Scalar fields coupled to gravity in (3+1) dimensions: Static spherically symmetric stiff matter solutions

Abstract

Einstein’s field equations for a spherically symmetric metric and a massless scalar field source are reduced to a system effectively of second order in time for the metric function μ. The solutions of this system split into two classes that we called the positive and negative branches, corresponding to scalar fields with spacelike and timelike gradients. Static solutions for the positive branch are well known, but the proof that μ=0 is a global attractor for the region μs+μ>0, μ<1∕2 is given for completeness. The negative branch of static solutions have the interpretation of a perfect fluid with pressure equal to mass density, called “stiff matter.” The main result of the paper is the characterization of static solutions of the negative branch by phase plane analysis. We prove that the solution μ=1∕4 is a global attractor for the region where μs+μ>0 and μ<1∕2. Two first integrals, one of which reducing to the solution μ=1∕4 in the static case, are also presented.