  Except for the presence of gravitational wave source term, the relativistic
perturbation equations of a zero-pressure irrotational fluid in a flat
Friedmann world model coincide exactly with the Newtonian ones to the second
order in perturbations. Such a relativistic-Newtonian correspondence is
available in a special gauge condition (the comoving gauge) in which all the
variables are equivalently gauge invariant. In this work we compare our results
with the ones in the synchronous gauge which has been used often in the
literature. Although the final equations look simpler in the synchronous gauge,
the variables have remnant gauge modes. Except for the presence of the gauge
mode for the perturbed order variables, however, the equations in the
synchronous gauge are gauge invariant and can be exactly identified as the
Newtonian hydrodynamic equations in the Lagrangian frame. In this regard, the
relativistic equations to the second order in the comoving gauge are the same
as the Newtonian hydrodynamic equations in the Eulerian frame. We resolve
several issues related to the two gauge conditions often to fully nonlinear
orders in perturbations.
