  We continue to study the 2nd-order cosmological perturbations in synchronous
coordinates in the framework of the general relativity (GR) during the
radiation dominated (RD) stage, and to focus on the scalar-tensor and
tensor-tensor couplings. The 1st-order curl velocity and the associated
1st-order vector metric perturbations are assumed to be vanishing. By
analytically solving the 2nd-order Einstein equation and the energy-momentum
conservation equations, we obtain the 2nd-order formal solutions (in the
integral form) of all the metric perturbations, density contrast and velocity;
perform the transformation between the synchronous coordinates; and identify
the residual gauge modes in the 2nd-order solutions. In addition, we present
the 2nd-order gauge transformations of the solutions from synchronous to
Poisson coordinates. To apply these formal solutions to concrete cosmological
study, one needs to choose proper initial conditions and do several numerical
integrals.
