Chapter 2 Differential geometry 2.1 Lengths and angles 2.1.1 Abstract index notation The notation ~v, !, . works well for multivectors and differential forms but is inadequate for more general tensors. Instead, we will use the abstract index notation in which a a vector ~v is written v and a covector ! is written !a a ~v $ v ;! $ !a (2.1.1) The abstract index a denotes the tensorial nature of the quantity by its position and does not take specific values. A contraction is denoted by repeated indices a ! · ~v $ !av (2.1.2) ab··· A tensor can have an arbitrary number of vector and covector indices Tcd··· and is expressed in components as ab··· αβ··· a b γ δ Tcd··· = Tγδ··· eαeβ : : : eced ::: (2.1.3) Note the difference between the two index notations. The abstract indices denote the ab··· αβ··· tensorial nature of Tcd··· , while the component indices label the components Tγδ··· and γ basis vectors ~eα and covectors e , and are summed over in the above equation. A tensor Tab can be decomposed into symmetric and antisymmetric parts Tab = T(ab) + T[ab] (2.1.4) where round brackets denote the symmetric part 1 T = (T + T ) (2.1.5) (ab) 2 ab ba and square brackets denote the antisymmetric part 1 T = (T − T ) (2.1.6) [ab] 2 ab ba 24 PH211 Physical Mathematics Fall 2019 An n-vector is an antisymmetric tensor with n vector indices and an n-form is an antisymmetric tensor with n covector indices, for example ~ [ab] ~v $ v ;! $ ![ab] (2.1.7) A differential form can be expressed in components as a general tensor α1 αn ![a1···an] = !α1···αn ea1 : : : ean (2.1.8) but is more naturally expressed in terms of a differential form basis 1 ! = ! eα1 ^ ::: ^ eαn (2.1.9) [a1···an] n! α1···αn a1 an and similarly for multivectors. The contraction of a differential form with a multivector is 1 1 ! · ~v~ $ ! v[ab] ;! · ~v~ $ ! v[bc] 2! [ab] 2! [abc] (2.1.10) ~ 1 ~ 1 ! · ~v~ $ ! v[abc] ;! · ~v~ $ ! v[abc] 2! [ab] 3! [abc] and more generally the contraction of an m-form ! with an n-vector v is 8 1 [b1···bmam+1···an] > ![b1···bm]v for m ≤ n < m! ! · v $ (2.1.11) 1 :> ! v[b1···bn] for m ≥ n n! [an+1···amb1···bn] The divergence of a multivector follows the same pattern [ba2···an] r · v $ rbv (2.1.12) The exterior product of differential forms is ! ^ σ $ !aσb − !bσa (2.1.13) ! ^ σ $ !aσ[bc] + !bσ[ca] + !cσ[ab] (2.1.14) and more generally the exterior product of an m-form ! with an n-form σ is (m + n)! ! ^ σ $ ! σ (2.1.15) m! n! [a1···am b1···bn] and similarly for multivectors. The exterior derivative follows the same pattern r ^ φ $ raφ (2.1.16) r ^ ! $ ra!b − rb!a (2.1.17) r ^ ! $ ra![bc] + rb![ca] + rc![ab] (2.1.18) and more generally the exterior derivative of a n-form ! is r ^ ! $ (n + 1)r[a!b1···bn] (2.1.19) Ewan Stewart 25 2019/11/21 PH211 Physical Mathematics Fall 2019 2.1.2 Metric ab The metric gab and inverse metric g define lengths and angles in a space. They are symmetric tensors ab ba gab = gba ; g = g (2.1.20) and related by ab a g gbc = δc (2.1.21) a where the identity tensor δb has the property a b a b δbv = v ; δa !b = !a (2.1.22) and similarly for other tensors. The metric gives the inner product of vectors and covectors a b ab ~u · ~v = gabu v ;! · σ = g !aσb (2.1.23) and more generally of n-forms 1 ! · σ = ga1b1 : : : ganbn ! σ (2.1.24) n! [a1···an] [b1···bn] and similarly for n-vectors. Note that these inner products depend on the metric, in contrast to the contraction of an n-vector with an n-form. The inner product of two n-vectors or n-forms can be expressed as u · v = juj jvj cos θ (2.1.25) where θ is the angle between u and v. The metric induces a metric duality between vectors and covectors b a ab va = gabv ; v = g vb (2.1.26) a ab b ! = g !b ;!a = gab! and more generally n-vectors and n-forms [b1···bn] v[a1···an] = ga1b1 : : : ganbn v (2.1.27) [a1···an] a1b1 anbn ! = g : : : g ![b1···bn] (2.1.28) For example, the traditional vector representation E~ of the electric field E is a ab E = g Eb (2.1.29) More generally, the metric can raise or lower indices on any tensor 1 a ac T b = gbcT (2.1.30) 1It is important to maintain the horizontal position of the indices if the tensor is not symmetric a ac ca a ac ca since T b = gbcT 6= gbcT = T b if T 6= T . Ewan Stewart 26 2019/11/21 PH211 Physical Mathematics Fall 2019 The inner product of basis vectors is a b ~eα · ~eβ = gabeαeβ = gαβ (2.1.31) α An orthonormal basis has metric components gαβ = δαβ. In a coordinate system x , the length ds of an infinitesimal displacement dxa can be expressed as 2 a b α β ds = gabdx dx = gαβdx dx (2.1.32) For example, in Cartesian coordinates in two dimensional Euclidean space ds2 = dx2 + dy2 (2.1.33) and so the components of the metric are gxx = gyy = 1 and gxy = 0. In polar coordinates ds2 = dr2 + r2dθ2 (2.1.34) 2 and so grr = 1, gθθ = r and grθ = 0. Cartesian bases are the only orthonormal coordinate bases and they exist only in flat spaces. The identity tensor on the space of n-forms and n-vectors δ[a1···an] = n! δ[a1 : : : δan] (2.1.35) [b1···bn] [b1 bn] has the property 1 1 [a1···an] [b1···bn] [a1···an] [b1···bn] δ v = v ; δ ![b ···b ] = ![a ···a ] (2.1.36) n! [b1···bn] n! [a1···an] 1 n 1 n and can be expressed in components as δ[a1···an] = δα1···αn ea1 : : : ean eβ1 : : : eβn (2.1.37) [b1···bn] β1···βn α1 αn b1 bn 1 = δα1···αn ea1 ^ ::: ^ ean eβ1 ^ ::: ^ eβn (2.1.38) (n!)2 β1···βn α1 αn b1 bn 1 = ea1 ^ ::: ^ ean eα1 ^ ::: ^ eαn (2.1.39) n! α1 αn b1 bn The metric on the space of n-forms and n-vectors is g = g : : : g δ[b1···bn] (2.1.40) [a1···an][c1···cn] a1b1 anbn [c1···cn] The metric and inverse metric are related by 1 [a1···an][b1···bn] [a1···an] g g[b ···b ][c ···c ] = δ (2.1.41) n! 1 n 1 n [c1···cn] The inner product of two n-forms is 1 ! · σ = g[a1···an][b1···bn]! σ (2.1.42) (n!)2 [a1···an] [b1···bn] and similarly for n-vectors, and metric duality between n-vectors and n-forms is 1 v = g v[b1···bn] (2.1.43) [a1···an] n! [a1···an][b1···bn] 1 ![a1···an] = g[a1···an][b1···bn]! (2.1.44) n! [b1···bn] Ewan Stewart 27 2019/11/21 PH211 Physical Mathematics Fall 2019 2.1.3 Metric and volume form The volume form determines volumes and orientations but not lengths or angles, while the metric determines lengths and angles, and hence volumes, but not orientations. Thus the volume form is determined up to its orientation by the metric. Explicitly, Eq. (1.4.1) generalizes to −1[a1···aN] = δ[a1···aN] (2.1.45) [b1···bN] [b1···bN] and lowering indices gives −1 = g (2.1.46) [a1···aN] [b1···bN] [a1···aN][b1···bN] therefore [a1···aN][b1···bN] = ( · ) g[a1···aN][b1···bN] (2.1.47) where · = ±1 is needed because of sign differences between and −1 [a1···aN] [a1···aN] that can arise in spacetimes. Now, taking components of Eqs. (2.1.40) and (2.1.47), the determinant of the metric components 2 g = g = 1···N (2.1.48) 1···N1···N · and so · = sgn g (2.1.49) and p 1···N = jgj (2.1.50) Contracting Eq. (2.1.45) gives 1 −1[a1···ancn+1···cN] [a1···an] [b ···b c ···c ] = δ (2.1.51) (N − n)! 1 n n+1 N [b1···bn] corresponding to the Levi-Civita Kronecker delta relation. Hodge duality Combining metric duality between n-forms and n-vectors with volume duality ? be- tween n-vectors and (N − n)-forms, see Section 1.4.1, gives a mapping between n-forms and (N − n)-forms called the Hodge dual ∗ = ? (2.1.52) ~ −1 For example, the tensors ~"0 and ~µ0 in Eqs. (1.3.2) and (1.3.3) are c dc "0[ab] = "0[abd]g (2.1.53) −1[bc] −1 db ec µ0 a = µ0 [ade]g g (2.1.54) Ewan Stewart 28 2019/11/21 PH211 Physical Mathematics Fall 2019 −1 so, in natural units in which "0 = µ0 = 1, Eqs. (1.3.2) and (1.3.3) become D = ∗E + P (2.1.55) H = ∗B − M (2.1.56) Also, following Eq. (1.4.7), the divergence of an n-form ! is 2 r · ! = r · ! = (−1)n−1 ∗−1 r ^ ∗! (2.1.57) In three dimensions, the volume ?, metric and Hodge ∗ dualities can be used to map any antisymmetric tensor into either a scalar or a vector, see Figure 2.1.1.