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Notes on the Weyl Tensor, Decomposition of Riemann Tensor, Ruse-Lanczos Identity and Duality of the Curvature Tensor

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Notes on the Weyl Tensor, Decomposition of Riemann Tensor, Ruse-Lanczos Identity and Duality of the Curvature Tensor Notes on the Weyl tensor, decomposition of Riemann tensor, Ruse-Lanczos identity and duality of the curvature tensor Wytler Cordeiro dos Santos Universidade de Bras´ılia,CEP 70910-900, DF, Brasil Abstract These notes have the pedagogical purpose of exposing mathematical accounts about of the Weyl tensor. The deve- lopment of calculations is presented to obtain the curvature tensor from the conformal transformation, consequently resulting in the Weyl conformal tensor and the Riemann tensor reveals itself decomposed into its parts. The Ruse- Lanczos identity is a useful mathematical tool to build duality identities in the two pairs of antisymmetric indices of the Riemann tensor components. Keywords: Weyl tensor, decomposition of the Riemann tensor, dual tensor 1. Introduction In accordance with tensorial approach of the Electromagnetic Theory, the electromagnetic field is given in terms of the four-vector Fµ = Eµ + iBµ, where Eµ = (0; E1; E2; E3) and Bµ = (0; B1; B2; B3) are electric and magnetic field observated by an observer in an inertial reference frame, where the four-velocity vector of the observer is a timelike µ ν ν vector u = (1; 0; 0; 0). Thus, the four-electromagnetico vector is given by Fµ = Fµνu + iF˜µνu , where the Fµν is the ˜ 1 κλ field-strength tensor of electromagnetism and the Fµν = 2 κλµνF is its dual field-strength tensor. In this formalism, ν ν we separete the eletric vector given by Eµ = Fµνu and magnetic vector given by Bµ = F˜µνu with aid of dual operation. In fact, in the inertial reference frame in which it is at rest, the vectors Eµ and Bµ, are spatial three-vectors, then we can put F = E + iB, where the six components of the field-strength tensor of electromagnetism Fµν are into complex three-vector F [1]. The duality of an antisymmetric tensor like field-strength tensor of electromagnetism can be used for split vectorial components. With this in mind we can apply the dual operation to the pair of antisymmetric indices of Riemann curvature tensor. In the literature that deals with duality of the Riemann curvature tensor, decomposed into its irreducible components, the Weyl tensor, the Ricci tensor and the curvature scalar, seldom makes explicit details of calculations involving the decomposition of the the Riemann curvature tensor. In the basic courses of General Relativity, presents Weyl tensor as a definition without exposing the details of the calculations that lead to it. Some books, for example [2, 3], present the conformal transformation and indicate the path to be followed to obtain the Weyl curvature tensor. In this paper we solved this exercise to obtain the Weyl tensor from conformal transformation and explained the decomposition of Riemann curvature tensor. In order to obtain the duality properties of Riemann curvature tensor, it is necessary to study the Ruse-Lanczos identity [4]. Consequently, in the same way as for the duality of electromagnetismo we present the formalism of duality and complexification of the parts of Riemann curvature tensor. κ According to General Relativity, the Riemann curvature tensor, R µλν, is related to the distribution of matter, κ specifically by Ricci tensor, Rµν = R µκν, directly related to the local energy-momentum tensor by Einstein’s field equations, 1 R − g R = 8πGT ; (1) µν 2 µν µν Email address: [email protected] (Wytler Cordeiro dos Santos) 1 where G is the gravitational Newton’s constant and we are overlooking the value of the cosmological constant. In free gravitational field, we have no matter, Tµν = 0 and consequently we have null Ricci tensor, Rµν = 0. Greek labels refer to four dimensional spacetime and we use metric signature (− + ++). 2. Conformal transformation Hermann Weyl tried to unify electromagnetism and gravitation, this attempt was carried out in 1919 [5], however this attempt failed, but Weyl introduced a new principle of invariance in his theory, a scale change of the metric tensor given by g¯ = e2! g. This principle later evolved through Quantum Field Theory being known as ‘gauge invariance’ [6, 7, 8]. A scale change of the metric tensor is classified as conformal transformation in Riemannian Geometry leading to the Weyl conformal tensor. Let (M; g) be a spacetime where M is pseudo-Riemannian manifold and g is a Lorentz metric tensor. A con- formal transformation [2, 3, 9] is not a change of coordinates but an actual change in the geometry under a ‘gauge transformation’ [10] as already mentioned, g¯ = e2! g; (2) where g e g¯ are metric tensors on the manifold M and ! = !(x) is a real function of the spacetime coordinates. The ‘gauge transformation’ in equation (2) is a rescaling of the metric tensor and it is named Weyl rescaling. In the Weyl rescaling we can state that g¯ is conformally related to g, because the gauge transformation (2) pre- serves the angle between two vectors U and V of the tangent space Tp(M). The inner product between the two vectors U and V, denoted by U · V = g(U; V) and the measure of the angle between two vectors follows from the definition, U · V g(U; V) cos θ = = p p : (3) jUjjVj g(U; U) g(V; V) The measure of the angle between the same two vectors under Weyl rescaling (2) is given by, g¯(U; V) e2! g(U; V) g(U; V) cos θ¯ = p p = p p = p p = cos θ; (4) g¯(U; U) g¯(V; V) e2! g(U; U) e2! g(V; V) g(U; U) g(V; V) therefore comparing the equation (3) and (4), the Weyl rescaling is a conformal transformation that preserves the angle but changes the scale. In accordance with this statement, a conformal transformation on spacetime (M; g) preserves the local light cone structure, where, timelike vectors remain timelike vectors, spacelike vectors remain spacelike vectors and null vectors remain null vectors. A very useful applications are the Penrose conformal diagrams that enable the whole of an infinite spacetime to be represented as a finite diagram, by applying a conformal transformation to the metric structure [9, 11]. 2.1. Calculation of the connections or Christoffel symbols and Riemann curvature tensor in a conformal space Let us compute the Riemann curvature tensor and for the first step, we calculate the Christoffel symbols in a conformal space starting with 1 Γ¯ ρ = g¯ρσ(¯g + g¯ − g¯ ): (5) µν 2 µσ,ν νσ,µ µν,σ with a comma denoting a partial derivative. To facilitate some calculations let us admit that e!(x) ! Ω(x) is some non-vanishing scalar funtion of position in the Weyl local rescaling metric of the metric tensor (2), 2 µν −2 µν g¯µν = Ω gµν andg ¯ = Ω g ; (6) ρ replacing above equation (6) into Christoffel symbols Γ¯ µν and calculating the derivatives, we have, 1 Γ¯ ρ = Ω−2gρσ[2Ω(Ω; )g + Ω2g + 2Ω(Ω; )g + Ω2g − 2Ω(Ω; )g − Ω2g ] µν 2 ν µσ µσ,ν µ νσ νσ,µ σ µν µν,σ 1 1 = Ω−2gρσ[Ω2(g + g − g )] + Ω−2gρσ[2Ω(Ω; g + Ω; g − Ω; g )]; 2 µσ,ν νσ,µ µν,σ 2 ν µσ µ νσ σ µν 2 that is ρ ρ −1 ρ ρ ρσ Γ¯ µν = Γ µν + Ω (δ µΩ;ν +δ νΩ;µ −g gµνΩ;σ ): (7) Now we can calculate the Riemann curvature tensor in a conformal space with metric tensor g¯, ρ ρ ρ κ ρ κ ρ R¯ µσν = Γ¯ µν,σ − Γ¯ µσ,ν + Γ¯ µνΓ¯ σκ − Γ¯ σµΓ¯ νκ (8) where we need to calculate the derivative of Christoffel symbol (7), that it will result in the first term of Riemann curvature tensor (8), ρ ρ ρ −1 ρ −1 ρκ −1 ρκ −1 Γ¯ µν,σ = Γ µν,σ + δ µ(Ω Ω;ν );σ +δ ν(Ω Ω;µ );σ −gµν,σ(g Ω Ω;κ ) − gµν(g Ω Ω;κ );σ : The second term of Riemann curvature tensor (8) is similar to the above first term, for just changing the indexes ν $ σ, ρ ρ ρ −1 ρ −1 ρκ −1 ρκ −1 Γ¯ µσ,ν = Γ µσ,ν + δ µ(Ω Ω;σ );ν +δ σ(Ω Ω;µ );ν −gµσ,ν(g Ω Ω;κ ) − gµσ(g Ω Ω;κ );ν : By subtracting the two terms above, we have that, ρ −1 ρ −1 ρ −2 −1 −2 −1 δ µ(Ω Ω;ν );σ −δ µ(Ω Ω;σ );ν = δ µ[(−Ω Ω;σ Ω;ν +Ω ;νσ ) − (−Ω Ω;ν Ω;σ +Ω ;σν )] = 0 and then we have ρ ρ ρ ρ ρ −1 ρ −1 ρκ −1 ρκ −1 Γ¯ µν,σ − Γ¯ µσ,ν = Γ µν,σ − Γ µσ,ν + δ ν(Ω Ω;µ );σ −δ σ(Ω Ω;µ );ν −gµν,σ(g Ω Ω;κ ) + gµσ,ν(g Ω Ω;κ ) ρκ −1 ρκ −1 +gµσ(g Ω Ω;κ );ν −gµν(g Ω Ω;κ );σ : (9) Let us calculate the third term of Riemann curvature tensor of equation (8), κ ρ κ −1 κ κ κλ ρ −1 ρ ρ ρλ Γ¯ µνΓ¯ σκ = [Γ µν + Ω (δ µΩ;ν +δ νΩ;µ −g gµνΩ;λ )][Γ σκ + Ω (δ σΩ;κ +δ κΩ;σ −g gσκΩ;λ )] κ ρ −1 κ ρ κ ρ ρ κλ = Γ µνΓ σκ + Ω (δ µΓ σκΩ;ν +δ νΓ σκΩ;µ −Γ σκg gµνΩ;λ ) −1 ρ κ ρ κ κ ρλ +Ω (δ σΓ µνΩ;κ +δ κΓ µνΩ;σ −Γ µνg gσκΩ;λ ) −2 κ ρ κ ρ κ ρλ +Ω (δ µδ σΩ;ν Ω;κ +δ µδ κΩ;ν Ω;σ −δ µg gσκΩ;ν Ω;λ κ ρ κ ρ κ ρλ +δ νδ σΩ;µ Ω;κ +δ νδ κΩ;µ Ω;σ −δ νg gσκΩ;µ Ω;λ κλ ρ κλ ρ κλ ρτ −g gµνδ σΩ;κ Ω;λ −g gµνδ κΩ;σ Ω;λ +g gµνg gσκΩ;λ Ω;τ ); the last term in above equation can be simplified, κλ ρτ λ ρτ ρτ g gµνg gσκΩ;λ Ω;τ = δ σgµνg Ω;λ Ω;τ = gµνg Ω;σ Ω;τ : so that κ ρ κ ρ −1 ρ ρ ρ κ ρ κλ ρ ρλ κ Γ¯ µνΓ¯ σκ = Γ µνΓ σκ + Ω Γ σµΩ;ν +Γ σνΩ;µ +δ σΓ µνΩ;κ +Γ µνΩ;σ −g gµνΓ σκΩ;λ −g gσκΓ µνΩ;λ −2 ρ ρ ρ ρ ρλ ρλ +Ω δ σΩ;ν Ω;µ +δ µΩ;ν Ω;σ +δ σΩ;ν Ω;µ +δ νΩ;µ Ω;σ −g gµσΩ;ν Ω;λ −g gσνΩ;µ Ω;λ κλ ρ ρλ ρλ −g gµνδ σΩ;κ Ω;λ −g gµνΩ;σ Ω;λ +gµνg Ω;σ Ω;λ ; we can see that the last two terms cancel each other resulting in κ ρ κ ρ −1 ρ ρ ρ ρ κ κλ ρ ρλ κ Γ¯ µνΓ¯ σκ = Γ µνΓ σκ + Ω Γ σµΩ;ν +Γ σνΩ;µ +Γ µνΩ;σ +δ σΓ µνΩ;κ −g gµνΓ σκΩ;λ −g gσκΓ µνΩ;λ −2 ρ ρ ρ ρλ ρ κλ +Ω 2δ σΩ;µ Ω;ν +δ µΩ;ν Ω;σ +δ νΩ;µ Ω;σ −g (gµσΩ;ν Ω;λ −gνσΩ;µ Ω;λ ) − gµνδ σg Ω;κ Ω;λ The fourth and final term of Riemann curvature tensor of equation (8), is similar to the above third term, for just changing the indexes ν $ σ, κ ρ κ ρ −1 ρ ρ ρ ρ κ κλ ρ ρλ κ Γ¯ σµΓ¯ νκ = Γ σµΓ νκ + Ω Γ νµΩ;σ +Γ νσΩ;µ +Γ µσΩ;ν +δ νΓ µσΩ;κ −g gµσΓ νκΩ;λ −g gνκΓ µσΩ;λ −2 ρ ρ ρ ρλ ρ κλ +Ω 2δ νΩ;µ Ω;σ +δ µΩ;σ Ω;ν +δ σΩ;µ Ω;ν −g (gµνΩ;σ Ω;λ −gσνΩ;µ Ω;λ ) − gµσδ νg Ω;κ Ω;λ : The subtraction between the third and the fourth terms obtained above results in, κ ρ κ ρ κ ρ κ ρ −1 ρ κ ρ κ κλ ρ ρ Γ¯ µνΓ¯ σκ −
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