In this paper we prove a number of exact relations between optical observables, such as trigonometric parallax, position drift, and the proper motion of a luminous source, in addition to the variations of redshift and the viewing angle. These relations are valid in general relativity for any spacetime, and they are of potential interest for astrometry and precise cosmology. They generalize the well-known Etherington’s reciprocity relation between the angular diameter distance and the luminosity distance. Similar to the Etherington’s relation, they hold independently of the spacetime metric, the positions, and the motions of a light source or an observer. We show that those relations follow from the symplectic property of the bilocal geodesic operator, i.e., the geometric object that describes the light propagation between two distant regions of a spacetime. The set of relations we present is complete in the sense that no other relations between those observables should hold, in general. In the meantime, we develop the mathematical machinery of the bilocal approach to light propagation in general relativity and its corresponding Hamiltonian formalism.

中文翻译：

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天空可观测值的代数对称性：可变发射器和观测器


在本文中，我们证明了光学可观测量之间的许多精确关系，例如三角视差、位置漂移和光源自行，以及红移和视角的变化。这些关系在广义相对论中对于任何时空都是有效的，并且它们对于天体测量和精确宇宙学具有潜在的意义。他们概括了著名的埃瑟林顿角直径距离和光度距离之间的互易关系。与埃瑟林顿关系类似，它们的成立与时空度量、光源或观察者的位置和运动无关。我们证明这些关系源自双局部测地算子的辛性质，即描述时空两个遥远区域之间光传播的几何对象。我们提出的关系集是完整的，因为一般来说，这些可观察量之间不应该存在其他关系。与此同时，我们开发了广义相对论中光传播双局域方法的数学机制及其相应的哈密顿形式主义。