Lagrangian theory of tensor fields over spaces with contravariant and covariant affine connections and metrics and its application to Einstein's theory of gravitation in (V)over-bar(4) spaces

摘要

The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and a metric [((L)over-bar(n),g)-spaces] is considered. The functional, the Lie, the covariant, and the total variations of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives, are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The covariant Euler-Lagrange equations are found on the basis of the MLCD. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities. As an application of the investigated general scheme, (pseudo) Riemannian spaces with contravariant and covariant affine connections (whose components differ not only by sign) ((V)over-bar(n)-spaces) are considered as a special case of ((L)over-bar(n),g)-spaces with Riemannian metric, symmetric covariant connection and a weaker definition of dual vector basis with 'conformal' noncanonical contraction operator S(dx(i), partial derivative(j)) = dx(i)(partial derivative(j)) = e(phi(x)).g(j)(i). The geodesic and autoparallel equations in (V)over-bar(4)-spaces are found as different equations in contrast to the case of V-4 spaces. The Euler-Lagrange equations as Einstein's field equations in (V)over-bar(4)-spaces and the corresponding energy-momentum tensors (EMTs) are obtained and compared with the Einstein equations and the EMTs in V-4-spaces. The geodesic and the auto-parallel equations are discussed. [References: 34]