Applying gradient expansion to a perfect fluid and higher dimensions

We examine the nonlinear evolution of two types of spacetime by solving the Hamilton-Jacobi equation by the gradient expansion method to investigate the validity and limitation of the method itself. The first type is the nonlinear evolution of spacetime for an irrotational perfect fluid, and the second type is for an irrotational dust or an scalar field with an exponential potential in n-dimensional space. We find a recursion relation for the generating functional. Taking the comoving coordinate, the three-metric for perfect fluid is found up to the third order. The expression for the three-metric is in agreement with that of Comer et al. but the numerical coefficient is slightly different because of the different choice of coordinate condition. For a scalar field with an exponential potential in higher dimension, inhomogeneities decay during inflationary phase. The (n+1)-dimensional axisymmetric Szekeres solution is easily found as a byproduct.