光度和面积距离的非微扰结果
Non-perturbative results for the luminosity and area distances
摘要：
光度距离的概念通常定义在纯粹的FLRW（Friedmann-Lemaitre-Robertson-Walker）宇宙时空中，或其小扰动。然而，光度距离的抽象概念实际上比这更稳健，并且可以在几乎任意时空中非微扰地定义。一些相当普遍的结果是已知的，在$dA_\mathrm{observer}/dΩ_\mathrm{source}$，横截面积每单位立体角的零测地线喷雾从一些源发射，随后检测到一些观察者。我们将重新制定一个适当的正规化零测地线仿射参数和范Vleck行列式，$Δ_{vV}$这些结果。零测地仿射参数的贡献实际上是光度的平方反比定律，而范弗莱克行列式可以被看作是提供了一个偏离平方反比定律的度量。这个公式与所谓的Jacobi行列式密切相关，但van Vleck行列式具有更好的分析性质，在广义相对论，量子物理和量子场论社区中具有更广泛和更深入的理论基础。在目前的文章中，我们将集中在非微扰的结果，离开近FLRW微扰调查为未来的工作。
英文摘要：
The notion of luminosity distance is most often defined in purely FLRW (Friedmann-Lemaitre-Robertson-Walker) cosmological spacetimes, or small perturbations thereof. However, the abstract notion of luminosity distance is actually much more robust than this, and can be defined non-perturbatively in almost arbitrary spacetimes. Some quite general results are already known, in terms of $dA_\mathrm{observer}/dΩ_\mathrm{source}$, the cross-sectional area per unit solid angle of a null geodesic spray emitted from some source and subsequently detected by some observer. We shall reformulate these results in terms of a suitably normalized null geodesic affine parameter and the van Vleck determinant, $Δ_{vV}$. The contribution due to the null geodesic affine parameter is effectively the inverse square law for luminosity, and the van Vleck determinant can be viewed as providing a measure of deviations from the inverse square law. This formulation is closely related to the so-called Jacobi determinant, but the van Vleck determinant has somewhat nicer analytic properties and wider and deeper theoretical base in the general relativity, quantum physics, and quantum field theory communities. In the current article we shall concentrate on non-perturbative results, leaving near-FLRW perturbative investigation for future work.
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