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 that yields the “orthogonal complement with the same magnitude,” or alternatively that “swaps the exterior and inner products.”

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 as we can see from the penultimate relation above prefixes our Hodge star by the factor \({(-1)^{s}}\).

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.

An Illustrated Handbook