Abstract
Bifurcation theory is used to analyze the space of solutions of Einstein's equations near a spacetime with symmetries. The methods developed here allow one to describe precisely how the symmetry is broken as one branches from a highly symmetric spacetime to nearby space-times with fewer symmetries, and finally to a generic solution with no symmetries. This phenomenon of symmetry breaking is associated with the fact that near symmetric solutions the space of solutions of Einstein's equations does not form a smooth manifold but rather has a conical structure. The geometric picture associated with this conical structure enables one to understand the breaking of symmetries. Although the results are described for pure gravity, they may be extended to classes of fields coupled to gravity, such as gauge theories. Since most of the known solutions of Einstein's equations have Killing symmetries, the study of how these symmetries are broken by small perturbations takes on considerable theoretical significance.
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