Proper Size of the Visible Universe in FRW Metrics with Constant
  Spacetime Curvature
  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. In so doing, we confirm earlier results showing
the same phenomenon in a broad range of cosmologies, including LCDM, based on
the numerical integration of null geodesic equations through an FRW metric.
