DefinitionDefinition: AI-adaptedProof: AI-generatedverified 2026-10-22 (gpt-6.1-sol)

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Lorentzian Hodge star and periods

Physical scope

Pure mathematics; no physical premises or physical claim.

Definition

Let $V = R^{4}$ with ordered coframe $e^{0} = d x^{0}, e^{1} = d x^{1}, e^{2} = d x^{2}, e^{3} = d x^{3}$ , bilinear form $η = diag (- 1, 1, 1, 1)$ , inverse $η^{- 1} = η$ , and volume form ${v o l}_{η} = e^{0} \land e^{1} \land e^{2} \land e^{3}$ . For $0 \leq k \leq 4$ , give $Λ^{k} V^{*}$ the induced bilinear pairing $⟨ α, β ⟩_{η} : = \frac{1}{k!} α_{μ_{1} \dots μ_{k}} β^{μ_{1} \dots μ_{k}}$ , with indices raised by $η^{- 1}$ . The Lorentzian Hodge star is the unique linear map $⋆_{η} : Λ^{k} V^{*} \to Λ^{4 - k} V^{*}$ satisfying $α \land ⋆_{η} β = ⟨ α, β ⟩_{η} {v o l}_{η}$ for all $α, β \in Λ^{k} V^{*}$ . For an increasing index set $I \subseteq {0, 1, 2, 3}$ , write $e^{I}$ for the corresponding wedge, $I^{c}$ for its increasing complement, $s (I, I^{c})$ for the sign of the concatenation $(I, I^{c})$ relative to $(0, 1, 2, 3)$ , and $ϵ_{I} : = \prod_{i \in I} ϵ_{i}$ , where $ϵ_{0} = - 1$ and $ϵ_{1} = ϵ_{2} = ϵ_{3} = 1$ . The equivalent basis formula is $⋆_{η} e^{I} = ϵ_{I} s (I, I^{c}) e^{I^{c}}$ .

For a smooth manifold $N$ , an integer $1 \leq k \leq \dim N$ , a smooth closed $k$ -form $ω \in Ω^{k} (N)$ , and a finite real smooth singular $k$ -cycle $z = \sum_{σ} c_{σ} σ$ with $\partial z = 0$ , define its period by ${Per}_{z} (ω) : = \int_{z} ω : = \sum_{σ} c_{σ} \int_{σ} ω$ . The form $ω$ is exact when $ω = d α$ for a globally defined smooth $(k - 1)$ -form $α \in Ω^{k - 1} (N)$ . These are mathematical definitions and use no physical constants.

Definition references

A smooth differential $k$ -form

The wedge product of differential forms

Wedge monomials in a dual basis form a basis

Closed and exact differential forms

De rham cohomology

Smooth singular chain and cochain complexes

Integral of a form over a smooth singular simplex

Facts & Assumptions

Given: The fixed oriented vector space $V = R^{4}$ , the nondegenerate form $η$ , its displayed volume form, and the increasing coframe-wedge basis.

[F1]

A smooth differential $k$ -form defines $Ω^{k} (N)$ as the smooth sections of $Λ^{k} T^{*} N$ .

[F2]

The wedge product of differential forms defines the wedge of forms pointwise from the wedge of alternating covectors.

[F3]

Wedge monomials in a dual basis form a basis says that the increasing wedges of a dual basis form a basis in every degree.

[F4]

Closed and exact differential forms defines $Z^{k} = \ker d$ and $B^{k} = im d$ , hence closed and exact forms.

[F5]

De rham cohomology defines $H_{d R}^{k} (N) = Z^{k} (N) / B^{k} (N)$ , with $[ω] = 0$ precisely when $ω$ is exact.

[F6]

Smooth singular chain and cochain complexes defines finite real chains of smooth singular simplices and their signed boundary.

[F7]

Integral of a form over a smooth singular simplex assigns an extension-independent integral to each smooth singular simplex.

Proof

Given: The finite-dimensional exterior spaces and pairings in the Definition.

1.1givenF1F3construct

In degree zero, use $Λ^{0} V^{*} = {Alt}^{0} (V) = R$ , the convention of Alternating $k$ -covectors in [F3]'s exterior-algebra prerequisites, with basis $e^{\emptyset} = 1$ . The empty product $ϵ_{\emptyset}$ and the signs $s (\emptyset, {0, 1, 2, 3})$ and $s ({0, 1, 2, 3}, \emptyset)$ are $1$ . For each increasing $I$ , set $⋆_{η} e^{I} = ϵ_{I} s (I, I^{c}) e^{I^{c}}$ . Thus $⋆_{η} 1 = {v o l}_{η}$ and $⋆_{η} {v o l}_{η} = - 1$ ; for scalars $a, b$ , the degree-zero pairing is $a b$ because $0! = 1$ , so $a \land ⋆_{η} b = a b {v o l}_{η}$ . In positive degrees the $e^{I}$ form a basis by [F3], and in degree zero this was just established. Hence the prescription extends uniquely by linearity to a map $Λ^{k} V^{*} \to Λ^{4 - k} V^{*}$ for every $0 \leq k \leq 4$ ; its coefficients are constant, so it also defines a smooth map on the affine space.

2.1F1F2F3step 1.1algebra

For increasing $I, J$ of size $k$ , $e^{I} \land e^{J^{c}} = 0$ when $I \neq J$ , because an index repeats. When $I = J$ , $e^{I} \land ⋆_{η} e^{I} = ϵ_{I} {v o l}_{η}$ . The component pairing has $⟨ e^{I}, e^{J} ⟩_{η} = 0$ for $I \neq J$ and $⟨ e^{I}, e^{I} ⟩_{η} = ϵ_{I}$ . Thus the characterizing identity holds on basis pairs and, by bilinearity, for every $α, β$ . Pairing with all complementary basis wedges also proves uniqueness.

3.1F4F5F6F7∎

The cycle integral in the Definition is a finite sum of the unique simplex integrals in [F7], so it is independent of how a finite chain is written after equal simplex coefficients are combined. The terms closed, exact, and $[ω] = 0$ have the quotient meanings in [F4] and [F5]. No physical assumption or choice principle is used.

Lorentzian Hodge star and periods

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources