  We present the third-order analytic solution of the matter density
fluctuation in the proper-time hypersurface of nonrelativistic matter flows by
solving the nonlinear general relativistic equations. The proper-time
hypersurface provides a coordinate system that a local observer can set up
without knowledge beyond its neighborhood, along with physical connections to
the local Newtonian descriptions in the relativistic context. The initial
condition of our analytic solution is set up by the curvature perturbation in
the comoving gauge, clarifying its impact on the nonlinear evolution. We
compute the effective non-Gaussian parameters due to the nonlinearity in the
relativistic equations. With proper coordinate rescaling, we show that the
equivalence principle is respected and the relativistic effect vanishes in the
large-scale limit.
