What tensor is used to describe spacetime curvature in the Einstein field equations?

The Einstein tensor is used to describe spacetime curvature in the Einstein field equations.

It is defined as Gμν = Rμν − ½Rgμν, combining the Ricci curvature tensor, the Ricci scalar, and the metric tensor. This combination captures a form of curvature suitable for relating geometry to the distribution of matter and energy.

In the field equations, the Einstein tensor appears on the geometric side: Gμν + Λgμν = κTμν. The stress-energy tensor Tμν represents energy density, momentum density, pressure, and stresses. The equation therefore expresses general relativity’s central idea that matter and energy influence spacetime geometry.

A crucial mathematical feature is that the Einstein tensor has zero covariant divergence, a consequence of the contracted Bianchi identities. This matches the covariant conservation of stress-energy and is one reason the tensor is used instead of the Ricci tensor alone.

The alternatives are related but not equivalent answers. The metric tensor describes spacetime intervals and determines curvature, the Ricci scalar is a single-number curvature contraction, and the stress tensor belongs to the matter side of the equations.