USEFUL FORMULAE IN DIFFERENTIAL GEOMETRY Differential forms: 1 α = α dxµ1 ^ · · · ^ dxµp ; α 2 ^p: (1) p! µ1...µp α ^ β = (−)pqβ ^ α; α 2 ^p; β 2 ^q: (2) Exterior derivative, d: 1 dα ≡ @ α dxv ^ dxµ1 ^ · · · ^ dxµp : (3) p! [ν µ1...µp ] d maps p-forms to (p + 1)-forms: d : ^p ! ^p+1; d2 = 0: (4) Defining the components of dα,(dα)µ1...µp+1 , by 1 dα ≡ (dα) dxµ1 ^ · · · ^ dxµp+1 ; (5) (p + 1)! µ1...µp+1 we have (dα)µ1...µp+1 = (p + 1)@[µ1 αµ2...µp+1]; (6) where 1 T ≡ T + even perms − odd perms : (7) [µ1...µq] q! µ1...µq Leibnitz rule: d(α ^ β) = dα ^ β + (−)pα ^ dβ; α 2 ^p; β 2 ^q: (8) Stokes' Theorem: Z Z d! = !; (9) M @M where M is an n-manifold and ! 2 ^n−1. Epsilon tensors and densities: "µ1...µn ≡ (+1; −1; 0) (10) if µ1 : : : µn is an (even, odd, no) permutation of a lexical ordering of indices (1 : : : n). It is a tensor density of weight +1. We may also define the quantity "µ1···µn , with components given numerically by µ1···µn t " ≡ (−1) "µ1···µn ; 1 where t is the number of timelike coordinates. NOTE: This is the only quantity where we do not raise and lower indices using the metric tensor. "µ1...µn is a tensor density of weight −1. We define epsilon tensors: 1 p µ1...µn µ1...µn µ ...µ = jgj "µ ...µ ; = " ; (11) 1 n 1 n pjgj µ ...µn where g ≡ det(gµν) is the determinant of the metric tensor gµν. Note that the tensor 1 is obtained from µ1...µn by raising the indices using inverse metrics. Epsilon-tensor identities: ν1...νn t ν1...νn µ1...µn = (−1) n! δµ1...µn : (12a) From this, contractions of indices lead to the special cases µ1...µrνr+1...νn t νr+1...νn µ1...µrµr+1...µn = (−1) r!(n − r)! δµr+1...µn ; (12b) where again t denotes the number of timelike coordinates. The multi-index delta-functions have unit strength, and are defined by ν1···νp [ν1 νp] δµ ···µ ≡ δ ··· δ : (13) 1 p [µ1 µp] (Note that only one set of square brackets is actually needed here; but with our \unit- strength" normalisation convention (7), the second antisymmetrisation is harmless.) It is worth pointing out that a common occurrence of the multi-ndex delta-function is in an ν1···νp expression like Bν1 Aν2···νp δµ1···µp , where Aν2···νp is totally antisymmetric in its (p−1) indices. It is easy to see that this can be written out as the p terms 1 B A δν1···νp = B A + B A + B A + ··· + B A ν1 ν2···νp µ1···µp p µ1 µ2···µp µ2 µ3···µpµ1 µ3 µ4···µpµ1µ2 µp µ1···µp−1 if p is odd. If instead p is even, the signs alternate and 1 B A δν1···νp = B A −B A +B A −· · ·−B A : ν1 ν2···νp µ1···µp p µ1 µ2···µp µ2 µ3···µpµ1 µ3 µ4···µpµ1µ2 µp µ1···µp−1 Hodge ∗ operator: 1 ∗(dxµ1 ^ · · · ^ dxµp ) ≡ µ1...µp dxν1 ^ · · · ^ dxνn−p : (14) (n − p)! ν1...νn−p The Hodge ∗, or dual, is thus a map from p-forms to (n − p)-forms: ∗ : ^p ! ^n−p: (15) Note in particular that taking p = 0 in (14) gives 1 p ∗1 = = dxµ1 ^ · · · ^ dxµn = jgj dx1 ^ · · · ^ dxn: (16) n! µ1...µn 2 This is the general-coordinate-invariant volume element pjgj dnx of Riemannian geometry. It should be emphasised that conversely, we have p dxµ1 ^ dxµ2 ^ · · · ^ dxµn = (−1)t "µ1µ2···µn dnx = (−1)t µ1µ2···µn jgj dnx : This extra (−1)t factor is tiresome, but unavoidable if we want our definitions to be such that ∗1 is always the positive volume element. From these definitions it follows that 1 ∗α ^ β = jα · βj , (17) p! where α and β are p-forms and µ1...µp jα · βj ≡ αµ1...µp β : (18) Applying ∗ twice, we get ∗ ∗ ! = (−)p(n−p)+t !; ! 2 ^p: (19) In even dimensions, n = 2m, m-forms can be eigenstates of ∗, and hence can be self- dual or anti-self-dual, in cases where ∗∗ = +1. From (19), we see that this occurs when m is even if t is even, and when m is odd if t is odd. In particular, we can have real self- duality and anti-self-duality in n = 4k Euclidean-signature dimensions, and in n = 4k + 2 Lorentzian-signature dimensions. Adjoint operator, δ: First define the inner product Z 1 Z (α; β) ≡ ∗α ^ β = jα · βj = (β; α); (20) M p! M where α and β are p-forms. Then δ, the adjoint of the exterior derivative d, is defined by (α; dβ) ≡ (δα; β); (21) where α is an arbitrary p-form and β is an arbitrary (p − 1)-form. Hence δα = (−)np+t∗d∗α; α 2 ^p: (22) (We assume that the boundary term arising from the integration by parts gives zero, either because M has no boundary, or because appropriate fall-off conditions are imposed on the fields.) δ is a map from p-forms to (p − 1)-forms: δ : ^p ! ^p−1; δ2 = 0: (23) 3 Note that in Euclidean signature spaces, δ on p-forms is given by δα = ∗d∗α if at least one of n and p even; (24) δα = −∗d∗α; if n and p both odd: The signs are reversed in Lorentzian spacetimes. In terms of components, the above definitions imply that for all spacetime signatures, we have 1 δα = − (r αν ) dxµ1 ^ · · · ^ dxµp−1 ; (25) (p − 1)! ν µ1...µp−1 where 1 p r ανµ1...µp−1 ≡ p @ gανµ1...µp−1 (26) ν g ν is the covariant divergence of α. Defining the components of δα,(δα)µ1...µp−1 , by 1 δα ≡ (δα) dxµ1 ^ · · · ^ dxµp−1 ; (27) (p − 1)! µ1...µp−1 we have ν (δα)µ1...µp−1 = −∇να µ1...µp−1 : (28) Hodge-de Rham operator: ∆ ≡ dδ + δd = (d + δ)2: (29) ∆ maps p-forms to p-forms: ∆ : ^p ! ^p: (30) On 0-, 1-, and 2-forms, we have λ 0-forms: ∆φ = −∇λr φ, λ ν 1-forms: ∆!µ = −∇λr !µ + Rµ !ν; (31) λ ρσ σ σ 2-forms: ∆!µν = −∇λr !µν − 2Rµρνσ! + Rµ !σν − Rν !σµ; where Rµνρσ is the Riemann tensor and ρ Rµν ≡ R µρν (32) is the Ricci tensor. Hodge's theorem: We can uniquely decompose an arbitrary p form ! as ! = dα + δβ + !H ; (33) p−1 p+1 where α 2 ^ , β 2 ^ and !H is harmonic, ∆!H = 0. 4 RIEMANNIAN GEOMETRY 2 µ ν a For a metric ds = gµνdx dx , we define a vielbein eµ as a \square root" of gµν: a b gµν = eµ eν ηab; (34) where ηab is a local Lorentz metric. Usually, we work with positive-definite metric signature, µ so ηab = δab. The inverse vielbein, which we denote by Ea , satisfies µ a µ µ b b Ea eν = δν ; Ea eµ = δa: (35) a a µ The \solder forms" e = eµ dx give an orthonormal basis for the cotangent space. Similarly, µ the vector fields Ea @µ give an orthonormal basis for the tangent space. Torsion and curvature a a µ a We define the spin connection ! b = !µbdx , the torsion 2-form T and the curvature a 2-form Θ b by a 1 a µ ν a a b T ≡ 2 Tµνdx ^ dx = de + ! b ^ e ; (36) a 1 a µ ν a a c Θ b ≡ 2 R bµνdx ^ dx = d! b + ! c ^ ! b: (37) Define a Lorentz-covariant and general-coordinate covariant derivative Dµ that acts on tensors with coordinate and Lorentz indices: νa νa a νc c νa Dµ Vρb = rµ Vρb + !µcVρb − !µbVρc ; (38) where rµ is the usual general-coordinate covariant derivative: ν ν ν σ σ ν rµ Vρ = @µ Vρ + ΓµσVρ − ΓµρVσ ; (39) µ and Γνρ is the Christoffel connection. Demanding metricity for gµν, i.e. Dµ gνρ = 0, implies µ 1 µσ Γνρ = 2 g @ν gσρ + @ρ gνσ − @σ gνρ : (40) Demanding metricity for ηab, i.e. Dµ ηab = 0, implies !ab = −!ba; (41) c where !ab ≡ ηac! b. Bianchi Identities Taking exterior derivative of (36) and (37) gives a a a b a b DT ≡ d T + ! b ^ T = Θ b ^ e ; (42) a a a c a c D Θ b ≡ d Θ b + ! c ^ Θ b − Θ c ^ ! b = 0: (43) 5 a In general, on Lorentz-valued p forms such as α b, we define the Lorentz-covariant exterior derivative by a a a c c a D α b ≡ d α b + ! c ^ α b − ! b ^ α c: (44) Torsion-free metric connection With the metricity assumption, implying (41), and the assumption that the torsion a vanishes, it follows that ! b is then uniquely determined by (36) and (41); a a b d e = −! b ^ e ; !ab = −!ba: (45) c c Defining cab = −cba by a 1 a b c d e = − 2 cbc e ^ e ; (46) it follows that !ab is given by 1 c !ab = 2 (cabc + cacb − cbca)e : (47) Note that the vielbein is constant with respect to the Lorentz- and general-coordinate co- a variant derivative defined by (38); Dµ eν = 0. Symmetries of the Riemann tensor It follows from its definition as 2-form (37) that it is always antisymmetric on the final index pair: Rabµν = −Rabνµ; Rabcd = −Rabdc; (48) where we can always freely convert coordinates indices to Lorentz indices, and vice versa, µ ν c d using the vielbein.