On the Analyticity Property of Static Non-Vacuum Solutions of Einstein's Eqs

The analyticity property of static, non vacuum solutions of Einstein's equations is discussed. We examine and compare differentiability properties of static solutions of two non vacuum set of eqs: The Einstein-Klein-Gordon massless minimally coupled to gravity scalar field equations and Einstein-Klein-Gordon massless conformally coupled to gravity scalar field eqs. All C³ solutions of the former system turn out to be analytic relative to a harmonic atlas covering the static region and this analyticity property holds true for both metric and the scalar field. However that is not the case for the conformal system. The coupled eqs become singular on static solutions (g, Φ) subject to vanishing of 1 - Φ² within the static region. Despite the occurrence of this singularity we show that the conformal system admits at least one class of static solutions for which even though 1 - Φ² = 0 within the static region, the solutions are real analytic in the vicinity of the degeneracy.