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.

Tensors are the natural objects of study in multilinear algebra, the branch of algebra that generalizes linear algebra to mappings between multiple vector spaces. The tensor product, which constructs a new vector space from two existing ones, is the fundamental operation that makes tensors composable: the product of a rank-m tensor and a rank-n tensor is a rank-(m+n) tensor. This closure under multiplication makes tensors the building blocks of geometric field theories, where local degrees of freedom at each point of a manifold combine into globally defined fields.