Hodge_star_operator

Hodge star operator

In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.

1 Dimensions and algebra
2 Extensions
3 Formal definition of the Hodge star of k-vectors
4 Index notation for the star operator
5 Examples
6 Inner product of k-vectors
7 Duality
8 Hodge star on manifolds
- 8.1 The codifferential
9 Derivatives in three dimensions
10 Notes
11 References

Dimensions and algebra

The Hodge star operator establishes a correspondence between the space of k-vectors and the space of (n −k)-vectors. The image of a k-vector under this isomorphism is called the Hodge dual of the k-vector. The former space, of k-vectors, has dimension

${n \choose k}$

while the latter has dimension

${n \choose n - k},$

and by the symmetry of the binomial coefficients, these two dimensions are in fact equal. Two vector spaces with the same dimension are always isomorphic; but not necessarily in a natural or canonical way. The Hodge duality, however, in this case exploits the inner product and orientation of the vector space. It singles out a unique isomorphism, that reflects therefore the pattern of the binomial coefficients in algebra. This in turn induces an inner product on the space of k-vectors. The 'natural' definition means that this duality relationship can play a geometrical role in theories.

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 dual sets up an isomorphism between the two spaces of dimension 3, which are V itself and the space of wedge products of two vectors from V. See the Examples section for details. In this case the content is just that of the cross product of traditional vector calculus. While the properties of the cross product are special to three dimensions, the Hodge dual is available in all dimensions.

Extensions

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 dual can be defined for these spaces as well. As with most constructions from linear algebra, the Hodge dual can then be extended to a vector bundle. Thus a context in which the Hodge dual is very often seen is the exterior algebra of the cotangent bundle (i.e. 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 of the Hodge star of k-vectors

The Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. It has the following property, which defines it completely: given an oriented orthonormal basis $e 1, e 2,..., e n$ we have

$*(e_1\wedge e_2\wedge ... \wedge e_k)= e_{k+1}\wedge e_{k+2}\wedge ... \wedge e_n.$

Index notation for the star operator

Using index notation, the Hodge dual is obtained by contracting the indices of a k-form with the n-dimensional completely antisymmetric Levi-Civita tensor. This differs from the Levi-Civita symbol by an extra-factor of (det g)^½, where g is an inner product. (one uses an absolute value around the determinant if g is not positive-definite, e.g. for tangent spaces to Lorentzian manifolds)

Thus one writes

$(*\eta)_{i_1,i_2,\ldots,i_{n-k}}= \frac{1}{k!} \eta^{j_1,\ldots,j_k}\,\sqrt {|\det g|} \,\epsilon_{j_1,\ldots,j_k,i_1,\ldots,i_{n-k}}$

where η is an arbitrary antisymmetric tensor in k indices. It is understood that indices are raised and lowered using the same inner product g as in the definition of the Levi-Civita tensor. Although one can take the star of any tensor, the result is antisymmetric, since the symmetric components of the tensor completely cancel out when contracted with the completely anti-symmetric Levi-Civita symbol.

Examples

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

$*\mathrm{d}x=\mathrm{d}y\wedge \mathrm{d}z$

and

$*\mathrm{d}y=\mathrm{d}z\wedge \mathrm{d}x$

and

$*\mathrm{d}z=\mathrm{d}x\wedge \mathrm{d}y$

where dx, dy and dz are the standard orthonormal differential one-forms on R³. The Hodge dual in this case clearly corresponds to the cross-product in three dimensions.

In case n = 4 the Hodge dual acts as an endomorphism of the second exterior power, of dimension 6. It is an involution, so it splits it into self-dual and anti-self-dual subspaces, on which it acts respectively as +1 and −1.

Another useful example is n=4 Minkowski spacetime with metric signature (+,-,-,-) and coordinates (t,x,y,z) where

$*\,\mathrm{d}t=\mathrm{d}x\wedge \mathrm{d}y \wedge\mathrm{d}z$

$*\,\mathrm{d}x=\mathrm{d}t\wedge \mathrm{d}y \wedge\mathrm{d}z$

$*\,\mathrm{d}y=-\mathrm{d}t\wedge \mathrm{d}x \wedge\mathrm{d}z$

$*\,\mathrm{d}z=\mathrm{d}t\wedge \mathrm{d}x \wedge\mathrm{d}y$

for one-forms while

$*\, \mathrm{d}t \wedge\mathrm{d}x = - \mathrm{d}y\wedge \mathrm{d}z$

$*\, \mathrm{d}t \wedge\mathrm{d}y = \mathrm{d}x\wedge \mathrm{d}z$

$*\, \mathrm{d}t \wedge\mathrm{d}z = - \mathrm{d}x\wedge \mathrm{d}y$

$*\, \mathrm{d}x \wedge\mathrm{d}y = \mathrm{d}t\wedge \mathrm{d}z$

$*\, \mathrm{d}x \wedge\mathrm{d}z = - \mathrm{d}t\wedge \mathrm{d}y$

$*\, \mathrm{d}y \wedge\mathrm{d}z = \mathrm{d}t\wedge \mathrm{d}x$

for two-forms.

Inner product of k-vectors

The Hodge dual 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

$\zeta\wedge *\eta = \langle\zeta, \eta \rangle\;\omega$

where ω is the normalised volume form. It can be shown that $\langle\cdot,\cdot\rangle$ is an inner product, in that it is sesquilinear and defines a norm. Conversely, if an inner product is given on $Λ k (V)$ , then this equation can be regarded as an alternative definition of the Hodge dual¹.

In essence, the wedge products of elements of an orthonormal basis in V forms an orthonormal basis of the exterior algebra of V. When the Hodge star is extended to manifolds, as shown in a later section, the volume form can be written as

$\omega=\sqrt{|\det g_{ij}|}\;dx^1\wedge\ldots\wedge dx^n$

where $g i j$ is the metric on the manifold.

Duality

The Hodge star defines a dual in that when it is applied twice, the result is an identity on the exterior algebra, up to sign. Given a k-vector $\eta \in \Lambda^k (V)$ in an n-dimensional space V, one has

$**\eta=(-1)^{k(n-k)}s\;\eta$

where s is related to the signature of the inner product on V. Specifically, s is the sign of the determinant of the inner product tensor. Thus, for example, if n=4 and the signature of the inner product is either (+,−,−,−) or (−,+,+,+) then s=-1. For ordinary Euclidean spaces, the signature is always positive, and so s=+1. In ordinary vector spaces, this is not normally an issue. When the Hodge star is extended to pseudo-Riemannian manifolds, then the above inner product is understood to be the metric in diagonal form.

Hodge star on manifolds

One can repeat the construction above for each tangent space of an n-dimensional oriented Riemannian or pseudo-Riemannian manifold, and get the Hodge dual n− k-form, of a k-form. The Hodge star then induces an L²-norm inner product on the differential forms on the manifold. One writes

$(\eta,\zeta)=\int_M \eta\wedge *\zeta$

for the inner product of space sections $η$ and $ζ$ of $Λ k (M)$ . (The set of sections is frequently denoted as $Ω k (M) = Γ(Λ k (M))$ . Elements of $Ω k (M)$ are called exterior k-forms).

More generally, in the non-oriented case, one can define the hodge star of a k-form as a (n− k)-pseudo differential form; that is, a differential forms with values in the canonical line bundle.

The codifferential

The most important application of the Hodge dual on manifolds to is to define the codifferential δ. Let

δ = ( - 1) n k + n + 1 * d *

where d is the exterior derivative. s=+1 for Riemannian manifolds.

$d:\Omega^k(M)\rightarrow \Omega^{k+1}(M)$

while

$\delta:\Omega^k(M)\rightarrow \Omega^{k-1}(M)$ .

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

(δζ,η) = (ζ, d η)

This identity follows from the fact that for the volume form ω one has dω=0 and thus

$\int_M d(\zeta \wedge *\eta)=0$

The Laplace-deRham operator is given by

Δ = δ d + d δ

and lies at the heart of Hodge theory. It is symmetric:

(Δζ,η) = (ζ,Δη)

and non-negative:

$(\Delta\eta,\eta) \ge 0$ .

The Hodge dual sends harmonic forms to harmonic forms. As a consequence of the Hodge theorem, 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

$\star : H^k_\Delta(M)\to H^{n-k}_\Delta(M),$

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

Derivatives in three dimensions

The combination of the $\ast$ operator and the exterior derivative $d$ generates the classical operators grad, curl, and div, in three dimensions. 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, $ω = f (x, y, z)$ , the first case written out in components is identifiable as the grad operator:

$d\omega=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz.$

The second case followed by $\ast$ is an operator on 1-forms ( $ω = A d x + B d y + C d z$ ) that in components is the $c u r l$ operator:

$d\omega=\left({\partial C \over \partial y} - {\partial B \over \partial z}\right)dy\wedge dz + \left({\partial C \over \partial x} - {\partial A \over \partial z}\right)dx\wedge dz+\left({\partial B \over \partial x} - {\partial A \over \partial y}\right)dx\wedge dy.$

Applying the Hodge star gives:

$\ast d\omega=\left({\partial C \over \partial y} - {\partial B \over \partial z}\right)dx - \left({\partial C \over \partial x} - {\partial A \over \partial z}\right)dy+\left({\partial B \over \partial x} - {\partial A \over \partial y}\right)dz.$

The final case prefaced and followed by $\ast$ , takes a 1-form ( $ω = A d x + B d y + C d z$ ) to a 0-form (function); written out in components it is the divergence operator:

$\ast\omega=Ady\wedge dz-Bdx\wedge dz+Cdx\wedge dy$

$d\ast\omega=\left(\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}\right)dx\wedge dy\wedge dz$

$\ast d\ast\omega=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}.$

One advantage of this expression is that the identity $d 2 = 0$ , which is true in all cases, sums up two others, namely that $\operatorname{curl}(\operatorname{grad}(f))=\operatorname{div}(\operatorname{curl}(\mathbf{F}))=0$ . 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.

Notes

^ Darling, R. W. R. (1994). Differential forms and connections, Cambridge University Press.

References

Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (Provides a basic review of differential geometry in the special case of four-dimensional space-time.)
Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 . (Provides a detailed exposition starting from basic principles, but does not treat the pseudo-Riemannian case).
David Bleecker, Gauge Theory and Variational Principles, (1981) Addison-Wesley Publishing, New York' ISBN 0-201-10096-7. (Provides condensed review of non-Riemannian differential geometry in chapter 0).

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