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The Hodge Star, Poincaré Duality, and Electromagnetism

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The Hodge Star, Poincaré Duality, and Electromagnetism THE HODGE STAR, POINCARE´ DUALITY, AND ELECTROMAGNETISM SCOTT MORRISON Abstract. Riemannian and pseudo- Riemannian manifolds are objects with rich geometric structure and considerable interest in physics. In this essay we consider the behaviour of differential forms on pseudo- Riemannian manifolds. In particular an inves- tigation is made of the properties of the Hodge star operator, an isomorphism between rank k differential forms and rank n-k dif- ferential forms on n dimensional manifolds. Two main applica- tions of this operator are then explored. Poincar´eDuality can be proved in a great variety of settings|for example in accordance with Poincar´e'soriginal ideas, in the setting of simplicial mani- folds, or in a much more general case as a corollary of Alexander Duality. With the aid of the Hodge Decomposition Theorem, we will see that the Hodge star in fact induces Poincar´eDuality for Riemannian manifolds. Finally, we use the Hodge star to express Maxwell's equations of electromagnetism in a simple and general form, and exhibit a short proof of Lorentz invariance. 1. pseudo- Riemannian manifolds We define pseudo- Riemannian manifolds in very much the same way as Riemannian manifolds are defined. If M is a smooth manifold, a pseudo- Riemannian metric is a smooth tensor field g : C1(TM) ⊗ C1(TM) C1(M), where C1(TM) denotes the set of vector fields −! on M, and C1(M) denotes the set of smooth functions on M, such that for all p M, gp : TpM TpM R defined by 2 ⊗ −! gp :(Xp;Yp) g(X; Y )(p) 7! is symmetric and nondegenerate. The pair (M; g) is then called a pseudo- Riemannian manifold. Riemannian manifolds are thus a spe- cialisation of pseudo- Riemannian manifolds, for which we demand that at every p M gp is positive in the sense that gp(Xp;Xp) 0 Xp 2 ≥ 8 2 TpM, that is, gp is an inner product on TpM. 2. The Hodge Star Operator 2.1. Multilinear Algebra. We at first restrict our attention to the oriented vector space E = Rn, the model for the tangent space at a point on an abstract orientable manifold. Let g : E E R be a symmetric and nondegenerate 2-tensor. A result from× linear−! algebra Date: November 8, 1999. 1 2 SCOTT MORRISON allows a choice of positively oriented vectors e1; e2; : : : ; en E such that 2 n g = c ei ei; X i i=1 ⊗ i where e E∗ denotes the dual vector to ei, and with ci = 1. Denote by index(g)2 the number of coefficients equal to -1. Such a set± of vectors we call a g-orthonormal basis. 1 n Choose a g-orthonormal basis e1; : : : ; en, and define µ = e e . ^· · ·^ Then for any other g-orthonormal basis f1; : : : ; fn, µ(f1; : : : ; fn) = 1, and so µ is independent of the particular g-orthonormal basis chosen, depending only on g. Call µ the g-volume. Using g, we can define a symmetric bilinear function on the space of alternating tensors, g :Λk(E) Λk(E) R, by its values on the k × −! standard basis for Λ (E). If σ1 < < σk and τ1 < < τk then ··· ··· cσ cσ if σ1 = τ1; ; σk = τk g(eσ1 eσk ; eτ1 eτk ) = 1 ··· k ··· ^ · · · ^ ^ · · · ^ 0 otherwise We say that this basis for Λk(E) is g-orthonormal. 2.2. Existence and uniqueness. We now define the Hodge star op- erator. k n k Lemma 2.1. There is a unique isomorphism ? :Λ (E) Λ − (E) satisfying −! α ?β = g(α, β)µ α, β Λk(E): ^ 8 2 Proof. We first prove uniqueness. Suppose we have such a ? and let σ1 σk τ1 τk β = e e and α = e e where σ; τ Sn and ^ · · · ^ ^ · · · ^ 2 τ1 < < τk. Then α ?β = 0 unless σ1 = τ1; : : : ; σk = τk. Thus ?β = ···aeσk+1 eσn for^ some a R. Using this, we calculate β ?β = σ ^· · ·^ 2 σ ^ a( 1) µ = g(β; β)µ, and g(β; β) = cσ1 cσn , so a = ( 1) cσ1 cσn . Therefore− ? is uniquely determined, since··· it is uniquely− determined··· on the basis of Λk(E), by σ1 σk σ σk+1 σn ?e e = ( 1) cσ : : : cσ e e ; ^ · · · ^ − 1 n ^ · · · ^ where σ1 < < σk and σk+1 < < σn. Because µ depends only on g, and not the··· g-orthonormal basis··· chosen, ? also depends only on g. Using this as our definition of ?, it is clear that it is an isomorphism, since it maps the g-orthonormal basis of Λk(E) to the g-orthonormal n k basis of Λ − (E). 2.3. Properties of the Hodge star operator. The Hodge star op- erator has the following easily verified properties. THE HODGE STAR, POINCARE´ DUALITY, AND ELECTROMAGNETISM 3 Lemma 2.2. For all α, β Λk(E), 2 α ?β = β ?α ^ ^ ?1 = µ ?µ = ( 1)index(g) − index(g)+k(n k) ? ? α = ( 1) − α − 2.4. The Hodge star operator on pseudo- Riemannian mani- folds. Let (M; g) be an n-dimensional pseudo- Riemannian manifold. Firstly, note than index(g) is constant. There exists a unique volume n element µ Ω (M) such that µ(X1; ;Xn) = 1 for all positively oriented g-orthonormal2 bases on the tangent··· spaces to M. Using this, k n k we define the Hodge star operator, ? :Ω (M) Ω − (M) pointwise, by −! (?α)(p) = ?(α(p)): It is clear that the properties proved in Subsection 2.3 carry across. In particular, index(g)+k(n k) ? ? α = ( 1) − α, k− n k and so ? is an isomorphism Ω (M) ∼= Ω − (M). If M is compact, we can define g :Ωk(M) Ωk(M) R by × −! g(α, β) = Z α ?β = Z g(α, β)µ. M ^ M If M is in fact Riemannian, that is, if index(g) = 0, then since at each point p M gp(α, α) 0, g(α, α) 0 also, so this g is an inner product on Ωk(M2). ≥ ≥ 2.5. The codifferential operator. Define the codifferential k 1 k δ :Ω − (M) Ω (M) −! by δ! = 0 if ! Ω0(M), and otherwise 2 δβ = ( 1)nk+1+index(g) ? d ?: − index(g)+k(n k) Note that since ? ? α = ( 1) − α, − index(g) k(n k) δδ = ?d ??d? = ( 1) ( 1) − ? dd? = 0: − − Lemma 2.3. g(dα, β) = Z dα ?β = Z α ?δβ = g(α, δβ) M ^ M ^ Proof. Using δβ = ( 1)nk+1+index(g) ? d ? β, − dα ?β α ?δβ = dα ?β + ( 1)nk+index(g)α ?? d ? β ^ − ^ ^ − nk+2index(g)+^k(n k) = dα ?β + ( 1) − α d ? β ^ − ^ = dα ?β + ( 1)kα d ? β ^ − ^ = d(α ?β); ^ 4 SCOTT MORRISON where we have used the fact that k2 + k is even. Integrating both sides over M, and applying Stokes' theorem, Z dα ?β Z α ?δβ = Z d(α ?β) M ^ − M ^ M ^ = Z α ?β @M ^ = Z α ?β 0 ^ = 0; we obtain the desired result. 3. Poincare´ Duality In this section we restrict our attention to compact Riemannian man- ifolds. Thus g :Ωk(M) Ωk(M) R is an inner product, and, in accordance with Lemma× 2.3, d and−!δ are adjoints with respect to g. 3.1. The Laplace-de Rham operator, and the Hodge Decom- position Theorem. Define ∆ : Ωk(M) Ωk(M) by ∆ = dδ + δd. If ∆α = 0 we say α is harmonic, and write−!k(M) for the vector space of harmonic forms on M. H Lemma 3.1. ∆α = 0 is equivalent to dα = δα = 0. Proof. If dα = δα = 0, then clearly ∆α = 0. If ∆α = 0, then g(∆α, α) = g(dδα, α) + g(δdα, α) = g(δα, δα) + g(dα, dα) = 0. Since g is positive, this implies dα = δα = 0. We now state without proof the following important theorem. Theorem 3.2 (Hodge Decomposition). Let ! Ωk(M). Then there k 1 k+1 2 k exists a unique α Ω − (M), β Ω and γ (M) such that ! = dα + δβ + γ: 2 2 2 H 3.2. Poincar´eDuality. Since δδ = 0, we can define associated ho- mology groups, k k 1 ker(δ :Ω (M) Ω − (M)) H0k = −! : image(δ :Ωk+1(M) Ωk(M)) −! Lemma 3.3. k ? : H (M) = H0n k ∼ − k n k Proof. We rely on the fact that ? :Ω (M) Ω − (M) is an isomor- phism. Firstly, if dα = β, then δ ? α = ( 1)−!p ? dα = ( 1)p ? β for some − − THE HODGE STAR, POINCARE´ DUALITY, AND ELECTROMAGNETISM 5 integer p. Similarly if δα = β, then d ? α = ( 1)q ? β for some integer q also. Thus − k k+1 n k n k 1 ? : ker(d :Ω (M) Ω (M)) ∼= ker(δ :Ω − (M) Ω − − (M)) k 1 ! k n k+1 ! n k ? : img(d :Ω − (M) Ω (M)) = img(δ :Ω − (M) Ω − (M)) ! ∼ ! and so taking quotients we have the required result. k k k Lemma 3.4. The maps , H (M) and , H0k(M) given by H ! H ! k inclusion followed by projection are isomorphisms, and so H (M) ∼= H0k(M). Proof. The proof for each map is very similar, so we only give the argument for Hk(M). If γ k, ∆γ = dγ = 0. Thus [γ] Hk(M). We first show that the map is2 injective.H Suppose [γ] = 0. Thus2 γ = dβ k 1 for some β Ω − (M). δγ = 0, so g(γ; γ) = g(γ; dβ) = g(δγ; β) = g(0; β) = 0,2 and therefore γ = 0. To prove the map is surjective, suppose [!] Hk(M). Using the Hodge Theorem, there exist α k 1 2 k+1 k 2 Ω − (M); β Ω (M) and γ so ! = dα + δβ + γ.
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