Black hole entanglement entropy regularized in a freely falling frame

TL;DR

This paper calculates the entanglement entropy of a black hole horizon for a scalar field in a freely falling frame, showing finiteness with certain cutoffs and dispersions, and discusses implications for black hole entropy.

Contribution

It introduces a new approach to regularizing black hole entanglement entropy in a freely falling frame, analyzing effects of dispersion relations on entropy scaling.

Findings

Entropy is finite with a hard cutoff or super-luminal dispersion.

Entropy scales with horizon area in all cases.

Hard cutoff entropy is linear in the cutoff, not quadratic.

Abstract

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.