Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)

Abstract

A simple derivation is given of the Einstein-Maxwell field equations from the 2 nd ordinary exterior differential of a precursor to the soldering form for n-dimensional differentiable manifolds having a general linear connection and in 5-dimensional general relativity in particular. PACS numbers: 02.40.-k, 04.20.Fy This paper presents a simple derivation of the Einstein-Maxwell field equations from the 2 nd ordinary exterior differential of a precursor to the soldering form for n-dimensional differentiable manifolds M having a general linear connection and in 5-dimensional general relativity in particular. The products e a ⊗ ω b = e a ω b = e c δ c a δ b d ω d (1) of the basis tangent vectors e a and basis 1-forms ω a of M are basis elements 1 of the tensor product 2 T(M) ⊗ T*(M) of the tangent and cotangent bundles T(M) and T*(M) of M and are a precursor to "the soldering form" 3-5 (also called "the Cartan-Maurer form," 6 "the canonical form," 7-8 and "the displacement vector" 9-10) d P given by d P = e a ⊗ ω a = e a ω a = e a δ a b ω b , (2) where δ a b is the Kronecker delta and where Latin indices represent anholonomic coordinates. The 1 st ordinary exterior differentials of e a are given by 11-13 d e a = e b ω a b , (3) the contractions of which with ω b are given by 〈ω b , d e a 〉 = ω a b (4) and in view of which the 1 st absolute exterior differentials of e a are given by D e a = d e a − e b ω a b (5) = 0, where ω a b is the connection 1-form of M as given in terms of the connection coefficients Γ a b c by 14-19 ω a b = Γ c b a ω c , (6) and where the contractions of ω b with e a are given by 〈ω b , e a 〉 = δ b a. (7) The 2 nd ordinary exterior differentials of e a are given by 20-29 d 2 e a = d d e a (8) = d e b ω a b = (d e b) ∧ ω a b + e b d ω a b = e c ω b c ∧ ω a b + e b d ω a b = e b (d ω a b + ω c b ∧ ω a c) = e b Ω a b , the contractions of which with ω b are given by 〈ω b , d 2 e a 〉 = Ω a b , (9) where Ω a b is the curvature 2-form of M as given by Ω a b = d ω a b + ω c b ∧ ω a c (10) = D ω a b + ω a c ∧ ω c b = d ω a b − ω a c ∧ ω c b = D ω a b − ω c b ∧ ω a c = d ω a b + 1 2 [ω c b , ω a c ] = D ω a b + 1 2 [ω a c , ω c b ] = d ω a b − 1 2 [ω a c , ω c b ] = D ω a b − 1 2 [ω c b , ω a c ] = 1 2 R cda b ω c ∧ ω d , 1 Goldberg, S.