Hodge-Star and Differential-Form Identities

Hodge identities depend on degree, dimension, metric index, and orientation; their integration formulas also depend on support and boundary data. This card fixes those inputs before stating the algebra. It uses the metric signature $(-,+,+,+)$ only in the four-dimensional specialization, while the general formulas apply to any fixed nondegenerate signature.

Helpful background. Tensor Fields and Natural Operations supplies metric contractions; Affine Connections and Covariant Derivatives supplies the geometric setting used in applications.

Hodge pairing and exterior algebra

Let $(M,g)$ be an oriented pseudo-Riemannian $n$-manifold, let $s$ be the number of negative metric directions, and let $\mathrm{vol}_g$ be the volume form selected by $g$ and the orientation. For $\alpha,\beta\in\Omega^p(M)$, define $\star:\Omega^p(M)\to\Omega^{n-p}(M)$ by

$$\alpha\wedge\star\beta =\langle\alpha,\beta\rangle_g\,\mathrm{vol}_g.$$

With this definition on real forms,

$$\star^2\big|_{\Omega^p(M)} =(-1)^{p(n-p)+s}.$$

The pairing may be indefinite. Reversing orientation reverses $\star$, and a degenerate metric supplies no Hodge star because it has no inverse. In Lorentzian coordinates, the determinant and oriented volume conventions are summarized in Tong 2019, §3.1, pp. 91–99.

The inner product on forms includes the usual inverse-factorial normalization and raises all $p$ indices with $g^{-1}$. Consequently, $\alpha\wedge\star\alpha$ is not positive in indefinite signature and can even vanish for a nonzero null form. Reversing orientation changes each single star but leaves $\star^2$ unchanged. These observations distinguish the algebraic square rule from any positivity claim and prevent an orientation choice from being mistaken for a metric-signature choice.

For $\alpha\in\Omega^p(M)$, $\beta\in\Omega^q(M)$, and a vector field $X$ with metric dual $X^\flat$, the main wedge and interior-product rules are

$$\begin{aligned} \alpha\wedge\beta &=(-1)^{pq}\beta\wedge\alpha,\\ \iota_X(\alpha\wedge\beta) &=(\iota_X\alpha)\wedge\beta +(-1)^p\alpha\wedge\iota_X\beta,\\ \iota_X(\star\alpha) &=\star(\alpha\wedge X^\flat),\\ \star(\iota_X\alpha) &=(-1)^{p-1}X^\flat\wedge\star\alpha. \end{aligned}$$

The first two identities are metric-free; the last uses the same metric and orientation on both sides. Alternating tensors, wedge products, contraction, and pullback are developed in Guillemin and Haine 2019, Ch. 1, §§1.4–1.9.

Moving $X^\flat$ to the left gives the equivalent rule $\star(X^\flat\wedge\alpha)=(-1)^p\iota_X\star\alpha$. The degree sign comes only from graded commutativity. It should not be guessed independently or supplemented by another signature sign: the metric index has already entered through the definition of $X^\flat$ and the Hodge star.

Codifferential and the boundary term

For a $k$-form $\beta$, declare the codifferential convention

$$\delta\beta =(-1)^{nk+n+s+1}\star d\star\beta.$$

This sign is part of the definition; it must not be copied from a positive-definite formula after changing signature. The square identity gives

$$\star\delta\beta=(-1)^k d\star\beta.$$

Let $\alpha\in\Omega^{k-1}(M)$. The graded Leibniz rule then yields

$$\begin{aligned} d(\alpha\wedge\star\beta) &=d\alpha\wedge\star\beta +(-1)^{k-1}\alpha\wedge d\star\beta\\ &=d\alpha\wedge\star\beta -\alpha\wedge\star\delta\beta. \end{aligned}$$

If $M$ is compact with smooth boundary, Stokes’ theorem with the outward-normal-first boundary orientation gives

$$\int_M\langle d\alpha,\beta\rangle_g\,\mathrm{vol}_g =\int_M\langle\alpha,\delta\beta\rangle_g\,\mathrm{vol}_g +\int_{\partial M}j^*(\alpha\wedge\star\beta),$$

where $j:\partial M\hookrightarrow M$ is the inclusion. Thus $\delta$ is the formal adjoint of $d$ when $\partial M$ is empty, when the forms are compactly supported away from the boundary, or when imposed boundary conditions make the pulled-back boundary form vanish. On a noncompact manifold, decay and convergence must replace compactness explicitly. These hypotheses and the induced orientation are covered in Wendl 2021–2022, §§9–13, pp. 69–118 and Guillemin and Haine 2019, Ch. 4, pp. 111–153.

The differential-form boundary term remains meaningful without introducing a unit normal. In particular, a null boundary should not be handled by importing a non-null unit-normal formula; Stokes’ pullback identity is the safe statement.

The displayed pullback is also the most economical boundary condition: requiring it to vanish states exactly what formal adjointness needs, without assuming a particular decomposition into normal and tangential components. On a non-null boundary one may translate it into absolute or relative conditions using a chosen outward unit normal and the induced metric. On a null boundary that translation is unavailable because the induced metric is degenerate, but the oriented pullback and Stokes formula remain well defined. Corners or nonsmooth boundaries require the corresponding piecewise-smooth version and consistent induced orientations on every face.

Four-dimensional Lorentzian signs

For $n=4$ and $s=1$, the codifferential exponent is even for every degree, so

$$\delta=\star d\star.$$

The square rule becomes $\star^2=+1$ on one- and three-forms and $\star^2=-1$ on zero-, two-, and four-forms. In particular,

$$\star1=\mathrm{vol}_g, \qquad \star\mathrm{vol}_g=-1, \qquad \star^2F=-F$$

for every real two-form $F$. The degree-zero and degree-four signs are as important as the familiar two-form sign: together they check both the metric index and the orientation-dependent normalization. On an interval, the same boundary formula reduces to integration by parts with the endpoint orientation $\{b\}-\{a\}$, fixing the sign of its boundary term.

Because the two-form square is negative, a real Lorentzian two-form cannot satisfy $\star F=+F$ or $\star F=-F$ unless it vanishes. After complexification, the star eigenvalues on two-forms are $+i$ and $-i$. This is signature sensitive: importing the real self-dual decomposition from four-dimensional Riemannian geometry would change the algebra.

Consistency checks and limits

Three short checks fix the convention package. Applying the star twice to $1$ must give $-1$, so $\star\mathrm{vol}_g=-1$ rather than $+1$. Applying it twice to a one-form must return that form, while applying it twice to a two-form must reverse its sign. Finally, expanding $d(\alpha\wedge\star\beta)$ must reproduce a plus boundary integral with outward-normal-first orientation. If any one check fails, the likely cause is a changed metric index, orientation, form degree, or codifferential exponent.

The codifferential formula is degree dependent: when it is applied after $d$, its exponent must be evaluated at degree $k+1$, not reused from degree $k$. The identities require a smooth nondegenerate metric and the stated orientation. They do not by themselves select boundary conditions, guarantee convergence on a noncompact domain, or identify the analytic adjoint of a closed operator before its function-space domain has been specified.

Detailed metric-duality checks appear in The Hodge Dual in Lorentzian Signature; volume and divergence conventions appear in Metric Volume, Divergence, and Integration; boundary hypotheses and orientation appear in The Generalized Stokes Theorem.

References

• Guillemin, Victor, and Peter J. Haine. Differential Forms. World Scientific, 2019, Ch. 1, §§1.4–1.9; Ch. 4, pp. 111–153. Open PDF.
• Tong, David. General Relativity. Cambridge Part III lecture notes, 2019, §3.1, pp. 91–99. Official PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022, §§9–13, pp. 69–118. Course PDF.