  We consider gravitational field equations which are Einstein equations
written in terms of embedding coordinates in some higher dimensional Minkowski
space. Our main focus is to address some tricky issues relating to the Cauchy
problem and possible non-equivalence with the intrinsic Einstein theory. The
well known theory introduced by Regge and Teitelboim in 9+1 dimensions is cast
in Cauchy-Kowalevskaya form and therefore local existence and uniqueness
results follow for \emph{analytic} initial data. In seeking a weakening of the
regularity conditions for initial data, we are led naturally to propose a 13+1
dimensional theory. By imposing an appropriate conserved initial value
constraint we are able, in the neighbourhood of a generic (free) embedding, to
obtain a system of nonlinear hyperbolic differential equations. The questions
of long time or global existence and uniqueness are formidable, but we offer
arguments to suggest that the situation is not hopeless if the theory is
modified in an appropriate way. We also present a modification of the
perturbation method of G\"{u}nther to weighted Sobolev spaces, appropriate to
noncompact initial data surfaces with asymptotic fall-off conditions.
