chapter  14
23 Pages

Infinite-dimensional symmetries in gravity

In these lectures we review the symmetry properties of Einstein’s theory when it is reduced from four to two dimensions. We explain how, in this reduction, the theory acquires an infinite-dimensional symmetry group, the Geroch group, whose associated Lie algebra is the affine extension of SL(2, R ). The action of the Geroch group, which is nonlinear and non-local, can be linearized, thereby permitting the explicit construction of many solutions of Einstein’s equations with two commuting Killing vectors ∂2 and ∂3. A non-trivial example of this method for a colliding plane wave metric is given.