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 n-dimensional vector space, the Hodge star is a one-to-one mapping of k-vectors to (n – k)-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 k-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 for k-vectors
Let V be an n-dimensional vector space with a nondegenerate symmetric bilinear form , referred to here as an inner product. This induces an inner product on k-vectors , for , by defining it on decomposable k-vectors and to equal the Gram determinant:14
extended to through linearity.
The unit n-vector is defined in terms of an oriented orthonormal basis of V as:
- for every pair of k-vectors
Dually, in the space of n-forms (alternating n-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 definition:
or equivalently, taking , , and :
This means that, writing an orthonormal basis of k-vectors as over all subsets of , the Hodge dual is the (n – k)-vector corresponding to the complementary set :
where is the sign of the permutation .
Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an isometry on the exterior algebra .
The Hodge star is motivated by the correspondence between a subspace W of V and its orthogonal subspace (with respect to the inner product), where each space is endowed with an orientation and a numerical scaling factor. Specifically, a non-zero decomposable k-vector corresponds by the Plücker embedding to the subspace with oriented basis , endowed with a scaling factor equal to the k-dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of inner products ). The Hodge star acting on a decomposable vector can be written as a decomposable (n − k)-vector:
where form an oriented basis of the orthogonal space . Furthermore, the (n − k)-volume of the -parallelepiped must equal the k-volume of the -parallelepiped, and must form an oriented basis of V.
A general k-vector is a linear combination of decomposable k-vectors, and the definition of the Hodge star is extended to general k-vectors by defining it as being linear.
In two dimensions with the normalized Euclidean metric and orientation given by the ordering (x, y), the Hodge star on k-forms is given by
On the complex plane regarded as a real vector space with the standard sesquilinear form as the metric, the Hodge star has the remarkable property that it is invariant under holomorphic changes of coordinate. If z = x + iy is a holomorphic function of w = u + iv, then by the Cauchy–Riemann equations we have that ∂x/∂u = ∂y/∂v and ∂y/∂u = −∂x/∂v. In the new coordinates
proving the claimed invariance.
A common example of the Hodge star operator is the case n = 3, 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 a is associated with a bivector A and vice versa, that is: . The Hodge star can also be interpreted as a form of the geometric correspondence between an axis and an infinitesimal rotation around the axis, with speed equal to the length of the axis vector. An inner product on a vector space gives an isomorphism identifying with its dual space, and the space of all linear operators 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: .
In case n = 4, the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since 4 − 2 = 2). 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, then application twice will return the argument up to a sign – see § Duality below. For example, in Minkowski spacetime where n = 4 with metric signature (+ − − −) and coordinates (t, x, y, z) where (using ):
for one-forms while
An easy rule to remember for the above 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 .[verification needed] An extra minus sign will enter only if does not contain . (The latter convention stems from the choice (+ − − −) for the metric signature. For (− + + +), one puts in a minus sign only if involves .)
The Hodge star is conformally invariant on n forms on a 2n dimensional vector space V, 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 d generates the classical operators grad, curl, and div on vector fields in three-dimensional Euclidean space. This works out as follows: d 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 inner 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 d2 = 0, which is true in all cases, sums up two others, namely that curl grad f = 0 and div curl F = 0. 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 Δ f = div grad f in terms of the above operations:
The Laplacian can also be seen as a special case of the more general Laplace–deRham operator where 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 plug and chug, one obtains the Laplacian acting on .
Applying the Hodge star twice leaves a k-vector unchanged except for its sign: for in an n-dimensional space V, one has
where s is the parity of the signature of the inner product on V, that is, the sign of the determinant of the matrix of the inner product with respect to any basis. For example, if n = 4 and the signature of the inner product is either (+ − − −) or (− + + +) then s = −1. For Riemannian manifolds (including Euclidean spaces), we always have s = 1.
The above identity implies that the inverse of can be given as
If n is odd then k(n − k) is even for any k, whereas if n is even then k(n − k) has the parity of k. Therefore:
where k is the degree of the element operated on.
For an n-dimensional oriented pseudo-Riemannian manifold M, we apply the construction above to each cotangent space and its exterior powers , and hence to the differential k-forms , the global sections of the bundle . The Riemannian metric induces an inner product on at each point . We define the Hodge dual of a k-form , defining as the unique (n – k)-form satisfying
for every k-form , where is a real-valued function on , and the volume form is induced by the Riemannian metric. Integrating this equation over , the right side becomes the (square-integrable) inner product on k-forms, and we obtain:
Computation in index notation
We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis in a tangent space and its dual basis in , having the metric matrix and its inverse matrix . The Hodge dual of a decomposable k-form is:
Here is the Levi-Civita symbol with , and we implicitly take the sum over all values of the repeated indices . The factorial accounts for double counting, and is not present if the summation indices are restricted so that . The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to Lorentzian manifolds.
An arbitrary differential form can be written:
The factorial is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component so that the Hodge dual of the form is given by
Using the above expression for the Hodge dual of , we find:
Although one can apply this expression to any tensor , the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor. It is thus equivalent to antisymmetrization followed by applying the Hodge star.
The unit volume form is given by:
The most important application of the Hodge star on manifolds is to define the codifferential on k-forms. Let
where is the exterior derivative or differential, and for Riemannian manifolds. Then
The codifferential is not an antiderivation on the exterior algebra, in contrast to the exterior derivative.
The codifferential is the adjoint of the exterior derivative with respect to the square-integrable inner product:
where is a (k + 1)-form and a k-form. This identity follows from Stokes' theorem for smooth forms:
provided M has empty boundary, or or has zero boundary values. (The proper definition of the above requires specifying a topological vector space that is closed and complete on the space of smooth forms. The Sobolev space is conventionally used; it allows the convergent of a sequence of forms (as ) to be interchanged with the combined differential and integral operations, so that and likewise for sequences converging to .)
Since the differential satisfies , the codifferential has the corresponding property
The Laplace–deRham operator is given by
and lies at the heart of Hodge theory. It is symmetric:
The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic k-forms, and so the Hodge star induces an isomorphism of cohomology groups
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