FORMULA SHEET FOR ASSIGNMENT 2 1 Tensor transformation rules Tensors are defined by their transformation properties under coordinate change. One distinguishes con- variant and contravariant indexes. Number of indexes is tensor’s rank, scalar and vector quantities are particular case of tensors of rank zero and one. Consider coordinate change xα = xα(x0α). Transformation rules are Scalar S = S0 − scalar (tensor of 0 rank) is invariant under transformations (1) Vector α 0 ∂x V α = V α − contravariant vector (tensor of rank 1) (2) ∂xα0 ∂xα0 V = V 0 − covariant vector (3) α α ∂xα Tensor α β0 α... α0... ∂x ∂x T = T 0 · · · − tensor of higher rank with mixed indexes (4) β... β ... ∂xα0 ∂xβ Contraction Contraction is a summation over a pair of one covariant and one contravariant indexes. It creates a tensor of rank less than original by two. We use shorthand that when two inderxes of different type are labeled by the same latter it implies a summation over them. α α αβ S = VαV ,V = T β (5) 2 The metric tensor Definition The metric tensor gαβ specifies the invariant interval (distance) between two neighbouring points (events) 2 α β ds = gαβdx dx (6) Lowering of indexes β γσ Aα = gαβA ,Tαβ = gαγgβσT (7) Defining gαβ γ γ γ γσ γ γ gαβ ≡ gαγgβ ⇒ gβ = δβ,... ⇒ . gβσg (≡ gβ) = δβ (8) Rising of indexes α αβ αβ αγ βσ A = g Aβ,T = g g Tγσ (9) 3 Affine connection (Christoffel symbols) α Affine connection Γβγ describes relation between vectors at two neighbouring points. α α β γ δV = −ΓβγV dx (10) Covariant derivatives We denote ordinary derivatives with comma and the covariant ones with semi- colon S;µ = S,µ − for scalars derivatives are equal. (11) α α α γ V;µ = V,µ + ΓµγV (12) γ Vα;µ = Vα,µ − ΓµαVγ (13) β β γ β β γ Tα;µ = Tα,µ − ΓµαTγ + ΓµγTα (14) α Relation between Γβγ and gαβ: In GR we use affine connection which is related to the first derivatives of the metric tensor by the requirement that gαβ;µ = 0 and restriction that connection is symmetric α α Γβγ = Γγβ. Then 1 Γα = gασ (g + g − g ) (15) βγ 2 σβ,γ σγ,β βγ,σ 4 Curvature, Riemann and Ricci tensors, Ricci scalar α α Covariant derivative in general is not commutative, V ;ν;µ − V ;µ;ν 6= 0. Namely α α α γ V ;ν;µ − V ;µ;ν ≡ R γµνV (16) α defines Riemann tensor R γµν which gives invariant measure of the curvature of the space. The space α is flat if R γµν = 0. α Riemann tensor R γµν α α α α γ α γ R βµν = Γβν,µ − Γβµ,ν + ΓγµΓβν − ΓγνΓβµ (17) Ricci tensor Rαβ is the contraction of the Riemann tensor γ Rαβ ≡ R αγβ (18) Ricci scalar R is the further contraction αβ α R ≡ g Rαβ = R α (19) 5 Useful Computational Formulae µ µ µ γ µ γ Rβν = Γβν,µ − Γβµ,ν + ΓγµΓβν − ΓγνΓβµ (20) γ √ Γαγ = ln −g ,α , g = |gαβ| (21).