deser and Misner, who consider the simpler distribution of a charged spherical shell. Qualitatively, the two situations share the same features. However, in the more realistic spherical distributions the formulae are correspondingly more complicated, and the minimum extension is found to be greater than that of the shell, as one might expect on physical grounds. Moreover, the correspondence between negative valued matter distributions and intrinsic singularities was not evident in the shell case.