  In all $n(\ge 3)$-dimensional gravitation theories whose Lagrangians are
functions of the Riemann tensor and metric, we show 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, if
back-reaction is taken into account to give a self-consistent solution,
ultra-dense regions with a semiclassical type-IV matter field cannot be static
even with higher-curvature correction terms. As a consequence, a static
Planck-mass relic is possible as a final state of an evaporating black hole
only if the semiclassical total energy-momentum tensor is of type I.
