LsdiffM and the Einstein equations

We give a formulation of the vacuum Einstein equations in terms of a set of volume-preserving vector fields on a four-manifold M. These vectors satisfy a set of equations which are a generalisation of the Yang-Mills equations for a constant connection on flat spacetime. It is known (Mason and Newman 1989) that the equations which describe self-dual Ricci-flat metrics can be derived from the self-dual Yang-Mills equations in flat space, with a specific choice of gauge group. In particular, consider the self-dual Yang-Mills equations on flat space, (M, η), Fij = 1 2 ǫij Fkl, (1) where Fij is the curvature of an algebra valued connection Ai. If we now impose that the connection Ai is constant on M, then equations (1) become a set of algebraic conditions on the connection [Ai, Aj ] = 1 2 ǫij kl [Ak, Al] . Mason and Newman showed (Mason and Newman 1989) that if we take the connection A to take values in LsdiffM, the volume-preserving diffeomorphisms of an auxiliary four-manifoldM, andw ̄ rite Ai = (e1, e2, e3, e4), then the (contravariant) metric g = η ei ⊗ej defines, up to a known conformal factor, a self-dual metric on the manifold M. If we consider the case where the connection A satisfies the full Yang-Mills equations, we are led to the equations η [ei, [ej , ek]] = 0, (2) which, if we allow the connection to have torsion, correspond to Einstein-Cartan theory (Mason and Newman 1989). The question we will consider here is whether it is possible to find a similar formulation of the full vacuum Einstein equations. We begin with four vectors, Vi, and an internal metric ηij , with inverse η such that the (contravariant) metric g = η Vi ⊗ Vj (3) satisfies the vacuum Einstein equations. (Letters i, j, . . . will denote internal indices, which will be raised and lowered using η and ηij . We will consider complex metrics, and will not discuss reality conditions.) Defining the structure functions of the vectors Vi by [Vi,Vj ] = C̃ k ij Vk, then, by an internal rotation, we impose the condition that C̃ j ij = −Vi(log f),