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Operators on Di Erential Forms Abstract Index Notation Leibniz Commuting Inner Products Differential forms Abstract index notation Coee & Chalk Press (p + q)! Ivica Smolić cbnd (α ^ β)a :::a b :::b = α β 1 p 1 q p!q! [a1:::ap b1:::bq ] 1 (?!) = ! a1:::ap ap+1:::am p! a1:::ap ap+1:::am M is a smooth m-manifold with a metric gab which has s negative b eigenvalues. Metric determinant is denoted by g = det(gµν ). (iX !)a1:::ap−1 = X !ba1:::ap−1 · Base of p-forms via wedge product, (d!)a1:::ap+1 = (p + 1)r[a1 !a2:::ap+1] b µ µ X µ µ (δ!)a :::a = r !ba :::a dx 1 ^ ::: ^ dx p = sgn(π) dx π(1) ⊗ · · · ⊗ dx π(p) 1 p−1 1 p−1 π2Sp p(m−p)+s 2 2 2 Thm. ?? = (−1) ; iX = d = δ = 0 · General p-form ! 2 Ωp, Thm. £X (f) = δ(fX) 1 µ1 µp ! = !µ ...µ dx ^ ::: ^ dx a a p! 1 p Thm. δ(fα) = iX α + fδα with X = r f · Unless otherwise stated, we assume: Thm. iX d?α = ?(X ^ δα) f 2 Ω0 ; α; ! 2 Ωp ; β 2 Ωq ; γ 2 Ωp−1 ; X; Y 2 TM Leibniz p · Generalized Kronecker delta iX (α ^ β) = (iX α) ^ β + (−1) α ^ (iX β) p δa1···an := n! δ [a1 δ a2 ··· δ an] d(α ^ β) = (dα) ^ β + (−1) α ^ (dβ) b1··· bn b1 b2 bn £X (α ^ β) = (£X α) ^ β + α ^ (£X β) 1 2 n 1 n µ1···µn det(A) := n! A [1A 2 ··· A n] = A µ1 ··· A µn δ1 ··· n 1 2 n 1 ··· n n! A [µ1 A µ2 ··· A µn] = det(A) δµ1··· µn ! a ···a a ···a m − n + k a ···a Commuting δ 1 k k+1 n = k! δ k+1 n a1···akbk+1··· bn k bk+1··· bn α ^ β = (−1)pqβ ^ α m p 1 m m(p+1)+s · Volume form 2 Ω , := jgj dx ^ ::: ^ dx iX ?α = ?(α ^ X) ; iX α = (−1) ?(X ^ ?α) s p 1··· m µ1···µm (−1) µ1···µm µ ···µ = jgj δµ ···µ ; = δ 1 m 1 m pjgj 1 ··· m [£X ; £Y ] = £[X;Y ] ; [£X ; iY ] = i[X;Y ] ; [£X ; d] = 0 a1···akak+1···am s ak+1···am a ···a b ···b = (−1) k! δ 1 k k+1 m bk+1···bm £X ?α − ? £X α = (δX) ?α + ?αb with α := p( gbc)g α ba1:::ap £X b[a1 jcja2:::ap] Operators on dierential forms d ? = (−1)p+1 ? δ ; δ ? = (−1)p ? d p p−1 ?∆ = ∆?; d∆ = ∆d ; δ∆ = ∆δ · Contraction with vector iX :Ω ! Ω iX f = 0 ; (iX !)(X1;:::;Xp−1) = !(X; X1;:::;Xp−1) Inner products · Hodge dual ? :Ωp ! Ωm−p 1 a1:::ap !µ ...µ (α j !) := αa1:::ap ! ?! = 1 p µ1...µp dxµp+1 ^ ::: ^ dxµm p! p!(m − p)! µp+1...µm Normalization of the volume form, ( j ) = (−1)s. s ?1 = ;? = (−1) ; iX = ?X Thm. (α j !) = α ^ ?! = ! ^ ?α = (−1)s (?α j ?!) · Exterior derivative d : Ωp ! Ωp+1 Thm. (X ^ γ j α) = (γ j iX α) 1 σ µ1 µp s d! = @σ!µ ...µ dx ^ dx ^ ::: ^ dx Thm. (X j Y )(α j α) = (−1) (i ?α j i ?α) + (i α j i α) p! 1 p X Y X Y p p · Lie derivative £X :Ω ! Ω Z hα; !i := α ^ ?! M £X ! = (iX d + diX )! Thm. If s = 0 then hα; αi ≥ 0 and hα; αi = 0 i α = 0. · Coderivative δ (or dy):Ωp ! Ωp−1 Thm. If s = 0 then −∆ is a positive operator in a sense that δf = 0 ; δ! = (−1)m(p+1)+s ? d ? ! h!; −∆!i = hd!; d!i + hδ!; δ!i ≥ 0 · Laplace–Beltrami operator (Laplacian) ∆ : Ωp ! Ωp Consequently ∆! = 0 i d! = 0 and δ! = 0. ∆ = dδ + δd = (d + δ)2 Pullback Maxwell’s equations (m; s) = (4; 1) For a C1 map φ : S ! M, we dene φ∗ :Ωp(M) ! Ωp(S), ∗ dF = 4π ?jm ; d ?F = 4π ?je (φ !)(X1;:::;Xp) := !(φ∗X1; : : : ; φ∗Xp) a a @(xµ1 ◦ φ) @(xµp ◦ φ) Electro/magnetic decomposition via X with N = XaX : ∗ (φ !)σ1...σp js = ··· σ !µ1...µp jφ(s) @yσ1 s @y p s E = −iX F ; B = iX ?F φ∗(α ^ β) = (φ∗α) ^ (φ∗β) ; φ∗d = dφ∗ −NF = X ^ E + ?(X ^ B) φ∗(?!) 6= ?(φ∗!) N.B. N(F j F ) = (E j E) − (B j B) N(F j ?F ) = −2 (E j B) Stokes’s theorem. Let M be a smooth orientable m-manifold boundary @M and inclusion { : @M ! M. Then for any compactly Maxwell’s equations via δ = −?d? and ! = −?(X ^ dX), supported ! 2 Ωm−1(M), (E j iX dX) (B j !) Z Z −δE = − + 4π(X j je) d! = {∗! N N M @M (B j i dX) (E j !) −δB = X + + 4π(X j j ) N N m Z −dE = £X F + 4π ?(X ^ jm) Corollary hdγ; αi + hγ; δαi = γ ^ ?α dB = £X ?F + 4π ?(X ^ je) @M Green’s identity for α; β 2 Ωp(M) Z Z ∆α^?β−∆β^?α = δα^?β−δβ^?α+β^?dα−α^?dβ M @M Killing vectors a a Thm. For a Killing vector K with N = KaK we have Frobenius’s theorem. Let M be a smooth m-manifold and D a k- dim distribution (subbundle of TM), dened by (m − k) linearly in- £K K = 0 ; δK = 0 ; d?dK = 2 ?R(K) ; dN = −iK dK dependent 1-forms fθ(1);:::; θ(m−k)g, so that at each point p 2 M T (i) £K α = δ(K ^ α) + K ^ δα ; £K ?α = ? £K α we have Djp = i Ker(θ jp). Then the following conditions are equivalent Thm. (m; s) = (4; 1). Let (M; gab) be a Lorentzian 4-manifold with • D is involutive, (8 X; Y 2 D):[X; Y ] 2 D a the Killing vector eld K . Then the twist 1-form !a, dened as • D is totally integrable: for any p 2 M there is a nbh Op and i b c d functions hj ; fj : Op ! R, such that ! := −?(K ^ dK) ;!a = abcdK r K m−k satises the following relations (i) X i θ = hj dfj j=1 d! = −2?(K ^ R(K)) ; Nδ! = 2 (! j dN) • the condition (! j !) = (dN j dN) − N∆N − 2NR(K; K) θ(1) ^ ::: ^ θ(m−k) ^ dθ(i) = 0 NdK = ?(K ^ !) − K ^ dN holds for any i 2 f1; : : : ; m − kg. N(dK j dK) = (dN j dN) − (! j !) K ! δ = 0 ; δ = 0 De Rham cohomology N N 2 dN (! j !) + 2NR(K; K) r r δ = − closed forms Z (M) = f! 2 Ω j d! = 0g N N 2 exact forms Br(M) = ! 2 Ωr j 9 γ 2 Ωr−1 : ! = dγ E (! j B) 4π δ = − (K j je) r r r N N 2 N HdR(M) = Z (M)=B (M) Pm r r B (! j E) 4π Euler-Poincaré characteristic χ(M) = r=0(−1) b with Bei δ = − − (K j jm) r r N N 2 N numbers b = dim HdR(M). For n ≥ 1 the de Rham cohomology n r n R groups of spheres S are HdR(S ) = (δr0 + δrn) . Thm. If M is a connected smooth manifold with nite fundamental r group, then HdR(M) = 0. Poincaré lemma. Every closed form on a contractible open set is exact. Hodge decomposition theorem. On any closed orientable Rieman- nian manifold M, for any ! 2 Ωp(M) there is a unique global de- composition ! = dα + δβ + χ with α 2 Ωp−1, β 2 Ωp+1 and χ 2 Ωp is harmonic, ∆χ = 0. ∗ C&CP.
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