ar X iv : g r - qc / 9 70 70 46 v 1 2 1 Ju l 1 99 7 Solving the Hamilton - Jacobi equation for gravitationally interacting electromagnetic and scalar fields ∗

The spatial gradient expansion of the generating functional was recently developed by Parry, Salopek, and Stewart to solve the Hamiltonian constraint in EinsteinHamilton-Jacobi theory for gravitationally interacting dust and scalar fields. I use the spatial gradient expansion to derive an order-by-order solution of the Hamiltonian constraint for gravitationally interacting electromagnetic and scalar fields. A conformal transformation and functional integral are used to derive the generating functional up to the terms fourth order in spatial gradients. The perturbations of a flat FriedmannRobertson-Walker cosmology with a scalar field, up to second order in spatial gradients are given. The application of this formalism is demonstrated in the specific example of an exponential potential.