Well-posed ADM equivalent of the Bondi-Sachs problem

Every well-posed hyperbolic problem has an associated characteristic representation. In the case of the Einstein equations, traditionally, characteristic problems have been stated in the Bondi-Sachs form, whereas initial-value problems have been represented in the ADM form, both being looked upon as independent versions of the Einstein equations. Under the restriction of spherical symmetry, we provide an ADM version of the Einstein equations that functions as the initial-value representation of the Bondi-Sachs equations. The ADM version allows us to interpret the Bondi-Sachs variables precisely in terms of characteristic fields of the Cauchy problem. The Bondi-Sachs version thus leads us to a version of the Cauchy problem that is first order in time (with no need for reduction) and automatically well posed