  An algebraic-hyperbolic method for solving the Hamiltonian and momentum
constraints has recently been shown to be well posed for general nonlinear
perturbations of the initial data for a Schwarzschild black hole. This is a new
approach to solving the constraints of Einstein's equations which does not
involve elliptic equations and has potential importance for the construction of
binary black hole data. In order to shed light on the underpinnings of this
approach, we consider its application to obtain solutions of the constraints
for linearized perturbations of Minkowski space. In that case, we find the
surprising result that there are no suitable Cauchy hypersurfaces in Minkowski
space for which the linearized algebraic-hyperbolic constraint problem is well
posed.
