Hodge star operator


In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.
For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients.
The naturalness of the star operator means it can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential -forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold.

Formal definition

Let be an -dimensional oriented vector space with a symmetric bilinear form, referred to here as an inner product. This induces an inner product on -vectors for, by defining it on simple -vectors and to equal the Gram determinant
extended to through linearity. The Gram matrix of Gram determinants is a matrix allowing the inner product on the whole of to be expressed as, where and are arbitrary multivectors represented by column matrices with entries corresponding to a fixed ordering of the basis elements.
The unit -vector is defined in terms of an oriented orthonormal basis of as:
where the sign is free to be chosen and fixed as plus or minus. With respect to, the right complement of a basis element is defined as the quantity such that, and this is extended to through linearity.
The Hodge star operator is a linear operator on the exterior algebra of, mapping -vectors to -vectors, for. It is defined for an arbitrary multivector by the constructive formula
which applies the Gram matrix and takes the right complement. The Hodge star has the following property, which can be derived from the definition:
Dually, in the space of -forms, the dual to is the volume form, the function whose value on is the determinant of the matrix assembled from the column vectors of in -coordinates. Applying to the above equation, we obtain the dual property
Equivalently, taking,, and :
This means that, writing an orthonormal basis of -vectors as over all subsets of, the Hodge dual is the -vector corresponding to the complementary set :
where is the sign of the permutation
and is the product
. In the Riemannian case,.
The Hodge dual of a multivector can be calculated with the geometric product using the identity
where the tilde denotes the reverse operation, and is the volume element, or pseudoscalar.
When the bilinear form is nondegenerate, the Hodge star operator takes an orthonormal basis to an orthonormal basis. In this case, it is an isometry on the exterior algebra.

Geometric explanation

The Hodge star is motivated by the correspondence between a subspace of and its orthogonal subspace, where each space is endowed with an orientation and a numerical scaling factor. Specifically, a non-zero decomposable -vector corresponds by the Plücker embedding to the subspace with oriented basis, endowed with a scaling factor equal to the -dimensional volume of the parallelepiped spanned by this basis. The Hodge star acting on a decomposable vector can be written as a decomposable -vector:
where form an oriented basis of the orthogonal space. Furthermore, the -volume of the -parallelepiped must equal the -volume of the -parallelepiped, and must form an oriented basis of.
A general -vector is a linear combination of decomposable -vectors, and the definition of Hodge star is extended to general -vectors by defining it as being linear.

Examples

Two dimensions

In two dimensions with the normalized Euclidean metric and orientation given by the ordering, the Hodge star on -forms is given by

Three dimensions

A common example of the Hodge star operator is the case, when it can be taken as the correspondence between vectors and bivectors. Specifically, for Euclidean R3 with the basis of one-forms often used in vector calculus, one finds that
The Hodge star relates the exterior and cross product in three dimensions: Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa, that is:.
The Hodge star can also be interpreted as a form of the geometric correspondence between an axis of rotation and an infinitesimal rotation around the axis, with speed equal to the length of the axis of rotation. A scalar product on a vector space gives an isomorphism identifying with its dual space, and the vector space is naturally isomorphic to the tensor product. Thus for, the star mapping takes each vector to a bivector, which corresponds to a linear operator. Specifically, is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis are given by the matrix exponential. With respect to the basis of, the tensor corresponds to a coordinate matrix with 1 in the row and column, etc., and the wedge is the skew-symmetric matrix, etc. That is, we may interpret the star operator as:
Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators:.

Four dimensions

In case, the Hodge star acts as an endomorphism of the second exterior power. If the signature of the metric tensor is all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian, then applying the operator twice will return the argument up to a sign – see below. This particular endomorphism property of 2-forms in four dimensions makes self-dual and anti-self-dual two-forms natural geometric objects to study. That is, one can describe the space of 2-forms in four dimensions with a basis that "diagonalizes" the Hodge star operator with eigenvalues .
For concreteness, we discuss the Hodge star operator in Minkowski spacetime where with metric signature and coordinates. The volume form is oriented as. For one-forms,
while for 2-forms,
These are summarized in the index notation as
Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature, for odd-rank forms and for even-rank forms. An easy rule to remember for these Hodge operations is that given a form, its Hodge dual may be obtained by writing the components not involved in in an order such that. An extra minus sign will enter only if contains.
Note that the combinations
take as the eigenvalue for Hodge star operator, i.e.,
and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in both mathematical and physical perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory.

Conformal invariance

The Hodge star is conformally invariant on -forms on a -dimensional vector space, i.e. if is a metric on and, then the induced Hodge stars
are the same.

Example: Derivatives in three dimensions

The combination of the operator and the exterior derivative generates the classical operators,, and on vector fields in three-dimensional Euclidean space. This works out as follows: takes a 0-form to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form. For a 0-form, the first case written out in components gives:
The scalar product identifies 1-forms with vector fields as, etc., so that becomes.
In the second case, a vector field corresponds to the 1-form, which has exterior derivative:
Applying the Hodge star gives the 1-form:
which becomes the vector field.
In the third case, again corresponds to. Applying Hodge star, exterior derivative, and Hodge star again:
One advantage of this expression is that the identity, which is true in all cases, has as special cases two other identities: , and . In particular, Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star. The expression is called the codifferential; it is defined in full generality, for any dimension, further in the article below.
One can also obtain the Laplacian in terms of the above operations:
The Laplacian can also be seen as a special case of the more general Laplace–deRham operator where in three dimensions, is the codifferential for -forms. Any function is a 0-form, and and so this reduces to the ordinary Laplacian. For the 1-form above, the codifferential is and after some straightforward calculations one obtains the Laplacian acting on.

Duality

When the bilinear form is nondegenerate, applying the Hodge star twice leaves a -vector unchanged up to a sign: for in an -dimensional space, one has
where is the parity of the signature of the scalar product on, that is, the sign of the determinant of the matrix of the scalar product with respect to any basis. For example, if and the signature of the scalar product is either or then. For Riemannian manifolds, we always have.
The above identity implies that the inverse of can be given as
If is odd then is even for any, whereas if is even then has the parity of. Therefore:
where is the degree of the element operated on.