  We discuss linearized gravitational perturbations of higher dimensional
spacetimes. For algebraically special spacetimes (e.g. Myers-Perry black
holes), we show that there exist local gauge invariant quantities linear in the
metric perturbation. These are the higher dimensional generalizations of the 4d
Newman-Penrose scalars that (in an algebraically special vacuum spacetime)
satisfy decoupled equations of motion. We show that decoupling occurs in more
than four dimensions if, and only if, the spacetime admits a null geodesic
congruence with vanishing expansion, rotation and shear. Decoupling of
electromagnetic perturbations occurs under the same conditions. Although these
conditions are not satisfied in black hole spacetimes, they are satisfied in
the near-horizon geometry of an extreme black hole.
