THE COPERNICAN QUANTIFICATION

This is the fourth of four essays, available from the forum library, 
which began with "Copernicus or Ptolemy: Cosmology Visualized" 
and proceeded through "A Pin for the Balloon" and "Comments on 
'Copernicus'."  The latter essay concluded with a verbal description of 
a mildly novel derivation of a cosmological line element, to which 
will now be given formal presentation in order to attempt a quantified 
resolution of the "mysteries" of both the "cosmological constant" and 
the "missing matter" of our unsettled theoretical cosmos:

ds^2 = - dt^2 + dr^2/((1 - r^2/R^2(t)) + (A/r)) + r^2 d(theta)^2 + 
r^2sin^2(theta) d(phi)^2

where r is a variable "absolute" radius, and the cosmological radius R 
is a function of the momentary time t.  A is the constant of integration 
introduced and rationalized in the previous essay, "Comments on 
'Copernicus'."

The cosmological constant (lambda) is a function of vacuum energy 
density and may be expressed as the inverse of the square of the 
cosmological radius.  For purposes of approximating the desired 
result, the constant A will be taken to be the Planck length, which 
permits a cosmological radius/constant of eight hundred kilometers to 
produce a "proper" radial interval of approximately ten billion light 
years.  This interval is the desired marginal "closure" radius/density, 
which accords with the observed "flatness" of the universe and which 
permits maximum spherical radius to the "Copernican" cosmos.

The "missing matter" difficulty is more than accounted for by taking 
the geometry of the universe as a product of vacuum energy density 
alone.  Given the Planck length as our constant, we have had to over-
/underestimate the absolute cosmological radius/constant by three/six 
orders of magnitude [see Abbott] so as to avoid too small a calculated 
proper radius.  We have, in compensation however, achieved a CC 
discrepancy reduction of some forty of forty-six orders of magnitude, 
and some hope of relief from tortuously contrived theories attempting 
to dispense with the otherwise troublesome vacuum energy altogether.  
The surviving discrepancy does not, at this early date, discourage 
continuing efforts at discovering "cancellations" in the particle field 
contributions to the vacuum energy, or even at some convenient 
fudging in order to bring the involved numbers into mutual 
compliance.

An obvious question and objection regarding the foregoing is to the 
combination of a seemingly absurd "absolute" radius of geographic-
scale kilometers with a "proper" interval reckoned in terms of billions 
of light years.  The two may be reconciled by imagining a "rod" of a 
few hundred kilometers length being "ejected" from the highly 
vacuum energetic sphere of our universe into the presumed 
surrounding void and "decompressing" (amidst its evanescence) to 
the equivalent of the cosmological radius.  Alternative estimations of 
the universal curvature radically reduced to the equivalent of the 
Planck length may be taken to be the particular victims of the isotropy 
confusion discussed in "A Pin for the Balloon," the second of the 
essays in this series.

[The acute student will have realized that the foregoing equation is 
unorthodox only to the extent of recovering the expression "(A/r)."  
Otherwise, it retains the premise of progressive reduction to "flat" 
space toward the origin and a hyperbolic density distribution 
consistent therewith.  A formula consistent with a progressive 
reduction to singularity at the origin and "Copernican" energy density 
is suggested by the following:  ds^2 = - dt^2 + (R/A)(dr^2/(r^2/R^2(t)) 
+ R^2 d(theta)^2 + R^2sin^2(theta) d(phi)^2)]

MICHAEL LAURENCE
JANUARY 17, 1995