Einstein tensor

In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner consistent with energy and momentum conservation.

The Einstein tensor ${\displaystyle \mathbf {G} }$ is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as

where ${\displaystyle \mathbf {R} }$ is the Ricci tensor, ${\displaystyle \mathbf {g} }$ is the metric tensor and ${\displaystyle R}$ is the scalar curvature. In component form, the previous equation reads as

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