Tensor

A tensor is a multilinear mathematical object that generalizes scalars, vectors, and matrices to arbitrary rank and dimension, transforming covariantly under coordinate changes. In general relativity and Riemannian geometry, the metric tensor encodes the distance structure of spacetime, while the Riemann curvature tensor measures its deviation from flatness.

Tensors are defined by their transformation law: under a change of coordinates, tensor components transform in a way that preserves the geometric meaning of the object itself. A vector — a rank-1 tensor — transforms linearly with the Jacobian matrix of the coordinate change. A rank-2 tensor, such as the metric, transforms with the product of two Jacobians. Higher-rank tensors generalize this pattern. This covariance ensures that tensor equations express geometric relationships that hold in all coordinate systems, not merely in a conveniently chosen one.

The tensor framework is not a notational convenience. It is the language in which physical laws are written when those laws must hold independently of the observer's coordinate choice. The Einstein field equations, Maxwell's equations in curved spacetime, and the stress-energy tensor that describes the flow of energy and momentum are all tensor equations. The requirement of general covariance — that physical laws take the same form in all coordinate systems — is, in practice, the requirement that they be expressed as tensor equations.