ON THE INSTABILITY OF THE N = 1 EINSTEIN-YANG-MILLS BLACK-HOLES AND MATHEMATICALLY RELATED SYSTEMS

The usual approach to analyze the linear stability of a static solution of some system of equations consists of searching for linearized solutions which satisfy suitable boundary conditions spatially and which grow exponentially in time. In the case of the n = 1 Einstein-Yang-Mills (EYM) black hole, an interesting situation occurs. There exists a perturbation which grows exponentially in time-and spatially decreases to zero at the horizon-but nevertheless is physically singular on the horizon. Thus, this unstable mode is unacceptable as initial data, and the question arises as to whether the n = 1 EYM black hole is stable. We analyze this issue here in the more general case of a scalar field-phi satisfying the wave equation partial derivative 2-phi/partial derivative t2 = (D(a)D(a) - V)phi on a manifold R X M, where D(a) is the derivative operator associated with a complete Riemannian metric on M and V is a bounded function on M whose derivatives also are bounded. We prove that if the operator A = - D(a)D(a) + V fails to be a strictly positive operator on the Hilbert space L2(M), then there exists smooth initial data of compact support in M which give rise to a solution which grows unboundedly with time. This implies that the n = 1 EYM black hole and other mathematically similar systems are unstable despite the nonexistence of physically acceptable exponentially growing modes. Rigorous criteria for linear stability are also obtained.