physics - Einstein tensor

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

$\mathbf{G}=\mathbf{R}-\frac{1}{2}R\mathbf{g}$

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

$G_{ab} = R_{ab} - \frac12 R g_{ab}$ ,

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

$\nabla_{a} G^{ab} = 0$ .

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

$G_{ab} = \frac{8\pi G}{c^4} T_{ab}$ .

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

$\nabla_{a} T^{ab} = 0$ .

Categories: Differential geometry | General relativity