Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 4 / b / ii

Varying the matter terms with respect to the inverse metric and writing the Einstein equation as $G_{ab}=T_{ab}$ givesThe factors reflect that the complete action, including its matter terms, carries the common prefactor $1/(16\pi )$. With the conventional definition in which a separate matter action satisfies $G_{ab}=8\pi T_{ab}^{\mathrm{conventional}}$, the corresponding tensor is rescaled accordingly.