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. 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 G {displaystyle mathbf {G} } is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as where R {displaystyle mathbf {R} } is the Ricci tensor, g {displaystyle mathbf {g} } is the metric tensor and R {displaystyle R} is the scalar curvature. In component form, the previous equation reads as