Critical Behavior in Gravitational Collapse of a Yang-Mills Field

Abstract

We present results from a numerical study of sphericallysymmetric collapse of a self-gravitating, SU(2) gauge field. Two distinct critical solutions are observed at the threshold of black hole formation. In one case the critical solution is discretely self-similar and black holes of arbitrarily small mass can form. However, in the other instance the critical solution is the n = 1 static Bartnik-Mckinnon sphaleron, and black hole formation turns on at finite mass. The transition between these two scenarios is characterized by the superposition of both types of critical behaviour. 04.25.Dm, 04.70.Bw In a recent numerical study of gravitational collapse of a massless scalar field, a type of critical behaviour was found at the threshold of black hole formation [1]. More precisely, in the analysis of spherically-symmetric evolution of various one-parameter families of initial data describing imploding scalar waves, it was observed that there is generically a critical parameter value, p = p ⋆ , which signals the onset of black hole formation. In the subcritical regime, p < p ⋆ , all of the scalar field escapes to infinity leaving flat spacetime behind, while for supercritical evolutions, p > p ⋆ , black holes form with masses well-fit by a scaling law, M BH ∝ (p−p ⋆ ) γ . Here, the critical exponent, γ ≃ 0.37, is universal in the sense of being independent of the details of the initial data. Thus, the transition between no-black-hole/black-hole spacetimes may be viewed as a continuous phase transition with the black hole mass playing the role of order parameter. In the intermediate asymptotic regime (i.e. before a solution "decides" whether or not to form a black hole) near-critical evolutions approach a universal attractor, called the critical solution, which exhibits discrete selfsimilarity (echoing). Using the same basic technique of studying families which "interpolate" between no-blackhole and black-hole spacetimes, similar critical behaviour has been observed in several other models of gravitational collapse [2][3][4].In this letter we summarize results from numerical study of the evolution of a self-gravitating non-abelian gauge field modeled by the SU (2) Einstein-Yang-Mills (EYM) equations. In addition to its intrinsic physical interest, we have chosen this model since, in contrast to all previously studied models, it contains static solutions which we suspected could affect the qualitative picture of critical behaviour. Let us recall that these static solutions, discovered by Bartnik and Mckinnon (BK) [5,6], form a countable family X n (n ∈ N ) of sphericallysymmetric, asymptotically flat, regular, but unstable, configurations.FIG. 1. Schematic representation of "phase-space" for spherically symmetric Yang-Mills collapse, showing possible end states of evolutions from a sufficiently general two-parameter family of initial conditions. The critical line OO ′ demarks the threshold of black hole formation. An interpolating family such as AA ′ exhibits Type I behaviour: the critical solution is the static BK s...

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