Narrow Results

The transformation laws for the electric and magnetic parts of the electromagnetic 2-form and the Weyl tensor under a boost are studied using the complex vector approach, which shows the close analogy between the two cases. For a nonnull electromagnetic field, one can always find an observer who sees parallel electric and magnetic fields (vanishing Poynting vector) and also sees a minimum electromagnetic energy density (and minimum electric and magnetic field magnitudes) compared to other observers. For Weyl fields of all Petrov types except III, and boosts along certain directions of the Weyl principal tetrad, the more complicated Weyl transformation closely mimics the electromagnetic boost transformation, allowing one to extend the electromagnetic results directly to the families of boosts along those directions. In particular for black hole spacetimes, the alignment of the electric and magnetic parts of the Weyl tensor (vanishing super-Poynting vector) leads to minimal gravitational super-energy as seen by the Carter observer within the family of all circularly rotating observers at each spacetime point outside the horizon.

This work is a study of the Weyl conformal tensor on manifolds of dimension $4$ and for each of the three possible signatures $(+,+,+,-)$, $(+,+,-,-)$ and $(+,+,+,+)$. In it, the properties of the Weyl tensor and its algebraic classification will be compared and contrasted.

In this paper, an algebraic classification of the Roter type spacetimes is given. It follows that the Roter type curvature condition is essentially equivalent with the pseudosymmetry condition on 4-dimensional Lorentzian manifolds.

The cylindrically symmetric static manifolds are classified for their Petrov types and metrics. This classification besides verifying the earlier result that such manifolds cannot be of petrov type II, III and N, gives a complete list of all static cylindrically symmetric metrics of Petrov type O. In the case of Petrov type D metrics, the results appear as three independent classes metrics.