The Goldberg-Sachs theorem in linearized gravity

The Goldberg-Sachs theorem has been very useful in constructing algebraically special exact solutions of Einstein vacuum equation. Most of the physical meaningful vacuum exact solutions are algebraically special. We show that the Goldberg-Sachs theorem is not true in linearized gravity. This is a remarkable result, which gives light on the understanding of the physical meaning of the linearized solutions. PACS numbers: 04.20-q, 04.25-g Solutions of the linearized Einstein vacuum equation are usually considered as approximations of solutions of the full vacuum equation. They are useful tools for describing physical systems. It is important to understand the relation between the full vacuum equation and the linearized one; in particular it is interesting to know which properties are common, or not, to both sets of solutions. An example of a common property is the Birkhoff’s theorem; which can be enunciated in the following way: vacuum spherically symmetric solutions are static. This statement remains true in linearized gravity (i.e., if we replace the vacuum equation by the linear one); as one can check by reconstructing the linear version of the standard proofs that appear in the literature. In this work we present an example of a property, the so called GoldbergSachs theorem[1], which is not common to both set of solution. The Goldberg-Sachs theorem for the Einstein vacuum equation relates algebraic properties of the Weyl tensor with the existence of a null, geodesic, shear-free congruence in the space-time. In the search of vacuum solutions, the existence of such a congruence leads to considerable simplification in the calculation. The Schwarzschild, Kerr and Robinson-Trautman space-times, which Fellowship holder of C.O.N.I.C.O.R E-mail: [email protected] Present address: Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, D-14476 Golm, Germany Member of CONICET