On purely gravito-magnetic vacuum space-times (Q2487287)

The authors consider Petrov type I exact solutions of the Einstein field equations for which there exists a family of observers with 4-velocity \(\vec u'\) (\(\vec u\cdot \vec u = 1\)) such that the gravito-electric component of the Weyl tensor vanishes. Such spacetimes are called algebraically general, purely gravito-magnetic and can be invariantly characterized. The authors present a somewhat different approach to the investigation of these spacetimes and as a consequence obtain several new nonexistence cases. It is shown that there are no purely magnetic, vacuum, spacetime metrics where any one of the following conditions holds: (a) the ratio of any two eigenvalues of the Weyl tensor is constant, (b) both of the Riemann principal null directions, defining the time-like blade, are nonrotating, (c) the shear tensor possesses an eigenvector \(\vec v\) which is defined by one of the space-like Riemann principal directions, (d) the vorticity is parallel to \(\vec v\), where \(\vec v\) is defined by one of the space-like Riemann principal directions.