12 th Marcel Grossman Meeting Paris, 17 th July 2009

Transcription

1 Department of Mathematical Analysis, Ghent University (Belgium) 12 th Marcel Grossman Meeting Paris, 17 th July 2009

2 Outline 1 2 The spin covariant derivative The curvature spinors Bianchi and Ricci identities 3 formalism to 5 4

3 The spacetime tensor algebra Let L be a 5-dimensional real endowed with a real scalar product g(, ) of Lorentzian signature. Use the L and its dual L to build a tensor algebra T(L) The spinorial tensor algebra Let S be a complex whose dimension is for the moment left unspecified. Use the S and its dual S to build a tensor algebra T(S) Notation We use abstract indices to denote tensorial quantities: small Latin letters a, b,... are indices for elements in T(L) and uppercase Latin letters A, B,... are indices for elements in T(S).

4 We introduce now a mixed quantity γ following algebraic property γ B aa C γbb + γba B γ C ab C ab C ab which fulfils the = δ C A g ab, This relation means that γ can be regarded as belonging to a representation on the S of the Clifford algebra Cl(L, g).

5 Theorem If the quantity γ of Cl(L, g), then: C ab the dimension of S is 4. belongs to an irreducible representation There exist two antisymmetric spinors ɛ AB, ɛ AB, unique up to a constant, such that ɛ AB ɛ CB = δ C A, γ ad A γ a C B = 1 2 δ D A δ C B δ C A δ D B + ɛ C D ɛ AB.

6 The quantities ɛ AB, ɛ AB can be used to raise and lower spinor indices. Example: ξ A ɛ AB = ξ B, ξ A = ɛ AB ξ B. From now on we set ɛ AB = ɛ AB. Previous theorem entails the relations γ AB a γ bab = 2g ab, γ a [AB] = γa AB, ɛ AB γ a AB = 0, γ acd γ a AB = ɛ AD ɛ BC ɛ AC ɛ BD ɛ ABɛ CD. Use these properties to transform tensors into spinors and back. Example: v AB = γ AB a v a, v a = 1 2 γa ABv AB. If v AB is given, then v a is completely determined. However, for a given v a the spinor v AB is not determined unless v AB is antisymmetric and trace-free.

7 The spin tetrad and the semi-null pentad Let {o A, ι A, õ A, ι A } be a basis in in S such that This entails ɛ AB = 2o [A ι B] + 2õ [A ι B]. o A ι A = 1 = õ A ι A, o A õ A = ι A ι A = o A ι A = ι A õ A = 0 The basis {o A, ι A, õ A, ι A } is a spin tetrad. From it we construct a semi-null pentad as follows l a γ a ABo A õ B, n a γ a ABι A ι B, m a o A γ a AB ι B, m a õ A γ a ABι B, u a 2o B γ a ABι A = 2õ B γ a AB ι A. Hence l a n a = 1, m a m a = 1, u a u a = 2.

8 Spin structures on a The spin covariant derivative The curvature spinors Bianchi and Ricci identities Let (M, g) be a and let T p (M) be the tangent space at a point p. We can introduce a B spin space S p and the quantity γaa p at each point p. Definition: Spin bundle The union S(M) p M S p, is a vector bundle called the spin bundle. The sections of S(M) are the contravariant rank-1 spinor fields on M. We can define tensor bundles with the tensor products of space-time tensors and spinors as building blocks. We use S(M) as the generic symbol to denote these tensor bundles.

9 Spin structures on a The spin covariant derivative The curvature spinors Bianchi and Ricci identities Definition: spin structure on a 5-dimensional B If the quantity γaa p varies smoothly on the M, then one can define a smooth section of the bundle S 0,1 1,1 (M), B denoted by γaa. When this is the case we call the smooth B section γaa a smooth spin structure on the Lorentzian (M, g). As a consequence, we can now define two smooth sections ɛ AB, ɛ AB of S(M) and use them to lower and raise spinor indices, just as we did before.

10 The spin covariant derivative The spin covariant derivative The curvature spinors Bianchi and Ricci identities Definition: (Spin covariant derivative) Suppose that S(M) admits a spin structure γaa. We say that a covariant derivative D a defined on S(M) is compatible B with the spin structure γ if aa D D a γbc = 0. B The covariant derivative D a is then called a spin covariant B derivative with respect to the spin structure γ Elementary properties of a spin covariant derivative are D a g bc = 0, D a ɛ AB = 1 ( ɛ CD ) D a ɛ CD ɛab. 4 aa

11 The spin covariant derivative The curvature spinors Bianchi and Ricci identities Theorem There is one and only one spin covariant derivative a on B S(M) with respect to the spin structure γaa which fulfils the property a ɛ AB = 0. From now on we will restrict ourselves to a. The spinor form of this operator is AB γ a AB a [AB] = AB, A A = 0

12 The curvature spinors The spin covariant derivative The curvature spinors Bianchi and Ricci identities Theorem The Riemann tensor R abcf of the covariant derivative a can be decomposed in the form where R abcf = Λ(g af g bc g ac g bf ) 1 2 G ab AB G cf C D Ψ ABC D G ab AB G cf C D Ω AC BD, G ab AC γ a (A B γ b C)B. The quantities Λ, Ω ABCD and Ψ ABCD are known collectively as the curvature spinors.

13 Algebraic properties of the curvature spinors The spin covariant derivative The curvature spinors Bianchi and Ricci identities Ψ ABCD Weyl spinor, Ω ABCD Ricci spinor, Λ scalar curvature. Ψ (ABCD) = Ψ ABCD Ψ ABCD has 35 independent components. Ω ABCD = Ω [AB]CD = Ω CDAB, Ω Ω ABCD + Ω BCAD + Ω CABD = 0. Ω ABCD has 14 independent components. AB C C = ΩA C CD = 0, = 50, Number of independent components of R abcd.

14 Bianchi and Ricci identities The spin covariant derivative The curvature spinors Bianchi and Ricci identities Proposition: spinor form of the second Bianchi identity The curvature spinors fulfill the differential identity W (Z Ψ V )BAW W (Z Ω V )ABW W (Z Ω V )BAW 2ɛ (A (V Z) B) Λ = 0. We introduce the linear differential operator AB G ab AB a b (AB) = AB. Proposition: spinor form of the Ricci identity For any spinor ξ B we have C Dξ B = Λɛ B(C ξ D) ξ A Ω B(C D)A 1 2 ξa Ψ BC DA. The action of C D can be extended to any spinor by means of the linearity and the Leibnitz rule.

15 The frame derivations: D l a a, n a a, δ m a a, δ m a a, D u a a. The spin coefficients: Do A = ɛo A + d ι A κι A + χõ A, o A = γo A + e ι A τι A + ωõ A, δo A = βo A σι A + ς ι A + φõ A, δoa = αo A ρι A + ς ι A + υõ A, Do A = θo A + ηι A + f ι A + ψõ A, Dι A = aõ A ɛι A + πo A χ ι A, ι A = bõ A γι A + νo A ω ι A, δι A = βι A + µo A + ξõ A ῡ ι A, δι A = αι A + λo A + ξõ A φ ι A, Dι A = cõ A + ζo A + θι A ψ ι A. Twelve 4-D spin coefficients α, β, γ, ɛ, κ, λ, µ, ν, π, ρ, σ, τ. Ten complex 5-D spin coefficients ζ, η, θ, χ, ω, φ, ξ, υ, ψ, ς. Six real 5-D spin coefficients a, b, c, d, e, f = 50 real Ricci rotation coefficients.

16 Components of the Weyl spinor: Ψ 0 Ψ ABCD o A o B o C o D, Ψ 0 Ψ ABCD o A o B o C õ D, Ψ 1 Ψ ABCD o A o B o C ι D, Ψ 1 Ψ ABCD o A o B ι C õ D, Ψ 1 Ψ ABCDo A o B o C ι D, Ψ 2 Ψ ABCD o A o B ι C ι D, Ψ 2 Ψ ABCD o A ι B ι C õ D, Ψ 2 Ψ ABCDo A o B ι C ι D, Ψ 3 Ψ ABCD o A ι B ι C ι D, Ψ 3 Ψ ABCD õ A ι B ι C ι D, Ψ 3 Ψ ABCDo A ι B ι C ι D, Ψ 4 Ψ ABCD ι A ι B ι C ι D, Ψ 4 Ψ ABCDι A ι B ι C ι D, Ψ 01 Ψ ABCD o A o B õ C ι D, Ψ 02 Ψ ABCD o A o B ι C ι D, Ψ 12 Ψ ABCD o A ι B ι C ι D. Ψ 00 Ψ ABCD o A o B õ C õ D, Ψ 11 Ψ ABCD o A ι B õ C ι D, Ψ 22 Ψ ABCD ι A ι B ι C ι D, Five 4-D components Ψ 0, Ψ 1, Ψ 2, Ψ 3, Ψ 4. Eleven 5-D complex components Ψ 0, Ψ 1, Ψ 1, Ψ 2, Ψ 2, Ψ 3, Ψ 3, Ψ 4, Ψ 01, Ψ 02, Ψ 12. Three 5-D real components Ψ 00, Ψ 11, Ψ (16 complex components)+3 real components=35.

17 Components of the Ricci spinor: Φ 00 Ω ABCD o A õ B o C õ D, Φ 11 Ω ABCD o A õ B ι C ι D, Φ 22 Ω ABCD ι A ι B ι C ι D, Ω Ω ABCD õ A ι B õ C ι D, Φ 01 Ω ABCD o A õ B õ C ι D, Φ 12 Ω ABCD ι A ι B õ C ι D, Φ 01 Ω ABCD o A õ B o C ι D, Φ 02 Ω ABCD o A ι B o C ι D, Φ 12 Ω ABCD o A ι B ι C ι D, Φ 02 Ω ABCD o A ι B õ C ι D. 4-D components Real: Φ 00, Φ 11, Φ 22, Complex: Φ 01, Φ 02, Φ D components Real: Ω, Φ 01, Φ 12, Complex: Φ (4 complex components)+6 real components=14

18 Commutation relations: D D = (γ + γ)d (ɛ + ɛ) (π + τ)δ ( π + τ) δ + (a + e)d, Dδ δd = (ᾱ + β + π)d + κ + (ɛ ɛ ρ)δ σ δ + (ς χ)d, DD DD = ( 2a + θ + θ)d + 2d + ( η 2 χ)δ + (η 2χ) δ + fd, δ δ = νd + (ᾱ + β + τ) + (γ γ + µ)δ + λ δ (ξ + ω)d, D D = 2bD + (2e θ θ) + (ζ 2 ω)δ + ( ζ 2ω) δ cd, δ δ δδ = ( µ + µ)d + ( ρ + ρ) + ( α + β)δ + (ᾱ β) δ + (υ ῡ)d, δd Dδ = ( ζ 2ξ)D + (η + 2ς) + (θ θ 2ῡ)δ 2φ δ + ψd. Ricci ( equations) and Bianchi identities.

19 The use of spinors might enable us to simplify tensorial computations in dimension 5. The extended formalism could be used to seek new exact exact solutions in dimension 5 with prescribed Petrov type. The extended formalism is the starting point for a generalization of the GHP formalism to dimension 5.