# Hodge star operator

Let V be a $n$-dimensional ($n$ finite) vector space with inner product $g$. The Hodge star operator (denoted by $\ast$) is a linear operator mapping http://planetmath.org/node/3050$p$-forms on $V$ to $(n-p)$-forms, i.e.,

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

In terms of a basis $\{e^{1},\ldots,e^{n}\}$ for $V$ and the corresponding dual basis $\{e_{1},\ldots,e_{n}\}$ for $V^{*}$ (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

 $\displaystyle\ast(e^{i_{1}}\wedge\cdots\wedge e^{i_{p}})$ $\displaystyle=$ $\displaystyle\!\!\!\!\frac{\sqrt{|g|}}{(n-p)!}g^{i_{1}l_{1}}\cdots g^{i_{p}l_{% p}}\varepsilon_{l_{1}\cdots l_{p}\,l_{p+1}\cdots l_{n}}e^{l_{p+1}}\wedge\cdots% \wedge e^{l_{n}}.$

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

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

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

where the notation $g(u,v)$ denotes the inner product on $p$-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 $M^{n}$ is a $n$-dimensional orientable manifold with a Riemannian (or pseudo-Riemannian) tensor $g$ and $V$ is a cotangent vector space of $M^{n}$. Also, one can extend this notion to antisymmetric tensor fields by computing Hodge star pointwise.

Title Hodge star operator HodgeStarOperator 2013-03-22 13:31:41 2013-03-22 13:31:41 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Definition msc 53B21 Hodge operator star operator hodge star operator