Second-Order Effects in Gravitational Wave Spacetime

In this paper, we investigate gravitational waves beyond the linear approximation, focusing on second-order contributions sourced by linearized waves in the transverse-traceless (TT) gauge. A general spacetime metric is constructed, and both timelike and null geodesic congruences are analyzed. For the timelike congruence, a non-vanishing expansion scalar and shear tensor are obtained, while the rotation tensor is found to vanish. In contrast, all these quantities vanish for the null congruence. Using a parallel-transported orthonormal tetrad, we derive the geodesic deviation equations up to second order in the wave amplitude $H$, showing that, as in the linear case, stretching and compression occur in the transverse $x$ and $y$ directions. However, when solving the geodesic equations numerically within the 3+1 formalism, we observe an additional effect: test particles undergo a small, second-order displacement along the direction of wave propagation.