Third-order geometrical null symmetries of gravitational fields (Q1122802)

By a third order geometrical symmetry is meant that there is a vector whose Lie derivative of the metric, and whose second order Lie derivative of the metric do not vanish, but the third derivative does. The Newman- Penrose formalism is used to find necessary and sufficient conditions for such symmetries. Electromagnetic fields are also considered and the third order null symmetries (when the vector is null) are investigated. It is found that the third order Lie derivative vanishes identically if the null vector is both the propagation vector for the source-free electromagnetic field and for some algebraically special free gravitational fields.