Exact static solutions for scalar fields coupled to gravity in (3+1)- dimensions - Bilge, Ayse H.; Daghan, Durmus (World Scientific Publishing Company, 2008)
Einstein's field equations for a spherically symmetric metric coupled to a massless scalar field are reduced to a system effectively of second order in time, in terms of the variables $=m/r$ and $y=(/ra)$, where $a$, $$, $r$ and $m$ are as in [W.M. Choptuik, ``Universality and Scaling in Gravitational Collapse of Massless Scalar Field", \textit{Physical Review Letters} {\bf{70}} (1993), 9-12]. Solutions for which $$ and $y$ are time independent may arise either from scalar fields with $_t=0$ or with $_s=0$ but $$ linear in $t$, called respectively the positive and negative branches having the Schwarzschild solution characterized by $=0 $ and $_s+=0$ in common. For the positive branch we obtain an exact solution which have been in fact obtained first in [I.Z. Fisher,``Scalar mesostatic field with regard for gravitational effects", \textit{Zh. Eksp. Teor. Fiz.} {\bf{18}} (1948), 636-640, gr-qc/9911008] and rediscovered many times (see D. Grumiller, ``Quantum dilaton gravity in two dimensions with matter", PhD thesis, \textit{Technische Universit$\ddot{a}$t, Wien} (2001), gr-qc/0105078) and we prove that the trivial solution $=0$ is a global attractor for the region $_s+>0 $, $<1/2$. For the negative branch discussed first in [M. Wyman, ``Static spherically symmetric scalar fields in general relativity", \textit{Physical Review D} {\bf{24}} (1981), 839-841] perturbatively, we prove that $=0$ is a saddle point for the linearized system, but the non-vacuum solution $=1/4$ is a stable focus and a global attractor for the region $_s+>0$, $<1/2$.

13 pages, 10 figures. Replaced by a revised version
Keywords: FOS: Physical sciences; General Relativity and Quantum Cosmology (gr-qc); General Relativity and Quantum Cosmology