nLab Einstein tensor

Definition

Given a pseudo-Riemannian manifold $(X,g)$, the Einstein tensor is the tensor field on $X$ given by

$$G \coloneqq Ric - \tfrac{1}{2}R g \,,$$

where

• $Ric$ is the Ricci tensor

• $R$ is t he scalar curvature

of the metric $g$.

In gravity

The Einstein tensor is the variational derivative of the pure Einstein-Hilbert action. Therefore the equations of motion of gravity (Einstein's equations) are the equation

$$G = T \,,$$

where $T$ is the energy-momentum tensor of the further force- and matter-fields of the theory.