On purely gravito-magnetic vacuum space-times

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 v^a which is defined by one of the space-like Riemann principal directions, (d) the vorticity is parallel to v^a, where v^a is defined by one of the space-like Riemann principal directions.