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Hodge star operator

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In mathematics, the Hodge star operator or Hodge star is a linear map introduced by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element of the algebra is called the element's Hodge dual.

For example, in 3-dimensional Euclidean space, every oriented plane has a unique normal vector, and every vector can be used to define a plane perpendicular to that vector. The Hodge star can be thought of as generalizing this relationship: in general, for an n-dimensional vector space, the Hodge star maps k-vectors to (nk)-vectors and vice versa.

Contents

Dimensions and algebraEdit

Suppose that n is the dimension of the oriented inner product space and k is an integer such that 0 ≤ kn, then the Hodge star operator establishes a one-to-one mapping from the space of k-vectors to the space of (nk)-vectors. The image of a k-vector under this mapping is called its Hodge dual. The former space, of k-vectors, has dimension  , while the latter has dimension  , which are equal by the symmetry of the binomial coefficients. Equal-dimensional vector spaces are always isomorphic, but not necessarily in a natural or canonical way. In this case, however, Hodge duality exploits the nondegenerate symmetric bilinear form, hereafter referred to as the inner product (though it might not be positive definite), and a choice of orientation to single out a unique isomorphism, which parallels the combinatorial symmetry of binomial coefficients. This in turn induces an inner product on the space of k-vectors. The naturalness of this definition means the duality can play a role in differential geometry.

The first interesting case is on three-dimensional Euclidean space V. In this context the relevant row of Pascal's triangle reads

1, 3, 3, 1

and the Hodge star sets up an isomorphism between the two three-dimensional spaces, which are V and the image space of the exterior product acting on pairs of vectors in V. See § Examples for details. In this case the Hodge star allows the definition of the cross product of traditional vector calculus in terms of the exterior product. While the properties of the cross product are special to three dimensions, the Hodge star applies to an arbitrary number of dimensions.

ExtensionsEdit

Since the space of alternating linear forms in k arguments on a vector space is naturally isomorphic to the dual of the space of k-vectors over that vector space, the Hodge star can be defined for these spaces as well. As with most constructions from linear algebra, the Hodge star can then be extended to a vector bundle. Thus a context in which the Hodge star is very often seen is the exterior algebra of the cotangent bundle, the space of differential forms on a manifold, where it can be used to construct the codifferential from the exterior derivative, and thus the Laplace–de Rham operator, which leads to the Hodge decomposition of differential forms in the case of compact Riemannian manifolds.

Formal definition for k-vectorsEdit

The Hodge star operator on a vector space V with an inner product is a linear operator on the exterior algebra of V, mapping k-vectors to (nk)-vectors where n = dim V, for 0 ≤ kn. It has the following property, which defines it completely: given two k-vectors α, β,

 

where   denotes the inner product on k-vectors and ω is the preferred unit n-vector.[1]:15

The inner product   on k-vectors is extended from that on V by requiring that for any decomposable k-vectors   and   it equals the determinant[1]:14

  sometimes called the Gram determinant.

The unit n-vector ω is unique up to a sign. The preferred choice of ω defines an orientation on V.

ExplanationEdit

Let W be a vector space, with an inner product  . The Riesz representation theorem states that for every continuous (even in the infinite-dimensional case) linear functional   there exists a unique vector v in W such that   for all w in W. The map   given by   is an isomorphism. (If W is complex, the map is conjugate linear as opposed to complex linear.) This holds for all vector spaces with an inner product, and can be used to explain the Hodge star.

Let V be an n-dimensional vector space with basis  . For 0 ≤ kn, consider the exterior power spaces   and  . For

 

we have

 

There is, up to a scalar, only one n-vector (an n-form), namely  . In other words,   must be a scalar multiple of   for all   and  .

Consider a fixed  . There exists a unique linear function

 

such that

 

This   is the scalar multiple mentioned in the previous paragraph. If   denotes the inner product on (nk)-vectors, then there exists a unique (nk)-vector, say

 

such that

 

This (nk)-vector λ is the Hodge dual of λ, and is the image of the   under the isomorphism induced by the inner product,

 

Thus,

 

Computation of the Hodge starEdit

Given an orthonormal basis   ordered such that  , for a positive-definite metric, we see that

 

where   is an even permutation of {1, 2, ..., n}.

Of these   relations, only   are independent. The first one in the usual lexicographical order reads

 

Expression in index notationEdit

Using tensor index notation, the Hodge dual of an arbitrary wedge product of one-forms is given by the following:

 

The symbol   is the Levi-Civita symbol defined so that   and   is the inverse metric. Note that the factorial   had to be inserted to account for double counting, and cancels out if one orders the summation indices so that  . The absolute value of the determinant is necessary since it may be negative, e.g. for tangent spaces to Lorentzian manifolds.

An arbitrary differential form can be expanded into its components as follows:

 

The factorial   is again included to account for double counting. We would like to define the dual of the component   so that the Hodge dual of the form is given by

 

Thus using the expression for the Hodge dual of   given at the beginning of this section we find[2]

 

It is understood that indices are raised and lowered using the same inner product   as in the definition of the Levi-Civita tensor. 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.

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, 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 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 star operator is the case n = 3, when it can be taken as the correspondence between the vectors and the skew-symmetric matrices of that size. This is used implicitly in vector calculus, for example to create the cross product vector from the wedge product of two vectors. Specifically, for Euclidean R3, one easily finds that

 

where dx, dy and dz are the standard orthonormal differential one-forms on R3. The Hodge star in this case clearly relates the cross-product to the wedge product in three dimensions. A detailed presentation not restricted to differential geometry is provided next.

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:[3]

 

where is the Hodge star. These dual relations can be implemented using multiplication by the unit pseudoscalar in Cl3(R),[4] i = e1e2e3 (the vectors {e} are an orthonormal basis in three dimensional Euclidean space) according to the relations:[5]

 

The dual of a vector is obtained by multiplication by i, as established using the properties of the geometric product of the algebra as follows:

 

and also, in the dual space spanned by {eem}:

 

In establishing these results, the identities are used:

 

and:

 

These relations between the dual and i apply to any vectors. Here they are applied to relate the axial vector created as the cross product a = u × v to the bivector-valued exterior product A = uv of two polar (that is, not axial) vectors u and v; the two products can be written as determinants expressed in the same way:

 

using the notation em = eem. These expressions show these two types of vector are Hodge duals:[3]

 

as a result of the relations:

 

with ℓ, m, n cyclic,

and:

 

also with , m, n cyclic.

Using the implementation of based upon i, the commonly used relations are:[6]

 

Four dimensionsEdit

In case n = 4, the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps two-forms to two-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 n = 4 Minkowski spacetime with metric signature (+ − − −) and coordinates (t, x, y, z) where (using  ):

 

for one-forms while

 

for two-forms. Because their determinants are the same in both (+ − − −) and (− + + +), the signs of the Minkowski space two-form duals depend only on the chosen orientation.[verification needed]

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  . 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 a minus sign only if   involves  .)

Inner product of k-vectorsEdit

The Hodge star induces an inner product on the space of k-vectors, that is, on the exterior algebra of V. Given two k-vectors η and ζ, one has

 

where ω is the normalised n-form (i.e. ω ∧ ⋆ω = ω). In the calculus of exterior differential forms on a pseudo-Riemannian manifold of dimension n, the normalised n-form is called the volume form and can be written as

 

where   is the matrix of components of the metric tensor on the manifold in the coordinate basis.

If an inner product is given on  , then this equation can be regarded as an alternative definition of the Hodge star.[7]

The ordered wedge products of k distinct orthonormal basis vectors of V form an orthonormal basis on each subspace   of the exterior algebra of V.

DualityEdit

Applying the Hodge star twice leaves a k-vector unchanged, up to sign. Given a k-vector η in k(V) in an n-dimensional space V, one has

 

where s is related to the signature of the inner product on V. Specifically, s is the sign of the determinant of the matrix representation of the inner product tensor with respect to any basis. Thus, for example, if n = 4 and the signature of the inner product is either (+ − − −) or (− + + +) then s = −1. For Riemannian manifolds (including Euclidean spaces), the signature is always positive, and so s = 1.

Note that the above identity implies that the inverse of can be given as

 

Note that 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

Applying the construction above to each cotangent space of an n-dimensional oriented Riemannian or pseudo-Riemannian manifold, we can obtain an object known as the Hodge dual of a k-form. To be explicit –

For any k-form ζ we define ζ as the unique (nk)-form satisfying

 

for every k-form η (here the inner product on forms and the volume form ω are induced by the Riemannian metric tensor in the usual way; greater understanding of these objects can be found by learning about the inner product on k-forms and the volume form).

The Hodge star is thus related to the L2 inner product on k-forms by the formula:

 

for k-forms η and ζ. (Note that we can also see this as an inner product on sections of  . The set of sections is frequently denoted as  . Each element of   is a k-form.)

More generally, in the non-oriented case, 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.

CodifferentialEdit

The most important application of the Hodge star on manifolds is to define the codifferential δ on k-forms. Let

 

where d is the exterior derivative or differential, and s = 1 for Riemannian manifolds.

 

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, in that

 

where ζ is a (k + 1)-form and η a k-form. This identity follows from Stokes' theorem for smooth forms, when

 

i.e. when M has empty boundary or when η or ζ has zero boundary values (of course, true adjointness follows after continuous continuation to the appropriate topological vector spaces as closures of the spaces of smooth forms).

Notice that since the differential satisfies d2 = 0, 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 the 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.

Derivatives in three dimensionsEdit

The combination of the operator and the exterior derivative d generates the classical operators grad, curl, and div, in three-dimensional Euclidean space. This works out as follows: d can take a 0-form (function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (applied to a 3-form it just gives zero). For a 0-form,  , the first case written out in components is identifiable as the grad operator:

 

The second case followed by is an operator on 1-forms ( ) that in components is the curl operator:

 

Applying the Hodge star gives:

 

The final case prefaced and followed by , takes a 1-form ( ) to a 0-form (function); written out in components it is the divergence operator:

 

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.

One can also obtain the Laplacian. Using the information above and the fact that Δ f = div grad f then for a 0-form,  :

 

NotesEdit

  1. ^ a b Harley Flanders (1963) Differential Forms with Applications to the Physical Sciences, Academic Press
  2. ^ Frankel, T. (2012). The Geometry of Physics (3rd ed.). Cambridge University Press. ISBN 978-1-107-60260-1.
  3. ^ 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.
  4. ^ Venzo De Sabbata, Bidyut Kumar Datta (2007). "The pseudoscalar and imaginary unit". Geometric algebra and applications to physics. CRC Press. p. 53 ff. ISBN 1-58488-772-9.
  5. ^ William E Baylis (2004). "Chapter 4: Applications of Clifford algebras in physics". In Rafal Ablamowicz, Garret Sobczyk (ed.). Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100 ff. ISBN 0-8176-3257-3.
  6. ^ David Hestenes (1999). "The vector cross product". New foundations for classical mechanics: Fundamental Theories of Physics (2nd ed.). Springer. p. 60. ISBN 0-7923-5302-1.
  7. ^ Darling, R. W. R. (1994). Differential forms and connections. Cambridge University Press.

ReferencesEdit