Connection between the nonlinearσmodel and the Einstein equations of general relativity

We show that the equations of general relativity contain an O(2,1) $\sigma$ model. This $\sigma$-model structure emerges from a 3 + 1 decomposition of the Einstein equations which holds irrespective of the presence of symmetries in the space-time. This includes, in particular, the stationary (one Killing vector) and the Ernst (two Killing vectors) formulations of the gravitational field. From this connection with the $\sigma$ model we find a new family of solutions of the Einstein equations. These solutions have (3,1) signature and one Killing vector. They are complex or real and they depend on two arbitrary functions (one holomorphic and one antiholomorphic). In particular, in the presence of two Killing vectors they give two different subclasses of solutions: one is associated with the instantons of the σ model and the other is of Taub-NUT (Newman-Unti-Tamburino) type.