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 naturality 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. (In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive-definite.) This induces an inner product on -vectors for , by defining it on simple -vectors and to equal the Gram determinantHarley Flanders (1963) Differential Forms with Applications to the Physical Sciences, Academic Press.
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. (Note: In the general pseudo-Riemannian case, orthonormality means for all pairs of basis vectors.) 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: for all -vectors
Dually, in the space of -forms (alternating -multilinear functions on ), 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 for all -vectors
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 (pseudoscalar) in the contravariant basis ( relating to , not the common in geometric algebra).
There is a left version of the Hodge dual that can have per-grade sign differences compared to the right version defined above in even numbers of dimensions. When a distinction needs to be made, the right Hodge dual is often denoted by a superscript star such that . The left Hodge dual is denoted by a subscript star and defined as , where the underline takes the left complement satisfying for any basis element . Using the geometric product, the left Hodge dual of an arbitrary multivector can be calculated with
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 (with respect to the scalar product), 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 (equal to the Gramian, the determinant of the matrix of scalar products ). 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 (see also: 3D rotation group#Lie algebra) 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 (i.e. it maps 2-forms to 2-forms, since ). 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 (or , depending on the signature).
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. One way to remember these Hodge duals is by noting that for any form , its Hodge dual satisfies , using the negative if contains and the positive otherwise. (For , one puts in a minus sign only if involves an odd number of the spatial forms , , and .)
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 (a function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (and takes a 3-form to zero). 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: (1) , and (2) . 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 (multiplied by an appropriate power of −1) 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 (including Euclidean spaces), 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.
On manifolds