The tensors we have constructed belong to multilinear algebra: they live on a single fixed vector space. In differential geometry and physics one works with tensor fields, tensors that vary smoothly from point to point of a manifold. This section is a brief account of how the algebra above connects to the classical tensor calculus of Ricci, Levi-Civita, and Einstein. We do not develop manifolds in this book, so the geometric parts are descriptive.
9.12.1 Tensors of Type
Let be an -dimensional real vector space with basis . In this section we write the dual basis with upper indices, , so that .
Definition 9.12.1 (Tensor of type ). A tensor of type on is an element of
Using the isomorphism of Theorem 6.2.19 on the first factors and then Proposition 9.9.6, so a tensor of type is the same as an -linear form A pure tensor corresponds to the form In coordinates, is specified by its components and it is recovered from them as because these tensors form a basis of (Theorem 9.9.2), and the form corresponding to each of them takes the value on one tuple of basis vectors and on all the others. We use the Einstein summation convention: an index that appears once as a superscript and once as a subscript is summed over. With it, the formula above reads .
Example 9.12.2 (Familiar objects as tensors).
- A vector is a tensor of type : .
- A covector (linear functional) has type : .
- A bilinear form has type : , with .
- A linear map has type , via (Propositions 9.8.10 and 9.8.8): , where . So is the entry of the matrix of .
The defining feature of tensors in physics, and the historical reason physicists care about them, is the way their components change under a change of basis. Let be a new basis with , so that is the change-of-basis matrix (row index , column index ). The new dual basis is , since .
Proposition 9.12.3 (Tensor transformation law). Under the change of basis , the components of a tensor of type transform as
Proof. By definition . Substitute and , and expand by multilinearity. ∎
Note. The transformation law expresses the fact that the tensor itself, as an element of , does not depend on a basis; only its components do. Many older texts defined a tensor as a collection of numbers , one for each basis, related by this law. With tensor products, the law becomes a theorem.
9.12.3 Tensor Fields on Manifolds
A smooth manifold is a space that looks locally like , such as a sphere or a torus. Each point has a tangent space , a real vector space of dimension . A tensor field of type on is a smooth assignment , where for each .
Local coordinates near give a basis of the tangent space , written , and the dual basis of its dual space, the cotangent space . A tensor field is then where the component functions are smooth. If are other local coordinates, the chain rule gives So the components transform by Proposition 9.12.3 with and . For a vector field this gives the familiar rule , and for a covector field .
9.12.4 Index Gymnastics: Raising, Lowering, and Contraction
Suppose carries a nondegenerate symmetric bilinear form , such as an inner product or the metric tensor below. Write , and let be the entries of the inverse matrix, so that . Then indices can be “raised” and “lowered”.
Definition 9.12.4 (Raising and lowering indices). Given a tensor of type :
- Lowering the upper index gives , of type .
- Raising the lower index gives , of type .
In general, any index can be raised with or lowered with .
In coordinate-free terms, lowering an index applies the map , , of Theorem 9.1.8 (an isomorphism because is nondegenerate, Definition 9.1.9) to one tensor factor, and raising applies its inverse. For a vector, lowering gives .
Definition 9.12.5 (Contraction of a tensor). Let , and . The contraction of the th upper index with the th lower index is the linear map with where a hat marks an omitted factor. In components, with in the th upper and the th lower position, summed over.
The map exists by the universal property, because the right side of its defining formula is multilinear in the and ; so contraction does not depend on a basis. The component formula follows by applying to the basis tensors and using . For type , contraction is the map of Definition 9.11.1 (with the two factors swapped), and contracting gives by Proposition 9.11.2.
9.12.5 The Metric Tensor and General Relativity
The central object of Riemannian geometry is the metric tensor: a smooth tensor field of type whose value at each point is an inner product on . In local coordinates, where is a symmetric positive-definite matrix at each point . The metric determines lengths, angles, volumes, curvature, and geodesics.
In Einstein’s general relativity, spacetime is a four-dimensional pseudo-Riemannian manifold : each is only required to be a nondegenerate symmetric bilinear form, and in general relativity it has signature (Definition 9.6.10). That is, in a basis that diagonalizes , one diagonal entry is positive and three are negative, and by Sylvester’s law of inertia (Theorem 9.6.9) these numbers do not depend on the basis. (Some authors use the opposite sign convention, signature .) So each tangent space, with , looks like Minkowski space: with the form . The Einstein field equations relate the geometry of spacetime, encoded in the Einstein tensor (a tensor field of type built from and its first and second derivatives), to its matter and energy content, encoded in the stress–energy tensor , also of type . Here is the cosmological constant (it has nothing to do with exterior powers), is Newton’s gravitational constant, and is the speed of light.
Note. The Einstein tensor is , where is the Ricci curvature tensor, a contraction of the Riemann curvature tensor of type , and is the scalar curvature, a further contraction. Tensor products, contraction, and raising and lowering indices with the metric are exactly the operations needed to write these equations.
Example 9.12.6 (Contraction in general relativity). The Riemann curvature tensor has type . Contracting its upper index with its second lower index gives the Ricci tensor , of type . Raising an index with the inverse metric and contracting again gives the scalar curvature . Each contraction removes one upper and one lower index. The Riemann tensor carries all the curvature information; the Ricci tensor measures how the volumes of small regions are distorted compared with flat space; and the scalar curvature is a single number at each point, an average of the curvature there.
Note. The whole apparatus of tensor calculus (index notation, Einstein summation, raising and lowering, contraction) is the coordinate form of the multilinear algebra developed in this chapter. The coordinate-free approach gives conceptual clarity and makes independence of the basis automatic; index notation is efficient for computation. Both perspectives are essential.