Well-posedness of (N = 1) classical supergravity

In this paper we investigate whether classical (N = 1) supergravity has a well-posed locally causal Cauchy problem. We define well-posedness to mean that any choice of initial data (from an appropriate function space) which satisfies the supergravity constraint equations and a set of gauge conditions can be continuously developed into a space-time solution of the supergravity field equations around the initial surface. Local causality means that the domains of dependence of the evolution equations coincide with those determined by the light cones. We show that when the fields of classical supergravity are treated as formal objects, the field equations are (under certain gauge conditions) equivalent to a coupled system of quasilinear nondiagonal second-order partial differential equations which is formally nonstrictly hyperbolic (in the sense of Leray--Ohya). Hence, if the fields were numerical valued, there would be an applicable existence theorem leading to well-posedness. We shall observe that well-posedness is assured if the fields are taken to be Grassmann (i.e., exterior algebra) valued, for then the second-order system decouples into the vacuum Einstein equation and a sequence of numerical valued linear diagonal strictly hyperbolic partial differential equations which can be solved successively