the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10 quasilinear second-order partial differential equations for the metric tensor. The contracted Bianchi identities can also be easily expressed with the aid of the Einstein tensor: The (contracted) Bianchi identities automatically ensure the covariant conservation of the stress–energy tensor in curved spacetimes: The physical significance of the Einstein tensor is highlighted by this identity. In terms of the densitized stress tensor contracted on a Killing vector , an ordinary conservation law holds: Uniqueness David Lovelock has shown that, in a four-dimensional differentiable manifold, the Einstein tensor is the only tensorial and divergence-free function of the and at most their first and second partial derivatives. However, the Einstein field equation is not the only equation which satisfies the three conditions: Resemble but generalize Newton–Poisson gravitational equation Apply to all coordinate systems, and Guarantee local covariant conservation of energy–momentum for any metric tensor. Many alternative theories have been proposed, such as the Einstein–Cartan theory, that also satisfy the above conditions. See also Contracted Bianchi identities Vermeil's theorem Mathematics of general relativity General relativity resources Notes References Tensors in general relativity Tensor