Critical Phenomena Inside Global Monopoles

Quite an industry has developed around the study of nonlinear phenomena occurring at the threshold of black hole formation first discovered by Choptuik [1]. A number of reviews detail the substantial progress in understanding these phenomena [2,3]. By specifying a form of initial data parameterized by some parameter p, numerical evolutions can determine whether the value of p is sufficient to form a black hole. Such a solution is considered super-critical, while for small p, no black hole forms and the solution is sub-critical. Separating supercritical solutions from sub-critical solutions is the critical solution parameterized by the critical value p∗. In analogy with statistical mechanics where the black hole mass represents the order parameter, cases where infinitesimal mass black holes are formed are called Type II, and cases in which only finite mass black holes are formed are called Type I. Here, I present a new Type II critical solution occurring within a model with a triplet scalar field. By assuming a hedgehog ansatz for the triplet, I find a critical solution which is, in a strict sense, non-spherically symmetric. Adding a potential allows the study of critical phenomena within a global monopole. No Type I critical behavior around the static monopole solutions is observed. As a generalization of Choptuik’s original model, the triplet scalar field contains the single scalar field model within a certain region of parameter space. However because of the different assumed symmetries of the fields, the hedgehog ansatz excludes this region. Hence, it is not surprising that I find a new solution, one which is discretely self-similar (DSS) with echoing period ∆ = 0.46, much smaller than that for either the original DSS (∆ = 3.44) or that of the Einstein Yang-Mills (EYM) model (∆ = 0.74) [4]. An ansatz similar to this one (an SU(2) valued hedgehog ansatz) was used to study the collapse of Skyrmions [5–7]. Both these models allow the consideration of non-spherically symmetric initial data, albeit quite special data, within the comparatively simple framework of one spatial dimension, namely a radius r. In Section II, I present the most general form of the model under consideration here. First, I choose a triplet scalar field Φ accompanied by the usual symmetrybreaking potential with vacuum value |Φa| = η and coupling λ. For certain evolutions discussed later, I also introduce a free, massless scalar field ψ(r, t) which couples to Φ only through gravity. With the most general form of the equations of motion, Section III then discusses various aspects of regions of the parameter space. In particular, Section III A presents a new critical solution obtained with the hedgehog ansatz, and Section III B discusses the stability of this solution with respect to the addition of ψ, a free scalar field. Section III C addresses the results obtained in the interior of a monopole. I then conclude in Section IV.