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We study the stability properties of the standard ADM formulation of the 3+1evolution equations of general relativity through linear perturbations of flatspacetime. We focus attention on modes with zero speed of propagation andconjecture that they are responsible for instabilities encountered in numericalevolutions of the ADM formulation. These zero speed modes are of two kinds:pure gauge modes and constraint violating modes. We show how the decoupling ofthe gauge by a conformal rescaling can eliminate the problem with the gaugemodes. The zero speed constraint violating modes can be dealt with by using themomentum constraints to give them a finite speed of propagation. This analysissheds some light on the question of why some recent reformulations of the 3+1evolution equations have better stability properties than the standard ADMformulation.Comment: 15 pages, 9 figures. Added a new section, plus incorporated many  comments made by refere

Towards an understanding of the stability properties of the 3+1 evolution equations in general relativity