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. This expansion is used here 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.