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.
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.
Let V be an n-dimensional oriented vector space with a nondegenerate symmetric bilinear form
(In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive-definite.)
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
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.
The Hodge star relates the exterior and cross product in three dimensions:[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 applying the operator 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 operator with eigenvalues
Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature,
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.
The Hodge star is conformally invariant on n-forms on a 2n-dimensional vector space
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.
(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:
Applying the Hodge star twice leaves a k-vector unchanged up to a sign: for
For an n-dimensional oriented pseudo-Riemannian manifold M, we apply the construction above to each cotangent space
We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis
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.
The most important application of the Hodge star on manifolds is to define the codifferential
The codifferential is the adjoint of the exterior derivative with respect to the square-integrable scalar product:
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 analogy to the Poincare lemma for exterior derivative, one can define its version for codifferential, which reads[4] A practical way of finding
make a direct sum decomposition[4] This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and the projector operators on the summands fulfills idempotence formulas:[4]