  We compute the black hole horizon entanglement entropy S_E for a massless
scalar field, first with a hard cutoff and then with high frequency dispersion,
both imposed in a frame that falls freely across the horizon. Using WKB
methods, we find that S_E is finite for a hard cutoff or super-luminal
dispersion, because the mode oscillations do not diverge at the horizon and the
contribution of high transverse momenta is cut off by the angular momentum
barrier. For sub-luminal dispersion the entropy depends on the behavior at
arbitrarily high transverse momenta. In all cases it scales with the horizon
area. For the hard cutoff it is linear in the cutoff, rather than quadratic.
This discrepancy from the familiar result arises from the difference between
the free-fall frame and the static frame in which a cutoff is usually imposed.
In the super-luminal case the entropy scales with a fractional power of the
cutoff that depends on the index of the dispersion relation. Implications for
the possible relation between regularized entanglement entropy and the
Bekenstein-Hawking entropy are discussed.
