2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217407In addition to the second-order Einstein equations on four-dimensional homogeneous isotropic background universe filled with the single perfect fluid, we also derived the second-order perturbations of the continuity equation and the Euler equation for a perfect fluid in gauge-invariant manner without ignoring any mode of perturbations. The consistency of all equations of the second-order Einstein equation and the equations of motion for matter fields is confirmed. Due to this consistency check, we may say that the set of all equations of the second-order are self-consistent and they are correct in this sense.4 pages, no figure, Prepared for the proceedings of the international conference JGRG18 (2008), Hiroshima, JapanGeneral Relativity and Quantum CosmologyCosmology and Nongalactic AstrophysicsHigh Energy Physics - TheoryMathematical Physicstext