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 that is consistent with conservation of energy and momentum. Definition The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as where is the Ricci tensor, is the metric tensor and is the scalar curvature, which is computed as the trace of the Ricci Tensor by . In component form, the previous equation reads as The Einstein tensor is symmetric and, like the on shell stress–energy tensor, and has zero divergence: Explicit form The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor. However, this expression is complex and rarely quoted in textbooks. The complexity of this expression can be shown using the formula for the Ricci tensor in terms of Christoffel symbols: where is the Kronecker tensor and the Christoffel symbol is defined as and terms of the form represent its partial derivative in the μ-direction, i.e.: Before cancellations, this formula results in individual terms. Cancellations bring this number down somewhat. In the special case of a locally inertial reference frame near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified: where square brackets conventionally denote antisymmetrization over bracketed indices, i.e. Trace The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor . In dimensions (of arbitrary signature): Therefore, in the special case of dimensions, . That is, the trace of the Einstein tensor is the negative of the Ricci tensor's trace. Thus, another name for the Einstein tensor is the trace-reversed Ricci tensor. This case is especially relevant in the theory of general relativity. Use in general relativity The Einstein tensor allows the Einstein field equations to be written in the concise form: where is the cosmological constant and is the Einstein gravitational constant. From the explicit form of the Einstein tensor, the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives of the metric. As a symmetric order-2 tensor,