A variational principle giving gravitational “superpotentials,” the affine connection, Riemann tensor, and Einstein field equations

Abstract

A first-order Lagrangian is given, from which follow the definitions of the fully covariant form of the Riemann tensorR _μνkλ in terms of the affine connection and metric; the definition of the affine connection in terms of the metric; the Einstein field equations; and the definition of a set of gravitational “superpotentials” closely connected with the Komar conservation laws [7]. Substitution of the definition of the affine connection into this Lagrangian results in a second-order Lagrangian, from which follow the definition of the fully covariant Riemann tensor in terms of the metric, the Einstein equations, and the definition of the gravitational “superpotentials”.