1 | For finite matrices this property indicates that the two density matrices have the same eigenvalues. | |

2 | If the boundary of is not empty there could be extra terms in the entropy proportional to the “area” of the boundary 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 are functions of [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 . This does not affect the conclusions of [111] since they are based on the relation rather than on the explicit form of . | |

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 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8 | As I have recently learned (private communication from Myers), Jacobson and Myers (unpublished) had similar ideas back in the 1990s.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 of [17] is related to as . |

Living Rev. Relativity 14, (2011), 8
http://www.livingreviews.org/lrr-2011-8 |
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