A study of the Riemann curvature tensor and Petrov classification in general relativity

In this article, we present and support by examples a criterion for the existence of linebreak gravitational radiation in terms of the invariants of the Riemann curvature tensor. We give a classification of spacetimes in terms of electric and magnetic parts of the Weyl tensor and discuss some examples of spacetimes having purely magnetic and purely electric Weyl tensors. The Lanczos potential is studied using the method of general observers and tetrad formalisms. We obtain the Lanczos potentials for perfect fluid spacetimes, Godel cosmological model, and Kerr black hole. The work also considers the space-matter tensor, introduced by Petrov, and the perfect-fluid spacetimes with the divergence-free space-matter tensor.