Black hole entanglement entropy regularized in a freely falling frame

We compute the black hole horizon entanglement entropy SE 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 SE is finite for a hard cutoff or superluminal 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 subluminal 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 superluminal 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. An appendix provides an explicit derivation of the entangled, thermal nature of the near-horizon free-fall vacuum for a dispersive scalar field in four dimensions.