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 μ = m/r and y = (α/ra), where a, α, r and m are as in [W.M. Choptuik, “Universality and Scaling in Gravitational Collapse of Massless Scalar Field”, Physical Review Letters 70 (1993), 9-12]. Solutions for which μ and y are time independent may arise either from scalar fields with φt = 0 or with φs = 0 but φ linear in t, called respectively the positive and negative branches having the Schwarzschild solution characterized by φ = 0 and μs+μ = 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”, Zh. Eksp. Teor. Fiz. 18 (1948), 636-640, gr-qc/9911008] and rediscovered many times (see D. Grumiller, “Quantum dilaton gravity in two dimensions with matter”, PhD thesis, Technische Universität, Wien (2001), gr-qc/0105078) and we prove that the trivial solution μ = 0 is a global attractor for the region μs + μ > 0, μ < 1/2. For the negative branch discussed first in [M. Wyman, “Static spherically symmetric scalar fields in general relativity”, Physical Review D 24 (1981), 839-841] perturbatively, we prove that μ = 0 is a saddle point for the linearized system, but the non-vacuum solution μ = 1/4 is a stable focus and a global attractor for the region μs + μ > 0, μ < 1/2.