Einstein tensor - Definition and Overview

In differential geometry, the Einstein tensor <math>\mathbf{G}<math> is a 2-tensor defined over Riemannian manifolds and which is defined in index-free notation as,

<math>\mathbf{G}=\mathbf{R}-\frac{1}{2}R\mathbf{g}<math>

where <math>\mathbf{R}<math> is the Ricci tensor, <math>\mathbf{g}<math> is the metric tensor and <math>R<math> is the Ricci scalar (or scalar curvature). In components, the above equation reads

<math> G_{ab} = R_{ab} - \frac12 R g_{ab} <math>,

The Bianchi identities can be easily expressed with the aid of the Einstein tensor:

<math> \nabla_{a} G^{ab} = 0 <math>.

In general relativity, the Einstein tensor allows a compact expression of the Einstein equations:

<math> G_{ab} = \frac{8\pi G}{c^4} T_{ab}<math>.

The Bianchi identities automatically ensure the conservation of the energy-momentum tensor in curved spacetimes:

<math> \nabla_{a} T^{ab} = 0 <math>.