J un 1 99 5 The Robinson - Trautman Type III Prolongation Structure Contains

The minimal prolongation structure for the Robinson-Trautman equations of Petrov type III is shown to always include the infinite-dimensional, contragredient algebra, K 2 , which is of infinite growth. Knowledge of faithful representations of this algebra would allow the determination of Bäcklund transformations to evolve new solutions. The general class of Robinson-Trautman solutions [1] to the vacuum Einstein field equations have been important examples of exact solutions for many years, albeit they seem to have various difficulties with respect to their interpretation [2]. They are solutions characterized by having a repeated principal null direction, which is of course geodesic and shearfree, and is required to be diverging but not twisting. The standard reference [3] gives the general form of the metric which any Einstein space must have if it permits such a repeated principal null direction, and notes that all possible algebraically-special Petrov types are allowed. In the case of Petrov type III, the field equations are [3] first reduced to K = ∆ log P ≡ 2P 2 ∂ ζ ∂ ζ log P = −3[f (ζ, u) + f (ζ, u)] , (1) where K is the Gaussian curvature of the 2-surface spanned by ζ and ζ. This equation determines the general RT-solution of Petrov type III. However, since u is nowhere explicitly mentioned within the partial differential equation (pde), it is well-known [3] that one could always simply ignore that dependence, perform a coordinate transformation sending f (ζ) → ζ, leaving the curvature completely invariant, and reducing our equation to the rather simple-appearing equation K = 2P 2 ∂ ζ ∂ ζ log P = − 3 2 (ζ + ζ) , or u xy = 1 2 (x + y)e −2u , where log P ≡ u , (2) the subscripts denote partial derivatives, and the symbols {x, y} have been introduced instead of {ζ, ζ}, both to simplify the typography and to normalize the equation so that the coefficient has a value which will prove convenient. As already pointed out, all Petrov type III solutions of the vacuum field equations with diverging , non-twisting null directions are determined by the general solution of Eq. (2). Nonetheless , only one rather trivial solution is available for study, namely P = (ζ + ζ) 3/2 , even though all its Lie symmetries have been found. [4] This unfortunate situation has caused us to apply the general methods …