Nonstaticity theorem for type II, III, and IV matter fields in $f(R_{mu urhosigma},g^{mu u})$ gravity

In all $n(ge 3)$-dimensional gravitation theories whose Lagrangians are functions of the Riemann tensor and metric, we prove that static solutions are absent unless the total energy-momentum tensor for matter fields is of type I in the Hawking-Ellis classification. In other words, there is no hypersurface-orthogonal timelike Killing vector in a spacetime region with an energy-momentum tensor of type II, III, or IV. This asserts that ultra-dense regions with a semiclassical type-IV matter field cannot be static even with higher-curvature correction terms.