Einsteintensor

The Einstein tensor, denoted Gμν, is a symmetric rank-2 tensor central to general relativity. It is defined from the spacetime metric gμν and its curvature by Gμν = Rμν − 1/2 gμν R, where Rμν is the Ricci tensor and R is the scalar curvature, R = gμν Rμν. The quantities Rμν and R are obtained from contractions of the Riemann curvature tensor, which encodes how spacetime is curved by matter and energy.

Properties of the Einstein tensor include its divergence-free nature, ∇μ Gμν = 0, a consequence of the contracted Bianchi identities. This property ensures compatibility with local energy-momentum conservation, ∇μ Tμν = 0, when the tensor is used in the Einstein field equations. Gμν is constructed to be a diffeomorphism-invariant object, determined entirely by the spacetime metric and its first and second derivatives.

Role in the Einstein field equations is to relate spacetime curvature to the matter-energy content of the universe. In the standard form, the equations are Gμν + Λ gμν = 8πG/c4 Tμν, where Λ is the cosmological constant, G is Newton’s gravitational constant, c is the speed of light, and Tμν is the stress-energy tensor. In the absence of a cosmological constant, or when moved to the other side, Gμν = (8πG/c4) Tμν − Λ gμν. Solutions to these equations describe gravitational phenomena such as cosmological expansion, black holes, and gravitational waves.

Computation of the Einstein tensor depends on the metric, requiring the Levi-Civita connection and the Riemann tensor to form Rμν and R, and to assemble Gμν.