Hodge star operator

Summary

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-vectorsEdit

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[1]: 14 

 

extended to   through linearity.

The unit n-vector   is defined in terms of an oriented orthonormal basis   of V as:

 

The Hodge star operator is a linear operator on the exterior algebra of V, mapping k-vectors to (nk)-vectors, for  . It has the following property, which defines it completely:[1]: 15 

  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  .

Geometric explanationEdit

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 (nk)-vector:

 

where   form an oriented basis of the orthogonal space  . Furthermore, the (nk)-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.

ExamplesEdit

Two dimensionsEdit

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

 
so that
 
proving the claimed invariance.

Three dimensionsEdit

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:[2]

 
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:[2]  .

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:  .

Four dimensionsEdit

In case  , 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, i.e., pseudo-Riemannian, then application twice will return the argument up to a sign – see § Duality 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 operation with eigenvalues   (or  , depending on the signature).

For concreteness, we discuss Hodge dual 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 for 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  .[verification needed] An extra minus sign will enter only if   contains  . (For (+ − − −), one puts in a minus sign only if   involves an odd number of the space-associated forms  ,   and  .)

Note that, the combinations

 
takes   as the eigenvalue for Hodge dual, 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 invarianceEdit

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 dimensionsEdit

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, has as special cases two other identities: 1) curl grad f = 0, and 2) 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   (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 Δf = 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 straightforward calculations one obtains the Laplacian acting on  .

DualityEdit

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(nk) is even for any k, whereas if n is even then k(nk) has the parity of k. Therefore:

 

where k is the degree of the element operated on.

On manifoldsEdit

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 (nk)-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:
 

More generally, if   is non-oriented, one can define the Hodge star of a k-form as a (nk)-pseudo differential form; that is, a differential form with values in the canonical line bundle.

Computation in index notationEdit

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 as follows:

 

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:[3]

 

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:

 

CodifferentialEdit

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
 
while
 

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 property is useful as it can be used to define the codifferential even when the manifold is non-orientable (and the Hodge star operator not defined). The identity can be proved 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 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:
 
and non-negative:
 

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

 
which in turn gives canonical identifications via Poincaré duality of H k(M) with its dual space

In coordinates, with notation as above, the codifferential of the form   may be written as

 
where here   denotes the Christoffel symbols of  .

CitationsEdit

  1. ^ a b Harley Flanders (1963) Differential Forms with Applications to the Physical Sciences, Academic Press
  2. ^ a b Pertti Lounesto (2001). "§3.6 The Hodge dual". Clifford Algebras and Spinors, Volume 286 of London Mathematical Society Lecture Note Series (2nd ed.). Cambridge University Press. p. 39. ISBN 0-521-00551-5.
  3. ^ Frankel, T. (2012). The Geometry of Physics (3rd ed.). Cambridge University Press. ISBN 978-1-107-60260-1.

ReferencesEdit