ELSEVIER 18 July 1996 Physics Letters B 381 (1996) 49-55 PHYSICS LETTERS B Evolution and fate of perturbed Bartnik-McKinnon spacetime Shuxue Ding ’ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO Department of Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 15.2, Japan Received 15 May 1995; revised manuscript received 2 October 1995 Editor: M. Dine zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Abstract The evolution and the fate of a bubble-perturbed Bartnik-McKinnon (BK) spacetime are investigated. The infinitesimal neighborhood of the origin in a BK spacetime is shown to be equivalent to a part of a Yang-Mills cosmology (YMC) spacetime. If an arbitrarily chosen small region near the origin in the BK spacetime is replaced by a YMC bubble with the same curvature, the YMC-BK system describes a small perturbed BK spacetime. The bubble can expand as a part of the YMC spacetime in the BK spacetime. There is a thin shell between the YMC bubble and the BK spacetime whose trajectory is non-spacelike. The instability of the BK spacetime is described by the evolution of the bubble. 1. Introduction Previously Bartnik and McKinnon (BK) found a series of particle-like solutions for the coupled Einstein-Yang-Mills system [ 11. One of properties of the solution is the instability [ 21. The instability can be attributed to a sphaleron with respect to the Yang-Mills field configuration space [3]. Besides the interest in the structure of the field configuration space, surprisingly, physical interests may also be relevant to these solutions in spite of their instability. As an example, fermion particles can be created in the BK spacetimes [ 41. Another question which is physically interesting is the evolution and the fate of the BK spacetime with a small perturbation. The issue has not been worked out completely. We shall engage in the discussions of the evolution of a specific kind of perturbed BK space- time and the fate in this letter. Zhou and Straumann previously discussed evolutions of two different kinds ’ E-mail: sding@th.phys.titech.ac.jp. of perturbed BK spacetimes numerically [ 51. They conjectured that a perturbed BK spacetime will either collapse into a Schwarzschild black hole or explode as a gauge boson “mini star”. However, the discussion is effective only in a short period; thus the discussion is far from complete. What is more, their numerical results gave no hope to the discussion of global prop- erties of an evolution of the solution. Then from the results we could understand little about the final fate of the evolution. However, the discussion of Zhou and Straumann gave us an important clue for our present discussions. The development of one of perturbed BK spacetimes dealt with by them acts very similar to a bubble with a Yang-Mills shell configuration. With time going, the configuration becomes thinner. This fact tells us that a thin shell formalism should be suitable for a discus- sion, even though it makes a direct numerical simula- tion difficult. In fact, Gibbons and Steif already mentioned this similarity [6]. They did not succeed in quantitative 0370-2693/96/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(96)00540-O