标题： 弗罗贝尼乌斯度量中可见宇宙的适当尺寸 显示英文标题

标题： Proper Size of the Visible Universe in FRW Metrics with Constant Spacetime Curvature

摘要： 在本文中，我们继续探讨弗里德曼-罗伯逊-沃尔克（FRW）度规的基本基础及其在宇宙学中的应用，特别是针对这样一个问题：可见宇宙的正确尺寸是多少？ 回答尺寸问题有几种方法，但通常对光从最遥远的来源到达我们今天的实际传播距离理解不完整。 困难通常源于坐标使用的不一致，或对诸如所谓固有距离R(t)=a(t)r等量的物理意义的过度解释，该式用不变的共动半径r和宇宙膨胀因子a(t)表示。 在本文中，我们使用五种非平凡的具有恒定时空曲率的FRW度规（即静态FRW度规，但排除闵可夫斯基度规）来证明，在从初始奇点开始膨胀的静态FRW时空中，今天的可见宇宙的固有尺寸等于R_h(t_0/2)，即其当前年龄一半时的引力视界。 例外情况是德西特和兰佐斯，它们的内容在原点之外已有预先存在的位置。 显示英文摘要

摘要： In this paper, we continue to examine the fundamental basis for the Friedmann-Robertson-Walker (FRW) metric and its application to cosmology, specifically addressing the question: What is the proper size of the visible universe? There are several ways of answering the question of size, though often with an incomplete understanding of how far light has actually traveled in reaching us today from the most remote sources. The difficulty usually arises from an inconsistent use of the coordinates, or an over-interpretation of the physical meaning of quantities such as the so-called proper distance R(t)=a(t)r, written in terms of the (unchanging) co-moving radius r and the universal expansion factor a(t). In this paper, we use the five non-trivial FRW metrics with constant spacetime curvature (i.e., the static FRW metrics, but excluding Minkowski) to prove that in static FRW spacetimes in which expansion began from an initial signularity, the visible universe today has a proper size equal to R_h(t_0/2), i.e., the gravitational horizon at half its current age. The exceptions are de Sitter and Lanczos, whose contents had pre-existing positions away from the origin.