Analysis of the static spherically symmetric SU\((n)\)-Einstein-Yang-Mills equations. (English) Zbl 0813.53049
Summary: The singular boundary value problem that arises for the static spherically symmetric SU\((n)\)-Einstein-Yang-Mills equations in the so- called magnetic case is analyzed. Among the possible actions of SU(2) on a SU\((n)\)-principal boundes over space-time there is one which appears to be the most natural. If one assumes that no electrostatic type component is present in the Yang-Mills fields and the gauge is suitably fixed a set of \(n-1\) second-order and two first-order differential equations is obtained for \(n-1\) gauge potentials and two metric components as functions of the radial distance. This system generalizes the one for the case \(n=2\) that leads to the discrete series of the Bartnick-Mckinnon and the corresponding black hole solutions. It is highly nonlinear and singular at \(r=\infty\) and at \(r=0\) or at the black hole horizon but it is known to admit at least one series of discrete solutions which are scaled versions of the \(n=2\) case. In this paper local existence and uniqueness of solutions near these singular points is established which turns out to be a nontrivial problem for general \(n\). Moreover, a number of new numerical soliton (i.e. globally regular) numerical solutions of the SU(3)-EYM equations are found that are not scaled \(n=2\) solutions.
MSC:
| 53Z05 | Applications of differential geometry to physics |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
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