Solutions of the spherically symmetric SU(2) Einstein-Yang-Mills equations defined in the far field (Q2738229)

The Einstein-Yang-Mills equations with \(SU(2)\) gauge group for static, spherically symmetric solutions in the magnetic ansatz [\textit{R. Barnik} and \textit{J. McKinnon}, Particle-like solutions of the EYM equations, Phys. Rev. Lett. 61, 141-144 (1988)] reduce to a system of three ordinary differential equations with unknowns \(A=A(r)\), \(w=w(r)\) and \(B=B(r)\), where \(r\) is the Schwarzschild coordinate. A solution \((A,w,B)\) is said to be defined in the \textit{far field} if there is an \(r_0 >0\) such that for all \(r>r_0\) the functions are defined and differentiable. In this paper, it is shown that any solution defined in the far field has finite ADM mass; in particular \(A(r)>0\) for large \(r\). Thus, such solution gives an asymptotically flat space-time. It is known that the zeros of \(A\) are isolated and can only accumulate at \(r=0\). The author proves that \(A\) can have at most two zeros. He also shows that the three types of possible solutions, Barnik-McKinnon particle-like solutions, Reissner-Nordström type solutions and black hole solutions having only one horizon can be distinguished by the behaviour of the function \(A(r)\) near \(r=0\).