A note on diffeomorphism-invariance in gravitational field equations

The notion of diffeomorphism invariance and general covariance are conceptually delicate issues for the field equations and the actions. A thorough study on the original Einstein field equation and its two modifications by Einstein is presented. It is concluded that the cosmological constant and the unimodular condition are independent concepts, and the unimodular condition has no direct role in the derivation of the traceless field equation. Unimodular gravity with unambiguous action does not exist, and the derivation of traceless equation from the action principle is an open question.