First-Order Structure of the Linearised Einstein Equations: Constant-Curvature Backgrounds

Abstract

Teukolsky's decoupled equations and Wald's adjoint method of metric reconstruction are built from second-order operators: the linearised Einstein operator G, the curvature-extraction operator T, the decoupled operator O and the reconstruction operator S^*, related by SG=OT. We show that on four-dimensional constant-curvature backgrounds with non-zero Lambda all of these operators factor into first-order maps through a single intermediate variable: the perturbation of the self-dual half of the spin connection. We also show that for non-zero Lambda the metric perturbation is itself obtained from the connection perturbation by a first-order map. The connection therefore gives a complete description of linearised gravity, and the fourth-order identity SG=OT follows by composition from a single second-order identity. Using conformal flatness, we choose the Hertz-potential map so that, after a conformal rescaling, the potential obeys the flat wave equation component by component. This leads to a new reconstruction scheme, in which the potential is sourced by the curvature psi, which in turn obeys a decoupled sourced equation. A prescribed curvature is then reached by a second-order rather than a fourth-order equation, and the corrector is local and algebraic in the stress-energy tensor. The price is that the full curvature, rather than a single component, must be determined, and that the reality of the reconstructed metric becomes a condition, which we argue holds for retarded solutions. We illustrate the scheme for a static point mass in de Sitter space, where it gives a real metric perturbation that agrees with linearised Schwarzschild-de Sitter up to a diffeomorphism. We view these results as a proof of concept for a first-order Teukolsky-Wald framework on Kerr.