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We compute the black hole horizon entanglement entropy S_E for a masslessscalar field, first with a hard cutoff and then with high frequency dispersion,both imposed in a frame that falls freely across the horizon. Using WKBmethods, we find that S_E is finite for a hard cutoff or super-luminaldispersion, because the mode oscillations do not diverge at the horizon and thecontribution of high transverse momenta is cut off by the angular momentumbarrier. For sub-luminal dispersion the entropy depends on the behavior atarbitrarily high transverse momenta. In all cases it scales with the horizonarea. For the hard cutoff it is linear in the cutoff, rather than quadratic.This discrepancy from the familiar result arises from the difference betweenthe 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 thecutoff that depends on the index of the dispersion relation. Implications forthe possible relation between regularized entanglement entropy and theBekenstein-Hawking entropy are discussed.Comment: 31 pages, 2 figures; v2: improvements to the presentation, typos  fixed, reference added, version published in PR

Black hole entanglement entropy regularized in a freely falling frame