(ed.). Gravitational waves (IOP, 2001)(422s).

Chapter 14Infinite-dimensional symmetries in gravityHermann Nicolai 1 and Alessandro Nagar 21 Max-Planck-Institut für Gravitationsphysik(Albert-Einstein-Institut), Mühlenberg 1, D-14476 Golm,Germany2 Dipartimento di Fisica, Università degli Studi di Parma14.1 Einstein theory14.1.1 IntroductionIn these lectures we review the symmetry properties of Einstein’s theory whenit 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 actionof the Geroch group, which is nonlinear and non-local, can be linearized, therebypermitting the explicit construction of many solutions of Einstein’s equations withtwo commuting Killing vectors ∂ 2 and ∂ 3 . A non-trivial example of this methodfor a colliding plane wave metric is given.The lectures review some well-known material at a pedagogical level.Therefore, rather than including references in the text, we have chosen to collectsome basic references at the end, which readers are invited to use as a guide to thevast literature on the subject of exact solutions, on the integrability of Einstein’sequations in the reduction to two dimensions, and finally on the generalization ofthese structures to other theories, including supergravity.14.1.2 Mathematical conventionsOur main interest is in studying the structural properties of Einstein’s theory andits generalizations. We will first formulate it in D dimensions, with coordinates245