Horizon surface gravity as 2d geodesic expansion
  The surface gravity of any Killing horizon, in any spacetime dimension, can be interpreted as a local, two-dimensional expansion rate seen by freely falling observers when they cross the horizon. Any two-dimensional congruence of geodesics invariant under the Killing flow can be used to define this expansion, provided that the observers have unit Killing energy. 
Ted Jacobson and Renaud Parentani