A note on diffeomorphism-invariance in gravitational field equations

TL;DR

This paper critically examines the role of diffeomorphism invariance in Einstein's gravitational field equations, clarifying misconceptions about the cosmological constant, unimodular gravity, and the derivation of traceless equations.

Contribution

It provides a detailed analysis of Einstein's original and modified equations, clarifies the independence of the cosmological constant and unimodular condition, and discusses open issues in deriving traceless equations from action principles.

Findings

Unimodular gravity with a well-defined action does not exist.

The cosmological constant and unimodular condition are independent.

Deriving traceless equations from the action principle remains unresolved.

Abstract

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.