On the instability of the n =1 Einstein--Yang--Mills black holes and mathematically related systems

Journal Article · Wed Jan 01 00:00:00 EST 1992 · Journal of Mathematical Physics (New York); (United States)

DOI:https://doi.org/10.1063/1.529957· OSTI ID:5558072

Wald, R M ^[1]

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1. Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois (United States). Department of Physics

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 {ital n}=1 Einstein--Yang--Mills (EYM) black hole, an interesting situation occurs. There exists a perturbation which grows exponentially in time{minus}and spatially decreases to zero at the horizon{minus}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 {ital 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}{sup 2}{phi}/{partial derivative}{ital t}{sup 2} = ({ital D}{sub {ital a}D}{sup {ital a}} {minus} {ital V}){phi} on a manifold R{times}{ital M}, where {ital D}{sub {ital a}} is the derivative operator associated with a complete Riemannian metric on {ital M} and {ital V} is a bounded function on {ital M} whose derivatives also are bounded. We prove that if the operator {ital A} = {minus}{ital D}{sub {ital a}D}{sup {ital a}} + {ital V} fails to be a strictly positive operator on the Hilbert space {ital L}{sup 2}({ital M}), then there exists smooth initial data of compact support in {ital M} which give rise to a solution which grows unboundedly with time. This implies that the {ital 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.

• OSTI ID: 5558072
• Journal Information: Journal of Mathematical Physics (New York); (United States), Vol. 33:1; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BLACK HOLES
STABILITY
YANG-MILLS THEORY
EINSTEIN FIELD EQUATIONS
GLOBAL ASPECTS
HILBERT SPACE
METRICS
RIEMANN SPACE
SCALAR FIELDS
SINGULARITY
SPACE-TIME
WAVE EQUATIONS
BANACH SPACE
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