In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of a physical space, in which case the field quantity acquires a unit of measurement. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor is defined on a vector fields set over a module , we call a tensor field on .O'Neill, Barrett. Semi-Riemannian Geometry With Applications to Relativity
A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the Riemann curvature tensor refers a tensor field, as it associates a tensor to each point of a Riemannian manifold, a topological space.
Intrinsic approach Geometric introduction
A scalar field is an assignment (mathematically a function) of a real or complex number to every point of a manifold. One example is the temperature field. The temperature at a point does not change even if the coordinate system changes.
Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface. Again, the length and direction of these vectors is coordinate-independent.
Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field , such that given any two vectors at point , their inner product is . The field could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is Tissot's indicatrix.
An intrinsic definition should specify tensor fields in a coordinate-independent way: independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates. One way to achieve this is using the concept of tensor bundles.
Tensor bundles
A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle. (There are vector bundles that are not tensor bundles: the Möbius band for instance.) Explicitly, a tensor bundle is the fiber bundle
where V is the tangent bundle of M and V∗ is the cotangent bundle of M, and its fibers are
where is the tangent space of x and is the cotangent space of x, for every x in M.
Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Starting with the tangent bundle (the bundle of tangent spaces) the whole apparatus explained at component-free treatment of tensors carries over in a routine way – again independently of coordinates, as mentioned in the introduction.
In the entirely analogous way that for the tangent bundle V over M, a section of the tangent bundle is a function that sends every point x of the manifold M to a vector v(x) which is an element of the fiber Vx of the tangent bundle V, and a vector field is defined to be such section, then a section of the tensor bundle sends every point x of the manifold M to a tensor T(x) which is an element of the fiber of the tensor bundle and we therefore can give a definition of tensor field as follows.
Definition Let be a manifold, for instance the Euclidean space .
Equivalently, a tensor field is an assignment of elements for every point , where now denotes the tensor product of vectors spaces, such that it constitutes a smooth map . The elements are called tensors of type . This is then guaranteed geometric content, since everything has been done in an intrinsic way.
Remark One can, more generally, take to be any vector bundle on , and its dual bundle. In that case can be a more general topological space. These sections are called tensors of or tensors for short if no confusion is possible. Given two tensor bundles E → M and F → M, a linear map A: Γ(E) → Γ(F) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A(fs) = fA(s), for each section s in Γ(E) and each smooth function f on M. Thus a tensor section is not only a linear map on the vector space of sections, but a C∞(M)-linear map on the module of sections. This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are. In coordinates Locally in a coordinate neighbourhood with coordinates we have a local basis (Vielbein) of vector fields , and a dual basis of 1 forms so that . In the coordinate neighbourhood we then have
where here and below we use Einstein summation conventions. Note that if we choose different coordinate system then and where the coordinates can be expressed in the coordinates and vice versa, so that
i.e.
The system of indexed functions (one system for each choice of coordinate system) connected by transformations as above are the tensors in the definitions below.
Via coordinate transitions Following and , the concept of a tensor relies on a concept of a reference frame (or coordinate system), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems.The term "affinor" employed in the English translation of Schouten is no longer in use.
For example, coordinates belonging to the n-dimensional real coordinate space may be subjected to arbitrary affine transformations:
(with n-dimensional indices, summation implied). A covariant vector, or covector, is a system of functions that transforms under this affine transformation by the rule
The list of Cartesian coordinate basis vectors transforms as a covector, since under the affine transformation . A contravariant vector is a system of functions of the coordinates that, under such an affine transformation undergoes a transformation
This is precisely the requirement needed to ensure that the quantity is an invariant object that does not depend on the coordinate system chosen. More generally, the coordinates of a tensor of valence (p,q) have p upper indices and q lower indices, with the transformation law being
The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be smooth (or differentiable, analytic, etc.). A covector field is a function of the coordinates that transforms by the Jacobian of the transition functions (in the given class). Likewise, a contravariant vector field transforms by the inverse Jacobian.
Notation
The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle TM = T(M) might sometimes be written as
to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M. This should not be confused with the very similar looking notation
in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold M.
Curly (script) letters are sometimes used to denote the set of infinitely-differentiable tensor fields on M. Thus,
are the sections of the (m,n) tensor bundle on M that are infinitely-differentiable. A tensor field is an element of this set.
Tensor fields as multilinear forms There is another more abstract (but often useful) way of characterizing tensor fields on a manifold M, which makes tensor fields into honest tensors (i.e. single multilinear mappings), though of a different type (although this is not usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (C∞) vector fields on M, (see the section on notation above) as a single space – a module over the ring of smooth functions, C∞(M), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over any commutative ring.
As a motivating example, consider the space of smooth covector fields (1-forms), also a module over the smooth functions. These act on smooth vector fields to yield smooth functions by pointwise evaluation, namely, given a covector field ω and a vector field X, we define
Because of the pointwise nature of everything involved, the action of on X is a C∞(M)-linear map, that is,
for any p in M and smooth function f. Thus we can regard covector fields not just as sections of the cotangent bundle, but also linear mappings of vector fields into functions. By the double-dual construction, vector fields can similarly be expressed as mappings of covector fields into functions (namely, we could start "natively" with covector fields and work up from there).
In a complete parallel to the construction of ordinary single tensors (not tensor fields!) on M as multilinear maps on vectors and covectors, we can regard general (k,l) tensor fields on M as C∞(M)-multilinear maps defined on k copies of and l copies of into C∞(M).
Now, given any arbitrary mapping T from a product of k copies of and l copies of into C∞(M), it turns out that it arises from a tensor field on M if and only if it is multilinear over C∞(M). Namely C∞(M)-module of tensor fields of type over M is canonically isomorphic to C∞(M)-module of C∞(M)-multilinear forms
This kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to a function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously.
A frequent example application of this general rule is showing that the Levi-Civita connection, which is a mapping of smooth vector fields taking a pair of vector fields to a vector field, does not define a tensor field on M. This is because it is only -linear in Y [in place of full C∞(M)-linearity, it satisfies the Leibniz rule, ]. Nevertheless, it must be stressed that even though it is not a tensor field, it still qualifies as a geometric object with a component-free interpretation.
Applications
The curvature tensor is discussed in differential geometry and the stress–energy tensor is important in physics, and these two tensors are related by Einstein's theory of general relativity.
In electromagnetism, the electric and magnetic fields are combined into an electromagnetic tensor field.
Differential forms, used in defining integration on manifolds, are a type of tensor field.
Tensor calculus
In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus. Even to formulate such equations requires a fresh notion, the covariant derivative. This handles the formulation of variation of a tensor field along a vector field. The original absolute differential calculus notion, which was later called tensor calculus, led to the isolation of the geometric concept of connection.
Twisting by a line bundle