  Beginning with a relativistic action principle for the irrotational flow of
collisionless matter, we compute higher order corrections to the Zel'dovich
approximation by deriving a nonlinear Hamilton-Jacobi equation for the velocity
potential. It is shown that the velocity of the field may always be derived
from a potential which however may be a multi-valued function of the space-time
coordinates. In the Newtonian limit, the results are nonlocal because one must
solve the Newton-Poisson equation. By considering the Hamilton-Jacobi equation
for general relativity, we set up gauge-invariant equations which respect
causality. A spatial gradient expansion leads to simple and useful results
which are local --- they require only derivatives of the initial gravitational
potential.
