PlanetMath: Hodge star operator

(more info)

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Hodge star operator

(Definition)

Let V be a -dimensional ( finite) vector space with inner product . The Hodge star operator (denoted by $\ast$ ) is a linear operator mapping -forms on to -forms, i.e.,

$\displaystyle \ast : \Omega^p (V)\to \Omega^{n-p}(V).$

In terms of a basis $\{e^1,\ldots, e^n\}$ for and the corresponding dual basis $\{e_1,\ldots, e_n\}$ for (the star used to denote the dual space is not to be confused with the Hodge star!), with the inner product being expressed in terms of components as $g = \sum_{i,j = 1}^n g_{ij} e^i\otimes e^j$ , the $\ast$ -operator is defined as the linear operator that maps the basis elements of $\Omega^p(V)$ as

$\displaystyle \ast(e^{i_1} \wedge \cdots \wedge e^{i_p})\!\!\!\!$

$\displaystyle =$

$\displaystyle \frac{\sqrt{\vert g\vert}}{(n-p)!} g^{i_1 l_1}\cdots g^{i_p l_p} ... ...l_1 \cdots l_p\, l_{p+1} \cdots l_n} e^{l_{p+1}}\wedge \cdots \wedge e^{l_{n}}.$

Here, $\vert g\vert=\det g_{ij}$ , and $\varepsilon$ is the Levi-Civita permutation symbol

This operator may be defined in a coordinate-free manner by the condition

$\displaystyle u \wedge *v = g (u, v) \, \mathop{\bf Vol}(g)$

where the notation

denotes the inner product on

-forms (in coordinates, $g(u,v) = g_{i_1 j_1} \cdots g_{i_p j_p} u^{i_1 \ldots i_p} v^{j_1 \ldots j_p}$ ) and $\mathop{\bf Vol}(g)$ is the unit volume form associated to the metric. (in coordinates, $\mathop{\bf Vol}(g) = \sqrt {\operatorname{det}(g)} e^1 \wedge \cdots \wedge e^n$ )

Generally $\ast \ast = (-1)^{p(n-p)} \operatorname{id}$ , where $\operatorname{id}$ is the identity operator in $\Omega^p (V)$ . In three dimensions, $\ast \ast = \operatorname{id}$ for all $p=0,\ldots,3$ . On $\mathbb{R}^3$ with Cartesian coordinates, the metric tensor is $g=dx\otimes dx + dy\otimes dy + dz\otimes dz$ , and the Hodge star operator is

$\displaystyle \ast dx = dy\wedge dz,\ \ \ \ \ \ \ast dy = dz\wedge dx,\ \ \ \ \ \ \ast dz = dx\wedge dy.$

The Hodge star operation occurs most frequently in differential geometry in the case where is a -dimensional orientable manifold with a Riemannian (or pseudo-Riemannian) tensor and is a cotangent vector space of . Also, one can extend this notion to antisymmetric tensor fields by computing Hodge star pointwise.