Exact Static Solutions for Scalar Fields Coupled to Gravity in $(3+1)$-Dimensions

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 $\mu =m\mathrm{/}r$ and $y=(\alpha \mathrm{/}ra)$, where $a$, $\alpha$, $r$ 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 $\mu $ and $y$ are time independent may arise either from scalar fields with $\phi_t=0$ or with $\phi_s=0$ but $\phi$ linear in $t$, called respectively the positive and negative branches having the Schwarzschild solution characterized by $\phi=0 $ and $\mu_s+\mu=0$ 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 $\mu=0$ is a global attractor for the region $\mu_s+\mu>0 $, $\mu<1/2$. For the negative branch discussed first in [M. Wyman, Static spherically symmetric scalar fields in general relativity", $\mathit{\text{Physical}}\mathit{\text{Review}}\mathit{\text{D}}$ {\bf{24}} (1981), 839-841] perturbatively, we prove that $\mu =0$ is a saddle point for the linearized system, but the non-vacuum solution $\mu =1\mathrm{/}4$ is a stable focus and a global attractor for the region ${\mu }_s+\mu >0$, $\mu <1\mathrm{/}2$.