On Effective Constraints for the Riemann-Lanczos System of Equations

There have been conflicting points of view concerning the Riemann--Lanczos problem in 3 and 4 dimensions. Using direct differentiation on the defining partial differential equations, Massa and Pagani (in 4 dimensions) and Edgar (in dimensions n > 2) have argued that there are effective constraints so that not all Riemann tensors can have Lanczos potentials; using Cartan's criteria of integrability of ideals of differential forms Bampi and Caviglia have argued that there are no such constraints in dimensions n < 5, and that, in these dimensions, all Riemann tensors can have Lanczos potentials. In this paper we give a simple direct derivation of a constraint equation, confirm explicitly that known exact solutions of the Riemann-Lanczos problem satisfy it, and argue that the Bampi and Caviglia conclusion must therefore be flawed. In support of this, we refer to the recent work of Dolan and Gerber on the three dimensional problem; by a method closely related to that of Bampi and Caviglia, they have found an 'internal identity' which we demonstrate is precisely the three dimensional version of the effective constraint originally found by Massa and Pagani, and Edgar.Comment: 9pages, Te

Kerr-Schild Approach to the Boosted Kerr Solution

Using a complex representation of the Debney-Kerr-Schild (DKS) solutions and the Kerr theorem we analyze the boosted Kerr geometries and give the exact and explicit expressions for the metrics, the principal null congruences, the coordinate systems and the location of the singularities for arbitrary value and orientation of the boost with respect to the angular momentum. In the limiting, ultrarelativistic case we obtain light-like solutions possessing diverging and twisting principal null congruences and having, contrary to the known pp-wave limiting solutions, a non-zero value of the total angular momentum. The implications of the above results in various related fields are discussed.Comment: 16 pages, LaTe