Propagators from characteristic surfaces

The Cauchy problem for hyperbolic equations is well known since many years ago. We can take the wave equation in Minkowski space-time as an example. If we choose an infinite space-like surface where it is known the function and its normal derivative (with respect to the surface) the solution is fully obtained in the rest of the space. The situation becomes more delicate when the surface where we know the data is light-like, as the light cones are. The problem is then called a Goursat or characteristic one, as the surfaces are the characteristics of the equation [1] and another different procedure is needed to obtain the solution. But if we consider only a future light cone, curiously, the solution inside this depends only on the function or on the normal derivative of the function on the cone, but not on both. This characteristic surface is such that the derivative is not independent of the function. This is remarkable, as the causal development of a field depends in half of the data compared with the Cauchy evolution. Although the previous considerations are purely mathematical, there might be a correspondence with physics, if we consider the holographic principle. Naively we can state that degrees of freedom in a certain space (called bulk) match one to one with the degrees of freedom on the boundary of this space. This situation is reproduced in our simple example, where the boundary is the light cone itself, and the degrees of freedom are carried by the fields in it. The idea of holography has been studied extensively in the last years, specially in scenarios with negative cosmological constant. The so-calledAdS/CFT correspondence is probably the most successful example, although the knowledge about it is still incomplete. Much less attention has been devoted to Ricci-flat geometries, with vanishing cosmological constant. In this work we will study a simple scenario of this kind and try to check a possible holographic relation. In order to do this we will need explicit classical solutions, so we will introduce a method to find them. But before we start with a description of the geometry and its properties, and how general can be this procedure.