the metric was used by the pioneers of differential geometry. While some relativists consider the notation to be somewhat old-fashioned, many readily switch between this and the alternative notation: The metric tensor is commonly written as a 4×4 matrix. This matrix is symmetric and thus has 10 independent components. Invariants One of the central features of GR is the idea of invariance of physical laws. This invariance can be described in many ways, for example, in terms of local Lorentz covariance, the general principle of relativity, or diffeomorphism covariance. A more explicit description can be given using tensors. The crucial feature of tensors used in this approach is the fact that (once a metric is given) the operation of contracting a tensor of rank R over all R indices gives a number - an invariant - that is independent of the coordinate chart one uses to perform the contraction. Physically, this means that if the invariant is calculated by any two observers, they will get the same number, thus suggesting that the invariant has some independent significance. Some important invariants in relativity include: The Ricci scalar: The Kretschmann scalar: Other examples of invariants in relativity include the electromagnetic invariants, and various other curvature invariants, some of the latter finding application in the study of gravitational entropy and the Weyl curvature hypothesis. Tensor classifications The classification of tensors is a purely mathematical problem. In GR, however, certain tensors that have a physical interpretation can be classified with the different forms of the tensor usually corresponding to some physics. Examples of tensor classifications useful in general relativity include the Segre classification of the energy–momentum tensor and the Petrov classification of the Weyl tensor. There are various methods of classifying these tensors, some of which use tensor invariants. Tensor fields in general relativity Tensor fields on a manifold are maps which attach a tensor to each point of the manifold. This notion can be made more precise by introducing the idea of a fibre bundle, which in the present context means to collect together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle. A tensor field is then defined as a map from the manifold to the tensor bundle, each point being associated with a tensor at . The notion of a tensor field is of major importance in GR. For