Lorentzian Hodge Star in Ten Dimensions

Let $M$ be an oriented smooth $10$-manifold. Its orientation is a smooth choice of ray in $\det T_{x}M$ at each $x$; an oriented chart has positively oriented coordinate frame. Let $g$ be a smooth symmetric covariant $2$-tensor which is nondegenerate with inertia $(9,1,0)$ at every point, so in an oriented pseudo-orthonormal coframe its matrix is $\mathrm{diag}(-1,+1,\dots ,+1)$. In a positively oriented chart $(x^0,\dots ,x^9)$ define the metric volume form by

${\mathrm{vol}}_g=\sqrt{\mathrm{\mid }\det (g_{\mu \nu })\mathrm{\mid }}d{x}^0\wedge \cdots \wedge d{x}^9\mathrm{.}$

For smooth real $p$-forms $\alpha ,\beta \in {\mathrm{\Omega }}^p(M)$, $0\le p\le 10$, define the induced (generally indefinite) pointwise pairing by

$⟨\alpha ,\beta {⟩}_g=\frac{1}{p!}{\alpha }_{{\mu }_1\cdots {\mu }_p}{\beta }^{{\mu }_1\cdots {\mu }_p},$

where every upper index is raised with $g^{-1}$. The Lorentzian Hodge star is the pointwise linear map ${\ast }_g:{\mathrm{\Lambda }}^p{T}_x^{\ast }M\to {\mathrm{\Lambda }}^{10-p}{T}_x^{\ast }M$ characterized by

$\alpha \wedge {\ast }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{vol}}_g\mathrm{.}$

The equation is required for every $\alpha ,\beta$ in each cotangent exterior power. The local theorem Existence and Five-Form Properties of the Lorentzian Hodge Star proves that the metric volume form is well-defined and smooth and that this equation determines a unique smooth bundle isomorphism. Positivity of the induced pairing is not assumed.

Existence and Five-Form Properties of the Lorentzian Hodge Star

Smooth manifolds and their smooth charts

Oriented smooth manifolds and oriented charts

A smooth differential $k$-form

A smooth tensor field

Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms

Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form

The exterior algebra of covectors

The wedge product of differential forms

The metric volume form uses $\mathrm{\mid }\det g\mathrm{\mid }$ because the metric is Lorentzian. The positive-definite Hodge-star definition does not supply this Lorentzian operator.