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Riemann Curvature Tensor Appendix A Riemann Curvature Tensor We will often use the notation (...), μ and (...); μ for a partial and covariant derivative respectively, and the [anti]-symmetrization brackets defined here by 1 1 T[ab] := (Tab − Tba) and T( ) := (Tab + Tba) , (A.1) 2 ab 2 for a tensor of second rank Tab, but easily generalized for tensors of arbitrary rank. We can define the Riemannian curvature tensor in coordinate representation by the action of the commutator of two covariant derivatives on a vector field vα [∇ , ∇ ] vρ = ρ vσ, μ ν R σμν (A.2) ρ with the explicit formula in terms of the symmetric Christoffel symbols μν ρ = ρ − ρ + ρ α − ρ α . R σμν ∂μ σν ∂ν σμ αμ σν αν σμ (A.3) ρ From this definition it is obvious that R σμν possesses the following symmetries Rαβγδ =−Rβαγδ =−Rαβδγ = Rγδαβ. (A.4) In addition, there is an identity for the cyclic permutation of the last three indices 1 R [ ] = (R + R + R ) = 0. (A.5) α βγδ 3 αβγδ αδβγ αγδβ An arbitrary tensor of fourth rank in d dimension has d4 independent components. Since a tensor is built from tensor products, we can think of Rαβγδ as being composed ( × ) 1 2 1 2 of two d d matrices Aαβ and Aγδ.From(A.4), it follows that A and A are antisymmetric, each having n(n − 1)/2 independent components. Writing Rαβγδ ≡ ( ) → 1 ( ) → 2 RA1 A2 , where we have collected the index pairs αβ A and γδ A , this corresponds to a matrix, in which each index A1 and A2 labels d(d − 1)/2 C. F. Steinwachs, Non-minimal Higgs Inflation and Frame Dependence 243 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 244 Appendix A: Riemann Curvature Tensor components. Taking into account the symmetry under pairwise exchange of (αβ) ( ) 1 2 and γδ , we can consider RA1 A2 as a symmetric matrix in A and A ,having A1(A1 + 1)/2 independent components. Altogether, we find 1 1 1 1 1 3 1 d (d − 1) d (d − 1) + 1 = d4 − d3 + d2 − d (A.6) 2 2 2 8 4 8 4 independent components. We still need to figure out how many components are related by the cyclic identity (A.5)ind dimensions. We can artificially write 1 R = R − R + R − R + (αβ) ↔ (γδ) (A.7) αβγδ 8 αβγδ αβδγ βαδγ βαγδ and similar expressions for Rαγδβ and Rαδβγ. Inserting these expressions into (A.5) leads to the condition R[αβγδ] = 0. This totally antisymmetric object is identical zero for two identical indices but gives one additional constraint for each choice of four distinct, orderless indices, reducing the number of independent components by one respectively. In d dimensions, the number of additional constraints corresponds to “choose 4 out of d”. For integers d, n there exists a product formula to calculate the binomial n d d − n + k = . (A.8) n k k=1 Inserting n = 4, we obtain d 1 1 1 11 1 = [d (d − 1)(d − 2)(d − 3)] = d4 − d3 + d2 − d. (A.9) 4 24 24 4 24 4 Thus, the number of independent components IRiem(d) of Rαβγδ is given by 1 1 d 1 I (d) := d (d − 1) d (d − 1) + 1 − = d2 (d2 − 1). (A.10) Riem 4 2 4 12 We consider IRiem(d) for different dimensions d: IRiem(1) = 0, IRiem(2) = 1, IRiem(3) = 6, IRiem(4) = 20. (A.11) This shows that gravity in one dimension is trivial, since there is no dynamical degree of freedom. It further shows that gravity in two dimensions can be described by the (3) (3) Ricci scalar R and in three dimensions by Rμν = Rνμ . In general, the d-dimensional := γδ ( ) := 1 ( + ) Ricci tensor Rμν g Rγμδν has IRic d 2 d d 1 independent components. In particular IRic(4) = 10. Thus, ten components of Rαβγδ not contained in Rμν still remain. They are contained in the Weyl tensor Appendix A: Riemann Curvature Tensor 245 2 2 C := R − g [ R ] − g [ R ] + g [ g ] R. αβγδ αβγδ d − 2 α γ δ β β γ δ α (d − 1)(d − 2) α γ δ β (A.12) It follows that the Weyl tensor has (for d > 3) 1 I (d) = I (d) − I (d) = d (d + 1)(d + 2)(d − 3) (A.13) Weyl Riem Ric 12 independent components, and in particular IWeyl (4) = 10. Appendix B Variations and Derivatives B.1 Covariant Differentiation in General Relativity The action of the metric compatible covariant derivative ∇μ gαβ = 0 with respect to ρ the Christoffel symbol μν on a general tensor is defined as ∇ α...γ = α...γ + α ρ...γ + ... + γ α...ρ μ T β...δ ∂μ T β...δ μρ T β...δ μρ T β...δ − ρ α...γ − ... − ρ α...γ , μβ T ρ...δ μδ T β...ρ (B.1) with the symmetric Christoffel symbol 1 α (g) := gαρ ∂ g + ∂ g − ∂ g = α (g). (B.2) μν 2 ν μρ μ νρ ρ μν νμ μ For scalar functions ∇μϕ = ∂μϕ. For vector fields v , a useful formula is 1 √ √ √ ∇ vμ = √ ∂ g vμ or g ∇ vμ = ∂ g vμ . (B.3) μ g μ μ μ B.2 Functional Derivative We use again the condensed DeWitt notation for a generalized field φi = φA(x), introduced in Sect. 4.4. We want to emphasize that an object with two DeWitt indices like ij corresponds to a two-point function or a generalized bi-tensor AB(x, x ) in conventional notation. The primed indices like j refer to the space-time point x. This means we can construct objects which behave as tensors of different rank at different space-time points. A particular case of a generalized bi-tensor is a bi-scalar which has no indices A, B,..at all. A special bi-scalar, in turn, is the Dirac Delta- Distribution δ(x, x) which is defined in a 2ω-dimensional curved space-time with A metric gμν(x) for a general test field (x) by the equation C. F. Steinwachs, Non-minimal Higgs Inflation and Frame Dependence 247 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 248 Appendix B: Variations and Derivatives / A(x) = d2ω x |g(x )|1 2 δ(x, x )A(x ), (B.4) For simplicity, we specify 2ω = 4. In the condensed notation (B.4) takes the form i = i j δ j with i =| ( )|1/2 A ( , ) = A ˜( , ). δ j g x δ B δ x x δ B δ x x (B.5) The quantity δ˜(x, x) := |g(x)|1/2δ(x, x) is no longer a bi-scalar, since it transforms as a scalar at the point x, but as a scalar density at the point x. The expansion of Z[] up to linear order in condensed notation is defined by i Z[ + ζ]=:Z[]+Z, i [] ζ , (B.6) with ζi := δψi and [( )] i 4 δZ x A Z, [] ζ = d x ζ (x ) (B.7) i δ A(x) in conventional notation. Therefore, it seems obvious to define the functional deriv- A ative Z, i = δZ[(x)]/δ (x ) by the variation [3] Z[(x)] 1 ( [ + ]− []) =: 4 δ A( ). lim Z ζ Z d x ζ x (B.8) →0 δ A(x ) For the special case Z = Id, we obtain from (B.8) A A δ (x) δ (x) ζ A = d4x ζ B(x ) or = δ A δ˜(x, x ). (B.9) δ B(x) δ B(x) B B.3 Lie Derivative L γ...δ μ The Lie derivative ξ of an arbitrary tensor Tα...β in the direction of the vector ξ is γ ...δ ( ) a measure of the difference between Tα...β x “dragged” along the integral curve μ γ...δ( ) γξ of ξ compared to the tensor Tα...β x at this point. Calculating the infinitesimal γ ...δ ( ) flow of Tα...β x along γξ back from x to x with the infinitesimal coordinate μ ν μ μ transformation matrix ∂x /∂x = δ ν + ξ, ν yields γ ...δ ( ) = γ...δ + γ ρ...δ + ... + δ γ...ρ − ρ γ...δ − ... − ρ γ...δ . Tα...β x Tα...β ξ,ρTα...β ξ,ρTα...β ξα, Tρ...β ξβ, Tα...ρ (B.10) γ...δ( ) Taylor expansion of Tα...β x around x yields Appendix B: Variations and Derivatives 249 γ...δ( + ) = γ...δ( ) + γ...δ ρ. Tα...β x ξ Tα...β x Tα...β, ρ ξ (B.11) Since the right-hand sides of (B.10) and (B.11) involve only quantities at x, we can compare these two objects at x and define the Lie derivative as the difference γ...δ γ...δ T (x ) − T (x ) L γ...δ ( ) := α...β α ...β ξ T ... x lim α β →0 = γ...δ ρ − ρ...δ γ − ... − γ...ρ δ Tα...β, ρ ξ Tα...β ξ,ρ Tα...β ξ,ρ + γ...δ ρ + ... + γ...δ ρ. Tρ...β ξα, Tα...ρ ξβ, (B.12) A special case is the Lie derivative of the metric tensor. The vanishing of the Lie L = μ derivative ξ g μν 0 in direction ξ signalizes a symmetry of the space-time manifold and means that the vector ξμ is a generator of an isometry ρ ρ ρ L = , + + = ; + ; = . ξ g μν gμν ρ ξ gρν ξ, μ gνρ ξ, ν ξμ ν ξν μ 0 (B.13) The vector fields ξμ obeying (B.13) are called Killing vector fields.
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