The model flat geometry for the ambient construction is the future null cone in Minkowski space, with the origin deleted.
The ambient construction in this flat model space then asks: if one is provided with such a line bundle, along with its degenerate metric, to what extent is it possible to extend the metric off the null cone in a canonical way, thus recovering the ambient Minkowski space?
This section provides an overview of the construction, first of the null line bundle, and then of its ambient extension.
Furthermore, any extension metric which is homogeneous of degree 2 can be written in these coordinates in the form: where the gij are n2 functions with g(x,0) = g(x), the given conformal representative.
After some calculation one shows that the Ricci flatness is equivalent to the following differential equation, where the prime is differentiation with respect to ρ: One may then formally solve this equation as a power series in ρ to obtain the asymptotic development of the ambient metric off the null cone.