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 (55 ), (56 ), (57 ), (58 ), and (59 ) are valid even if subleading terms (as in Eq. (51 )) in the expansion of the metric near the singular surface $Σ$ are functions of $𝜃$ [111 ]. Such more general metrics describe what might be called a “local Killing horizon”.
4		Note that in [111 ] there is a typo in Eq. (3.9) defining $cp$ . This does not affect the conclusions of [111 ] since they are based on the relation $cp−1 = 8πpcp$ rather than on the explicit form of $cp$ .
5		Eqs. (93 ), (94 ) and (95 ) correct some errors in Eqs. (27) – (29) of [197 ].
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 [110 ] 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8		As I have recently learned (private communication from Myers), Jacobson and Myers (unpublished) had similar ideas back in the 1990s.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 [17 ] is related to $a3$ as $b6 = − a3$ .