List of Footnotes

1 For finite matrices this property indicates that the two density matrices have the same eigenvalues.
2 If the boundary of Σ is not empty there could be extra terms in the entropy proportional to the “area” of the boundary ∂Σ as was shown in [108]. We do not consider this case here.
3 It should be noted that formulas (55View Equation), (56View Equation), (57View Equation), (58View Equation), and (59View Equation) are valid even if subleading terms (as in Eq. (51View Equation)) in the expansion of the metric near the singular surface Σ are functions of 𝜃 [111Jump To The Next Citation Point]. Such more general metrics describe what might be called a “local Killing horizon”.
4 Note that in [111Jump To The Next Citation Point] there is a typo in Eq. (3.9) defining cp. This does not affect the conclusions of [111Jump To The Next Citation Point] since they are based on the relation cp−1 = 8πpcp rather than on the explicit form of cp.
5 Eqs. (93View Equation), (94View Equation) and (95View Equation) correct some errors in Eqs. (27) – (29) of [197Jump To The Next Citation Point].
6 This statement should be taken with some care. Entanglement entropy is a small correction compared to the Bekenstein–Hawking entropy if the UV cutoff 1∕𝜖 is, for example, on the order of a few GeV (energy scale of the standard model). However, the two entropies are of the same order if the cutoff is at the Planck scale. I thank ’t Hooft for his comments on this point.
7 Among other things the authors of [110Jump To The Next Citation Point] observe certain non-smooth behavior of the heat kernel coefficients for the spin-3/2 and spin-2 fields in the limit of vanishing angle deficit.
8 As I have recently learned (private communication from Myers), Jacobson and Myers (unpublished) had similar ideas back in the 1990s.
9 This is true for minimally-coupled matter fields. In the presence of non-minimal couplings there appear extra terms in the thermodynamic entropy, which are absent in the entanglement entropy, as we discussed earlier in Section 3.16.
10 Note that the coefficient b6 of [17Jump To The Next Citation Point] is related to a3 as b6 = − a3.