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 .

 

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