Exact static solutions for scalar fields coupled to gravity in (3+1)- dimensions

Abstract

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