We address questions regarding construction and implications of gauge-invariant "dressed" observables in nontrivial background geometries such as that of a black hole. Formally, such observables can be constructed, e.g. by locating points with geodesics launched from infinity. However, practical complications arise in non-trivial geometries, and in particular for observables behind black hole horizons. Greater simplicity can be achieved by considering null constructions where the dressing lies along a null geodesic, or null surface such as a cone. We first investigate basic properties of these null dressings in the simpler context of electromagnetism. Since null constructions provide simple dressings for gauge-invariant observables inside black holes, they also allow us to investigate the question of compatibility of observables inside and outside black holes, and in particular the idea of black hole complementarity. While such observables in general have non-vanishing state-dependent commutators, the failure to commute does not appear particularly enhanced by the presence of the horizon.

中文翻译：

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重力和电磁学中的规范不变可观测量：黑洞背景和零修整

我们解决了关于非平凡背景几何（例如黑洞）中规范不变的“修饰”可观测量的构造和含义的问题。形式上，可以构建这样的可观察量，例如通过从无穷远处发射测地线来定位点。然而，实际的复杂性出现在非平凡的几何形状中，特别是对于黑洞视界后面的可观测物。通过考虑修整沿零测地线或零表面（如圆锥）的零结构，可以实现更大的简单性。我们首先在简单的电磁学背景下研究这些零敷料的基本特性。由于零结构为黑洞内的规范不变可观测量提供了简单的修饰，它们还使我们能够研究黑洞内外可观测物的兼容性问题，特别是黑洞互补性的想法。虽然这些可观察对象通常具有非消失的状态相关换向器，但由于地平线的存在，换向失败似乎并没有特别增强。