The Hodge star

A pseudo inner product determines orthonormal bases for $${V}$$, among which we can choose a specific one $${\hat{e}_{\mu}}$$. The ordering of the $${\hat{e}_{\mu}}$$ determines a choice of orientation. This orientation uniquely determines an orthonormal basis (i.e. a unit “length” vector) for the one-dimensional vector space $${\Lambda^{n}V}$$, namely the unit n-vector (AKA orientation $${n}$$-vector, volume element)

$$\displaystyle \Omega\equiv \hat{e}_{1}\wedge\dotsb\wedge \hat{e}_{n}.$$

 Δ Many symbols are used in the literature for the unit $${n}$$-vector and related quantities, including $${\varepsilon}$$, $${i}$$, $${I}$$, and $${\omega}$$; to avoid confusion with the other common uses of these symbols, in this book we use the (non-standard) symbol $${\Omega}$$.

Since $${\Lambda^{n}V}$$ is one-dimensional, every element of $${\Lambda^{n}V}$$ is a real multiple of $${\Omega}$$. Thus $${\Omega}$$ sets up a bijection (dependent upon the inner product and choice of orientation) between $${\Lambda^{n}V}$$ and $${\Lambda^{0}V}$$ = $${\mathbb{R}}$$. In general, $${\Lambda^{k}V}$$ and $${\Lambda^{n-k}V}$$ are vector spaces of equal dimension, and thus we can also set up a bijection between them.

The Hodge star operator (AKA Hodge dual) is defined to be the linear map $${*\colon\Lambda^{k}V\to\Lambda^{n-k}V}$$ that acts on $${A\in\Lambda^{k}V}$$ such that for any $${B\in\Lambda^{n-k}V}$$, we have

$$\displaystyle A\wedge B=\left\langle *A,B\right\rangle \Omega.$$

An equivalent requirement is that $${\left\langle C\wedge*A,\Omega\right\rangle =\left\langle C,A\right\rangle }$$ for any $${C\in\Lambda^{k}V}$$. In particular, for signature $${(r,s)}$$ we immediately obtain

$$\displaystyle A\wedge*A=\left\langle *A,*A\right\rangle \Omega=(-1)^{s}\left\langle A,A\right\rangle \Omega.$$

 ◊ These relations allow one to think of the Hodge star $${*}$$ as an operator that roughly “swaps the exterior and inner products,” or alternatively that yields the “orthogonal complement with the same magnitude.”

The star operator is dependent upon a choice of inner product and orientation, but beyond that is independent of any particular basis. In particular, for any orthonormal basis $${\hat{e}_{\mu}}$$ oriented with $${\Omega}$$, we can take $${A\equiv \hat{e}_{1}\wedge\dotsb\wedge \hat{e}_{k}}$$ and $${B\equiv \hat{e}_{k+1}\wedge\dotsb\wedge \hat{e}_{n}}$$, in which case $${*A=\left\langle B,B\right\rangle B}$$, i.e. $${*A}$$ is constructed from an orthonormal basis for the orthogonal complement of $${A}$$; in fact, this relation can be used as an equivalent definition of the Hodge star.

Below we list some easily derived facts about the Hodge star operator, where $${V}$$ is $${n}$$-dimensional with unit $${n}$$-vector $${\Omega}$$ and a pseudo inner product of signature $${(r,s)}$$:

• $${*\Omega=1\Rightarrow\left(*A\right)\Omega=A}$$, if $${A\in\Lambda^{n}V}$$
• $${*1=\left(-1\right)^{s}\Omega\Rightarrow\left\langle \Omega,\Omega\right\rangle =\left(-1\right)^{s}}$$
• $${**A=\left(-1\right)^{k\left(n-k\right)+s}A=\left(-1\right)^{k\left(n-1\right)+s}A}$$, where $${A\in\Lambda^{k}V}$$
• $${A\wedge*B=B\wedge*A=(-1)^{s}\left\langle A,B\right\rangle \Omega}$$ if $${A,B\in\Lambda^{k}V}$$
• $${\left\langle A,B\right\rangle =(-1)^{s}*\left(A\wedge*B\right)}$$ if $${A,B\in\Lambda^{k}V}$$
 Δ Some authors instead define the Hodge star by the relation $${A\wedge*B=\left\langle A,B\right\rangle \Omega}$$, which differs by a sign in some cases from the more common definition we use; in particular, with this definition $${*\Omega=(-1)^{s}}$$ and $${*1=\Omega}$$.

Note that $${*A}$$ is not a basis-independent object, since it reverses sign upon changing the chosen orientation. Such an object is prefixed by the word pseudo-, e.g. $${*v}$$ is called a pseudo-vector (AKA axial vector, in which case $${v}$$ is called a polar vector) and $${\Omega}$$ itself is a pseudo-scalar.

 Δ The use of “pseudo” to indicate a quantity that reverses sign upon a change of orientation should not be confused with the use of “pseudo” to indicate an inner product that is not positive-definite. There are also other uses of “pseudo” in use. In particular, in general relativity the term “pseudo-tensor” is sometimes used, where neither of the above meanings are implied; instead this signifies that the tensor (to be defined) is not in fact a tensor.