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

TL;DR

This paper derives exact static solutions for spherically symmetric scalar fields coupled to gravity in four dimensions, analyzing their stability and identifying attractor solutions within the system.

Contribution

It provides explicit static solutions for scalar fields coupled to gravity and analyzes their stability, including the identification of attractors and saddle points.

Findings

Exact static solutions for scalar fields in 4D gravity

Trivial solution $=0$ is a global attractor in certain regions

Non-vacuum solution $=1/4$ is a stable focus and attractor

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 $\mu=m/r$ and $y=(\alpha/ra)$, 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", \textit{Physical Review 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 $\mu=1/4$ is a stable focus and a global attractor for the region $\mu_s+\mu>0$, $\mu<1/2$.