Minimum size of dense source distributions in general relativity

Abstract

The properties of very dense sources of the gravitational field are examined. The sources are spherically symmetric initially static distributions of neutral or charged dust, treated as dynamical systems with no phenomenological pressure term. The exterior Schwarzschild solution is analyzed from this point of view as generated by a realistic "bare" matter stress-tensor with appropriate initial Cauchy data. A coordinate frame is introduced in which the radial coordinate coincides with the invariant distance (grr = 1), and the initial value equations are solved in this frame. The systems considered become, in the limit, models for neutral and charged particles, and so provide solutions for the problems of interaction between "point" particles and the gravitational field. It is shown that, at the instant of time symmetry, there is a minimum invariant extension for a particle, below which no solution of the field equations exists; this fact emerges especially clearly in the invariant frame used here. The clothed (or exterior) mass and invariant properties of the interacting systems as a function of the bare mass, charge, and extension are given. The results are in agreement with those obtained previously in another frame, in terms of whose radial coordinate the particle had vanishing extension in the limit. The significance of the necessary departure of a particle from strictly pointlike structure as a result of its gravitational coupling is discussed.

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