Hodge dual
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In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. Note that the former space is n choose k-dimensional while the latter is n choose (n-k) dimensional, and by the symmetry of the combination function, these two dimensions are equal. Although any two vector spaces with the same dimension are isomorphic, they are not canonically isomorphic. The Hodge dual however leverages the inner product and orientation of the vector space to single out a unique isomorphism. This in turn induces an inner product on the space of k-vectors.
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-deRham operator, which leads to the Hodge decomposition of differential forms in the case of compact Riemannian manifolds.
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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 <math>e_1,e_2,...,e_n<math> we have
- <math>*(e_1\wedge e_2\wedge ... \wedge e_k)= e_{k+1}\wedge e_{k+2}\wedge ... \wedge e_n.<math>
Using index notation, the Hodge dual is obtained by contracting the indices of a k-form with the n-dimensional completely antisymmetric Levi-Civita symbol. Thus one writes
- <math>*\eta_{a_1,a_2,\ldots,a_k}=
\frac{1}{k!}\epsilon_{a_1,\ldots,a_n} \eta^{a_{k+1},\ldots,a_n}<math>
where η is an arbitrary antisymmetric tensor in k indices. When written in this form, it becomes clear that the vector space must have an inner product defined; it is needed to raise and lower the indeces of the contracted tensor. Using the inner product g to raise and lower indeces is equivalent (although not obviously so) to prefixing the Levi-Civita symbol with the square root of the determinant of the inner product <math>\sqrt {|g|}<math> in order to form the volume element. Note also, that 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
- <math>*dx=dy\wedge dz<math>
and
- <math>*dy=dz\wedge dx<math>
and
- <math>*dz=dx\wedge dy<math>
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 an endomorphism of the second exterior power, of dimension 6; it splits it into self-dual and anti-self-dual subspaces, on which it acts respectively as +1 and −1.
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 <math>\eta<math> and <math>\zeta<math>, one has
- <math>\zeta\wedge *\eta = \langle\zeta, \eta \rangle\;\omega<math>
where ω is the normalised volume form. It can be shown that <math>\langle\cdot,\cdot\rangle <math> is an inner product, in that it is sesquilinear and defines a norm. 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
- <math>\omega=\sqrt{|\det g_{ij}|}\;dx^1\wedge\ldots\wedge dx^n<math>
where <math>g_{ij}<math> 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 <math>\eta \in \Lambda^k (V)<math> in an n-dimensional space V, one has
- <math>**\eta=(-1)^{k(n-k)}s\;\eta<math>
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. More generally, in the non-oriented case, one can define the hodge star of a k-form as a n− k- pseudo differential form.
In the oriented case, the Hodge star then induces an L2-norm inner product on the differential forms on the manifold. One writes
- <math>(\eta,\zeta)=\int_M \eta\wedge *\zeta<math>
for the inner product of space sections <math>\eta<math> and <math>\zeta<math> of <math>\Lambda^k(M)<math>. (Note that the set of sections is frequently denoted as <math>\Omega^k(M)=\Gamma(\Lambda^k(M))<math>. Elements of <math>\Omega^k(M)<math> are called exterior k-forms).
Derivatives in three dimensions
The combination of the * operator and the exterior derivative d generates the classical operators div, grad and curl, 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, or a 2-form to a 3-form (applied to a 3-form it just gives zero). The first case written out in components is identifiable as the grad operator. The second followed by * is an operator on 1-forms that in components is curl. The final case prefaced and followed by *, so *d*, takes a 1-form to a 0-form (function); written out in components it is div. One advantage of this expression is that the identity d2 = 0, which is true in all cases, sums up two others, namely curl of a grad and div of a curl are identically zero.
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 codifferential
The most important application of the Hodge dual on manifolds to is to define the codifferential δ. Let
- <math>\delta = (-1)^{n(k+1)+1} s\; *d*<math>
where d is the exterior derivative. Note s=+1 for Riemannian manifolds. Note that
- <math>d:\Omega^k(M)\rightarrow \Omega^{k+1}(M)<math>
while
- <math>\delta:\Omega^k(M)\rightarrow \Omega^{k-1}(M)<math>.
The codifferential is the adjoint of the exterior derivative, in that
- <math>(\delta \zeta, \eta)= (\zeta, d\eta)<math>
This identity follows from the fact that for the volume form ω one has dω=0 and thus
- <math>\int_M d(\zeta \wedge *\eta)=0<math>
The Laplace-deRahm operator is given by
- <math>\Delta=\delta d + d\delta<math>
and lies at the heart of Hodge theory. It is symmetric:
- <math>(\Delta \zeta,\eta) = (\zeta,\Delta \eta)<math>
and non-negative:
- <math>(\Delta\eta,\eta) \ge 0<math>.
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 3-540-4267-2 . (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).