We prove the achronal averaged null energy condition for general quantum field theories in the near horizon geometry of spherical extremal black holes (i.e. AdS 2 × S d− 2 ), de Sitter and anti-de Sitter. The derivation follows from monotonicity of relative entropy after computing the modular hamiltonian of a null deformed region. For incomplete (but maximally extended) achronal null geodesics in AdS 2 × S d− 2 , we prove the positivity of a different light-ray operator for arbitrary CFTs. This agrees with a constraint recently derived for the Lorentzian cylinder.

中文翻译：

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极值视界和 (A)dS 的时间平均零能量条件

我们证明了球形极值黑洞（即 AdS 2 × S d- 2 ）、德西特和反德西特的近视界几何中一般量子场论的时间平均零能量条件。在计算零变形区域的模哈密顿量之后，该推导遵循相对熵的单调性。对于 AdS 2 × S d− 2 中不完整（但最大扩展）的非时空测地线，我们证明了不同光线算子对任意 CFT 的正性。这与最近为洛伦兹圆柱导出的约束一致。