Ricci flat metrics in various dimensions, depending from 2 light-cone parameters, and the Lagrangian for the 2 dimensional reduction of gravity

Abstract

We consider d-dimensional Riemanian manifolds which admit d-2 commuting space-like Killing vector fields, orthogonal to a surface, containing two one-parametric families of light-like curves. The condition of the Ricci tensor to be zero gives Ernst equations for the metric. We write explicitly a family of local solutions of this equations corresponding to arbitrary initial data on two characteristics in terms of a series. These metrics describe scattering of 2 gravitational waves, and thus we expect they are very interesting. Ernst equations can be written as equations of motion for some 2D Lagrangian, which governs fluctuations of the metric, constant in the Killing directions. This Lagrangian looks essentially as a 2D chiral field model, and thus is possibly treatable in the quantum case by standart methods. It is conceivable that it may describe physics of some specially arranged scattering experiment, thus giving an insight for 4D gravity, not treatable by standart quantum field theory methods. The renormalization flow for our Lagrangian is different from the flow for the unitary chiral field model, the difference is essentially due to the fact that here the field is taking values in a non-compact space of symmetric matrices. We investigate the model and derive the renormalized action in one loop.