Global SSS space-time models: $M_a$ and $Q$

Abstract

To make sense of a global space-time model and to give a meaning to the coordinates that we use, a choice of a constant curvature space-metric of reference it is as much necessary as it is a choice of units of mass, length and time. The choice we make leads to contradict the belief that the exterior domain of a Static Spherically Symmetric (SSS) space-time model of finite radius $R$ depends only on the active mass $M_a$ of the source. In fact it depends on two parameters $M_a$ and a new one $Q$. We prove that both can be calculated as volume integrals extended over the whole space. We integrate Einstein's equations numerically in two simple cases: assuming either that the source of perfect fluid has constant proper density or that the pressure depends linearly on the proper density. We confirm a preceding paper showing that very compact objects can have active masses $M_a$ much greater than their proper masses $M_p$, and we conjecture that the mass point Fock's model can be understood as the limit of a sequence of compact models when both Q and its radius shrink to zero and the pressure equals the density.