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 (?ω) = ω 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 ω)a1...ap−1 = X ωba1...ap−1

· Base of p-forms via wedge product, (dω)a1...ap+1 = (p + 1)∇[a1 ωa2...ap+1] b µ µ X µ µ (δω)a ...a = ∇ ωba ...a dx 1 ∧ ... ∧ dx p = sgn(π) dx π(1) ⊗ · · · ⊗ dx π(p) 1 p−1 1 p−1

π∈Sp

p(m−p)+s 2 2 2 Thm. ?? = (−1) , iX = d = δ = 0 · General p-form ω ∈ Ωp, Thm. £X (f) = δ(fX)  1 µ1 µp ω = ωµ ...µ dx ∧ ... ∧ dx a a p! 1 p Thm. δ(fα) = iX α + fδα with X = ∇ f

· Unless otherwise stated, we assume: Thm. iX d?α = ?(X ∧ δα)

f ∈ Ω0 ; α, ω ∈ Ωp ; β ∈ Ωq ; γ ∈ Ωp−1 ; X,Y ∈ 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  ∈ Ω ,  := |g| dx ∧ ... ∧ dx iX ?α = ?(α ∧ X) , iX α = (−1) ?(X ∧ ?α)

s p 1··· m µ1···µm (−1) µ1···µm µ ···µ = |g| δµ ···µ ,  = δ 1 m 1 m p|g| 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 |c|a2...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 ω)(X1,...,Xp−1) = ω(X,X1,...,Xp−1) Inner products · Hodge dual ? :Ωp → Ωm−p 1 a1...ap ωµ ...µ (α | ω) := αa1...ap ω ?ω = 1 p µ1...µp dxµp+1 ∧ ... ∧ dxµm p! p!(m − p)! µp+1...µm Normalization of the volume form, ( | ) = (−1)s. s ?1 =  ,? = (−1) , iX  = ?X Thm. (α | ω)  = α ∧ ?ω = ω ∧ ?α = (−1)s (?α | ?ω)  · Exterior derivative d : Ωp → Ωp+1 Thm. (X ∧ γ | α) = (γ | iX α)

1 σ µ1 µp s dω = ∂σωµ ...µ dx ∧ dx ∧ ... ∧ dx Thm. (X | Y )(α | α) = (−1) (i ?α | i ?α) + (i α | i α) p! 1 p X Y X Y

p p · Lie derivative £X :Ω → Ω Z hα, ωi := α ∧ ?ω M £X ω = (iX d + diX )ω Thm. If s = 0 then hα, αi ≥ 0 and hα, αi = 0 i α = 0. · Coderivative δ (or d†):Ωp → Ωp−1 Thm. If s = 0 then −∆ is a positive operator in a sense that δf = 0 , δω = (−1)m(p+1)+s ? d ? ω hω, −∆ωi = hdω, dωi + hδω, δωi ≥ 0

· Laplace–Beltrami operator (Laplacian) ∆ : Ωp → Ωp Consequently ∆ω = 0 i dω = 0 and δω = 0. ∆ = dδ + δd = (d + δ)2 Pullback Maxwell’s equations (m, s) = (4, 1) For a C∞ map φ : S → M, we dene φ∗ :Ωp(M) → Ωp(S),

∗ dF = 4π ?jm , d ?F = 4π ?je (φ ω)(X1,...,Xp) := ω(φ∗X1, . . . , φ∗Xp) a a ∂(xµ1 ◦ φ) ∂(xµp ◦ φ) Electro/magnetic decomposition via X with N = XaX : ∗ (φ ω)σ1...σp |s = ··· σ ωµ1...µp |φ(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∗(?ω) 6= ?(φ∗ω) N.B. N(F | F ) = (E | E) − (B | B) N(F | ?F ) = −2 (E | B) Stokes’s theorem. Let M be a smooth orientable m-manifold boundary ∂M and inclusion ı : ∂M → M. Then for any compactly Maxwell’s equations via δ = −?d? and ω = −?(X ∧ dX), supported ω ∈ Ωm−1(M), (E | iX dX) (B | ω) Z Z −δE = − + 4π(X | je) dω = ı∗ω N N M ∂M (B | i dX) (E | ω) −δB = X + + 4π(X | 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 ∈ 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 {θ(1),..., θ(m−k)}, so that at each point p ∈ M T (i) £K α = δ(K ∧ α) + K ∧ δα , £K ?α = ? £K α we have D|p = i Ker(θ |p). Then the following conditions are equivalent

Thm. (m, s) = (4, 1). Let (M, gab) be a Lorentzian 4-manifold with • D is involutive, (∀ X,Y ∈ D):[X,Y ] ∈ D a the Killing vector eld K . Then the twist 1-form ωa, dened as • D is totally integrable: for any p ∈ M there is a nbh Op and i b c d functions hj , fj : Op → R, such that ω := −?(K ∧ dK) , ωa = abcdK ∇ K

m−k satises the following relations (i) X i θ = hj dfj j=1 dω = −2?(K ∧ R(K)) , Nδω = 2 (ω | dN)

• the condition (ω | ω) = (dN | dN) − N∆N − 2NR(K,K)

θ(1) ∧ ... ∧ θ(m−k) ∧ dθ(i) = 0 NdK = ?(K ∧ ω) − K ∧ dN holds for any i ∈ {1, . . . , m − k}. N(dK | dK) = (dN | dN) − (ω | ω)

 K   ω  δ = 0 , δ = 0 De Rham cohomology N N 2  dN  (ω | ω) + 2NR(K,K) r r δ = − closed forms Z (M) = {ω ∈ Ω | dω = 0} N N 2 exact forms Br(M) = ω ∈ Ωr | ∃ γ ∈ Ωr−1 : ω = dγ  E  (ω | B) 4π δ = − (K | je) r r r N N 2 N HdR(M) = Z (M)/B (M)   Pm r r B (ω | E) 4π Euler-Poincaré characteristic χ(M) = r=0(−1) b with Bei δ = − − (K | 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 ω ∈ Ωp(M) there is a unique global de- composition ω = dα + δβ + χ with α ∈ Ωp−1, β ∈ Ωp+1 and χ ∈ Ωp is harmonic, ∆χ = 0.

∗ C&CP