Corrections to the thin-wall approximation in general relativity

We consider the question of whether the thin-wall formalism of Israel applies to the gravitating domain walls of a $\lambda {\phi }^4$ theory. The coupled Einstein-scalar equations that describe the thick gravitating wall are expanded in powers of the thickness of the wall. The solutions of the zeroth-order equations reproduce the results of the usual Israel thin-wall approximation for domain walls. The solutions of the first-order equations provide corrections to the expressions for the stress energy of the wall and to the Israel thin-wall equations. The modified thin-wall equations are then used to treat the motion of spherical and planar domain walls.