Ricci Curvature - Trace-free Ricci Tensor

In Riemannian geometry and general relativity, the trace-free Ricci tensor of a pseudo-Riemannian manifold is the tensor defined by

where is the Ricci tensor, is the scalar curvature, is the metric tensor, and is the dimension of . The name of this object reflects the fact that its trace automatically vanishes:

If n 3, the trace-free Ricci tensor vanishes identically if and only if

for some constant . In mathematics, this is the condition for to be an Einstein manifold. In physics, this equation states that is a solution of Einstein's vacuum field equations with cosmological constant.