Instant of Relativity Revised Copyright 2012 to 2016 by Del John Ventruella

Copyright © 2012, 2013 by Del John Ventruella. All Rights Reserved.
DRAFT COPY – NOT REVIEWED – MAY CONTAIN SERIOUS ERRORS!
DO NOT CIRCULATE! DO NOT USE!
An Instant of Relativity
Del John Ventruella
Abstract: This discussion focuses on the
application of Baez' and Bunn's infinitesimal
expression1 for the Einstein equation at the
core of general relativity to a pair of simple,
expansionary models of the universe to
estimate the universe's present scale. The
analysis is guided by current insights into the
famous, “big bang”38,40,41,43,44 cosmological
model of Lemaître43,44 offered by modern,
(WMAP) astronomical data. A simple proof
begins the analysis by comparing two
equations (the Baez and Bunn1 infinitesimal
Einstein equation expression and an Einstein
equation expression associated with the Hubble
flow2) that have been referenced by Baez and
Bunn (and S. Carroll2) to establish the basis for
their equality.
The content is inspired by a much more
detailed discussion of the related
conceptualization of Einstein's equation for
infinitesimal application written by Baez and
Bunn1 themselves. Anyone who has not yet
encountered general relativity via tensor
calculus will likely read the introduction to
relativity as presented by Baez and Bunn in
their paper with great pleasure. What is
offered here is intended principally as
encouragement to do so.
A very simple, exponential, “big bang” model
is first produced in the course of this discussion
that considers, in a conveniently crude manner,
the effect of a cosmological constant 27, 36
within a homogeneous and isotropic universe
that evolved to an essentially “flat” state
relative to space-time curvature early in its
long history. This is offered to demonstrate the
intriguing nature of the infinitesimal Einstein
equation conceptualized by Baez and Bunn
with regard to its capacity to provide basic
insight into cosmology18 and gravity when
considering vast scales of time. The “Baez and
Bunn” Einstein equation is applied to the data
set of the derived exponential expansionary
model that is developed here to illustrate this.
(An exponential model is one of the first, crude
models produced.)
A direct, numerical integration model
employing only the Einstein equation of Baez
and Bunn over fifteen billion years is also
compared to the predictions of the widely
accepted Friedmann-Walker-Robertson
equation in terms of computing the expansion
of the universe from the “big bang” to the
present time. The source data is substantially
derived from the WMAP survey data and
conclusions drawn from it by others.83, 84
This discussion is unlikely to provide new
perspective to anyone with an advanced grasp
of relativity. Dense fields of equations
summarizing a new insight and intended for
experts in the field are not present. Instead,
considerable effort is made to explain what is
done. Although simple integration is used,
tensor mathematics and differential equations
are not. In other words, what follows is meant
to be fun and easy to read.
The content may also help, in a very general
sense, to illustrate the usefulness of “first
approximation” models and a “state-space”
approach when seeking solutions associated
with physical systems via a method that does
not balk at the possibility that one should
“think crude” in the initial stages of pursuit of
a scientific or mathematical model in a manner
capable of offering a sense of basic
understanding by recognizing that the
opportunity to refine the solution in the future
1

may only present itself after a reasonable level
of understanding of a concept is confirmed via
such a simple model. Limited prior exposure
my be useful to ideas such as a four
dimensional68 universe and a quantum model
that considers particles to be geometric
points67 but is not required.
Introduction
Special relativity is now more than a century
old. General relativity followed as an extension
of special relativity but was originally
introduced by Einstein via a differential
equation that lacked a closed form solution. It
was not until a Prussian artillery officer
produced his namesake Schwarzschild (in very
rough, phonetic terms, SHVARTS-SHIELD)
solution for Einstein's equation as it pertains to
a stationary mass during World War One that
Einstein's differential field equation relating
energy density to changes in the curvature of
space-time as asserted in his theory of general
relativity had a solution expressed in four,
Euclidean dimensions that was applicable to a
general problem in physics.10
The Einstein equation is usually explored using
tensor variables.11 Tensor mathematics is not a
common element of undergraduate engineering
curriculums (which suggests one prospective
audience for this discussion). Non-physics (or
mathematics) majors may shrink from their first
encounter with Einstein's most famous insight
when presented in the context of its tensorial
roots. They may then seek a more
straightforward presentation (as did the author),
such as the Einstein equation of Baez and
Bunn.1
A great deal of patience can overcome this
mathematical obstacle if one is willing to delve
into tensor calculus, and many textbooks seem
to confirm that nothing will fully correct for
want of a solid foundation in tensor
mathematics if one wishes to consider the shape
of space-time in detail relative to matters
germane to cosmology. (An alternative path to
understanding general relativity, focused on
LaGrangian mechanics34, has been previously
presented elsewhere. LaGrangian mechanics
was a favorite domain of the famous, twentieth
century physicist, Richard Feynman, and he has
written an introduction to the subject for
undergraduates, which may appeal to those
with suitable curiosity.)
The obstacle erected by tensor mathematics is
usually sufficient to limit undergraduate
engineering students to momentary contact with
the subject of relativity. This may occur within
a broader, statistical and modern physics course
in which only special relativity is briefly
discussed, perhaps, too often, while being
perceived by the instructor as a subject that a
particular section of engineering
undergraduates will rarely if ever find to be
applicable to their future work.
Such an attitude is not historically surprising.
Einstein himself noted that general relativity
was a field that was not given a great deal of
emphasis in college curriculums, perhaps in
part due to its specialized application to
astronomy and cosmology before the
introduction of global positioning system based
navigation and analysis of the decay rates of
muons79, which, together, add only very
narrow, relevant applications in engineering and
particle physics. Of course, Einstein was in
favor of correcting the want of attention
regarding general relativity and authored a
book, The Meaning of Relativity, to enhance
such possibilities.
Other books, such as Introduction to the Theory
of Relativity, by Peter Gabriel Bergmann, (an
older text), and Gravity, An Introduction to
Einstein's General Relativity, by James B.
Hartle, have acquired some renown among past
students and some teachers of the subject for
individuals seeking more accessible treatments
available in paperback. Even with such
resources, relativity is not commonly
2

encountered in a broad range of scientific
fields, including many associated with physics,
although its consideration may enrich analysis.
(Brian Greene offers some interesting insight
into this from the perspective of a theoretical
physicist in his new introduction to The
Meaning of Relativity.)
Even quantum theory was somehow originally
conceived without the need to integrate
Einstein's older, general relativity. With
relativity commonly perceived as limited in
relevance to somewhat obscure cosmological
and astronomical questions focused on
gravitational interactions beyond the
expectations of the laws of Newton, the
founders of quantum physics originally turned
simply to the clock on the mantle instead of
Einstein's relativistic theory of space and
time.24
We should recall that in the long march of
human history both relativity and quantum
theory were new to the first half of the
twentieth century. Near the middle of the
nineteenth century many physicists had come to
believe that all that remained for science to
discover could be achieved through the
measurement of a few more significant digits
associated with the constants employed by
classical physical theories then held sacrosanct
by government, industry, and teaching
institutions, whose funding they substantially
controlled.60
The black body radiation spectrum and inability
to resolve other conundrums via classical
physical theories, including problems such as
the so-called, “occultation of Mercury”,
(leading to the assertion of an unobserved and
non-existent planet, given the name,
“Vulcan”,88
to explain Mercury's orbital
irregularities through Newton's gravitational
theory) suggested the need for scientific
advances, but acceptance of movement away
from classical models as the ultimate
expressions of natural theory came slowly in a
world of experts known specifically for their
prowess with those same theories. Determined
attempts to apply classical perspectives to
seemingly unsolvable scientific problems
continued (unsuccessfully relative to both of the
puzzles just described) into the twentieth
century.
It was then that Albert Einstein, as an
undergraduate in a state run college intended to
produce Switzerland's next generation of sub-
college level scientific teachers, chose to seek
to define the profession of a “theoretical
physicist” and, by working resolutely toward
that goal, overcame the strict notion of science
as no more than the practical servant of industry
and, in certain instances, the chosen field of
peculiar aristocrats, in a world in which
everything was perceived to be very old, and
thus a place in which everything must already
have been very well defined. With bold new
scientific perspectives rising to confront the
new problems that much older ideas were
proving powerless to address, there arose a
corresponding potential for eager young
“theoretical physicists” to be perceived as an
annoyance or simply as misguided by elder
scientists, whose reputations had been built
upon their mastery of classical theories.
In the ensuing conflict between the old and the
new there arose not a whisper of the specter of
the harsh lesson learned by Galileo (1564-
1642)61 when presenting his evidence in favor
of displacing the earth from the center of the
universe before the foot soldiers of a
disapproving religious power and under the
influence of governments controlled by princes,
who claimed to rule by the will of the favored
deity, and who had grown fond of having their
crowns bestowed in publicly acclaimed
ordinations by revered, religious elites to
impress those they ruled in a world in which the
lives of many, amid the ravages of disease and
war, were sufficiently harsh that heaven could
easily seem their sole hope for happiness.
Galileo's ideas did not suit the intellectual
3

paradigm that empowered glorification of “god
and country” from the perspective of those who
jealously guarded their intellectual influence
over the masses, their associated standing as a
deity's chosen leaders, and thus, their positions
as rulers, whom most dared not question on
pain of torture, death, and hell fire. Perhaps it
is not surprising that the Galilean
transformations are the first attempt in history
to establish relativistic theory (see Galilean
Relativistic Theory).
Baez and Bunn's Approach
The apparent link between the infinitesimal
statement of the Einstein equation of Baez and
Bunn and a variation on that equation focused
on the Hubble flow that is published by Carroll
is initially considered in the context of a proof
to clearly establish the mathematical connection
between one version of the Einstein equation
described in terms of volume and another
expressed relative to “scale factor”, which, as
previously indicated, is a term closely linked to
the concepts of Hubble and spatial expansion
via an outward flow. With the connection
between the two equations scrutinized in the
early segment of this discussion via the
referenced proof to plainly establish
equivalence and to further explore the
theoretical foundations of the Einstein equation
of Baez and Bunn before we apply it, a simple
mathematical problem is next pursued in a very
crude manner via an exponential (“EXP”)
model of expansion to test the concepts related
to Baez' and Bunn's Einstein equation in the
context of that equation's ability to predict
fifteen billion years of history via an instant of
relativity.
A second model is next considered that seeks to
apply the Baez and Bunn (“B-B”) model to the
“big bang” expansion of the universe via a
numerical integration (using an amusing, “time
jump” technique). The difference between the
two approaches developed here is in use of the
“Baez and Bunn” Einstein equation only with
regard to an instant of time at the beginning of
the universe (the “big bang”) in the exponential
(“EXP”) model, but the application of the
Einstein equation of Baez and Bunn (without
the pressure term) over the entire interval of
expansion from the “big bang” to the present
via a “time jump” numerical integration.
Warning Summary
Warnings have already been issued to note that
the first part of the discussion that follows
amounts to no more than a crude proof of the
identical nature of two, superficially different
Einstein equations published on the web sites of
Baez1 and Carroll2. (As has been described,
one is simply expressed in terms of volume,
and the other in terms of scale factor.) This
may seem like a mechanical means of
beginning a discussion that attempts to be
“fun”, but it leads to a conclusion that
establishes equality and thus provides grounds
for the Baez and Bunn (“B-B”) model to later
be applied to “big bang” expansion in a
numerical integration.
The second element of the discussion is an
illustration of the application of the Einstein
equation of Baez and Bunn to the “big bang”
theory of Lemaître via the development of a
crude, bifurcated, state-space, exponential
(“EXP”) model that has already been
referenced. The Friedmann-Walker-Robertson
(“F-W-R”) model56 of cosmological expansion
is then considered to provide a basis for
comparison to the results of the immensely
crude, exponential model produced here (that
relies upon general relativity only for
acceleration data at the first instant of the “big
bang”) and the “time jump” numerical
integration results.
No new scientific insights should be expected
from any of what has been proposed. Those
with some interest in the prospect of seeing
space-time considered by means of a simple,
geometrical equation based upon no more than
4

differential and integral calculus and a little
algebra might enjoy the ride provided by
consideration of the infinitesimal Einstein
equation of Baez and Bunn1 in a “big bang”
context.
Those who truly want to have a little fun can
stop reading here and revert to the paper by
Baez and Bunn1 (referenced in this document's
bibliography and found on the website of
Doctor Baez) if they have not yet experienced
it. (Readers with well-developed tensor math
skills seeking a review of general relativity
focused on past experience with it might find
the relativity notes or text of S. Carroll or the
internet videos of L. Susskind more useful to
them if some or all of these resources have not
vanished from the internet or become too
restricted to easily access over time.)
(A Few, Crude Notes on the Einstein
Equation of Baez and Bunn for Those
Disinclined to Find Their Own Copy of the
Paper by Baez and Bunn Describing in
Detail the Version of the Einstein Equation
Used Here)
This paper was never intended to introduce the
reader to the theoretical foundations of
Einstein's Theory of Relativity. There are many
fine discussions of that topic, and the paper by
Baez and Bunn is the most relevant to this
effort to extend the ideas contained within that
introduction to Einstein's universe with a
glimpse of the “big bang”.
Einstein's basic idea is commonly associated
with the notion of simultaneity. Keep in mind
that he studied in Berne, Switzerland at a time
when trains were the principle means of
traveling over long distances, and worked there
in the patent office. The train stations at that
time, when linked to a mode of travel that took
long periods of time, were subject to reasonable
interest by passengers and those operating the
train system with regard to whether two train
station clocks could remain simultaneously
synchronized within the train system over long
distances.
For Einstein, what are commonly presented as
his youthful “thought experiments” focused
upon an effort to consider what it would be like
to attempt to catch up to and travel along with a
photon of light may well have been
unconsciously enriched by the train station
clocks of the Swiss. (The problem was
eventually solved using the technology of the
telegraph.)
Einstein's ultimate insight, perhaps not
surprising given his studies in a land well
known for watch making, was based on the
impression that the photon was the ideal
timepiece of the universe, always traveling the
same distance in the same amount of time.
Even today we use the photon as a measuring
stick and describe the “light year” as the
distance that light travels in a year.
It was this knowledge of the constancy of the
velocity of light that caused Einstein to
recognize that there must be something more at
work in the universe than classical physics was
prepared to predict. (Actually, light, over very
short distances, is not constant in speed, as
Feynman has observed in his QED: The
Strange Theory of Light and Matter, varying
above and below the so-called speed of light
along its path ever so slightly. This paper takes
the perspective that light is closely approximate
in its average speed around “c”, the so-called
“speed of light”, so the “speed of light” is
approximately a constant. In this perspective,
this paper is, perhaps, outmoded. Nevertheless,
it is consistent with how the subject is
approached.)
If light must always be perceived as traveling at
the same speed, what does that tell us about the
observer? No matter how fast an observer is
moving (below the speed of light), the velocity
of light is always the same. The observer
cannot ever catch up with the photon of light, or
5

measure any variation in the speed of light,
regardless of the observer's velocity.
From Einstein's perspective, the only
reasonable conclusion was that observers in an
accelerated reference frame cannot experience
the passage of time in the same manner as
observers in a stationary reference frame. If
that were not true, an accelerated observer
would measure the speed of a photon traveling
on the observer's path in terms of the sum or
difference of the observer's velocity and the
speed of light. This does not occur in reality.
The notion of absolute simultaneity of clocks
traveling at different velocities (and as Einstein
ultimately established, over local terrain with
different mass densities) was thus rendered an
impossibility.
We know that if an observer is traveling at
nearly the speed of light, the photon must still
appear to travel at the speed of light to that
observer. The same must be true for an
observer whose velocity is zero. That could
only occur if something had happened to the
accelerated observer to cause the passage of
time in his reference frame to shift and make
the observation of the velocity of light the same
as for a person who was not moving at all.
The near light speed observer's clock must thus
slow dramatically compared to the stationary
observer's clock to make this possible. How
could this be? If the near light speed observer's
clock were running slower than the stationary
observer's clock as perceived by the stationary
observer due to slower passage of time in the
reference frame of the near light speed observer
(see “twin paradox” in discussions of relativity
that use two twin's as biological clocks), but
both observers' clocks were operating perfectly,
the only possible conclusion would be that the
near light speed observer was truly
experiencing a different rate of passage of time
itself. The near light speed observer had
somehow fallen behind the stationary observer's
reference for the passage of time.
This presentation, although crude, is not
excessively removed from the fundamental
ideas of the theory of relativity as initially
conceived by Einstein. The concept of a four
dimensional, space-time “continuum” as a
model for our universe was asserted only after
the introduction of relativity and was conceived
by Hermann Minkowski in 1908 rather than
Einstein himself.87
(Einstein initially received
the idea of a four-dimensional space-time
continuum as an idea that amounted to so much
“excessive egg-headedness”, until he realized it
was necessary for his next step, general
relativity.)
What Einstein's theory of relativity asserts,
when expressed mathematically in the form of
the equation of Baez and Bunn, is a
straightforward link between the rate at which a
certain volume of space defining a body of
mass (and cosmological constant) is expanding
or contracting and the types and magnitudes of
energy contained within that volume of interest.
The mass density term is associated with
inducement of a contraction of volume. The
“cosmological constant” term provides for a
corresponding expansionary term, contrary to
gravity.
The pressure terms are the least consistent with
a simple, “gut feeling” interpretation of the
model. We have to embrace the concept that if
mass is moving through space-time, a pressure
is produced that has the same effect as
additional mass with regard to expansion or
contraction of a spatial volume. In short, the
pressure terms correlate to relativistic mass or a
“flux of momentum”. (Today we might be
inclined to consider a Higg's field.) We will
eventually find that the pressure terms are
proportional to the square of the velocity of a
particle with mass.
If we link all of the terms on the right of the
Baez and Bunn Einstein equation (Equation
1.0) to a specific object comprised of a
6

spherical cloud of mass (and ignore
cosmological constant), it becomes plain that
by multiplying the result of the right hand side
of the equation by the initial volume over some
infinitely small time interval, we will produce a
value for the second derivative of the volume
(the acceleration of the volume) that is directly
proportional to the sum of the mass within the
spatial volume and the pressure due to the
velocity of the volume. As the volume shrinks
in proportion to this acceleration, we would, as
observers, perceive a force at work. That force
is gravity.
Pressure Terms...in an Einstein Equation...in
the Vacuum of Space?
As has been observed, the pressure terms of the
Einstein equation of Baez and Bunn are
probably the most shocking element of the
equation for those who first encounter it. As
human beings we are familiar with life on what
we perceive to be a giant planet and the effect
of the mass of that planet in the context of
gravity. “Cosmological constant” sounds like
an odd notion, but it isn't too difficult to move
beyond the initial inclination to raise an
eyebrow and recognize that the energy of a
“cosmological constant” has the opposite effect
on gravity as the energy of the mass of the
planet with which we are so familiar. The
pressure terms aren't so easily embraced. After
all, what does pressure have to do with gravity?
We know that the pressure of the weight of the
overhanging atmosphere is the cause of the air
pressure on the surface of the earth. We know
that the weight of the water in the oceans is the
basis for the crushing pressures in the depths of
the seas. What seems odd is the prospect of
matter moving in empty space producing a
pressure. At least it seems odd when
considered solely in the context of the Einstein
equation of Baez and Bunn, because being new
to relativity, we are considering it from a
perspective focused on our prior, classical
concept of gravity.
We'll continue our crude consideration of the
Einstein equation of Baez and Bunn with regard
to the pressure terms by asking ourselves how
we might have dealt with the notion of the
constancy of the velocity of light when
observed by a party traveling at any velocity
when developing this equation. We've
recognized that the Einstein equation of Baez
and Bunn contains only terms related to mass,
cosmological constant, and pressure.
We know that the value of the mass term
associated with mass density of what we take to
be a spherical volume containing mass and
cosmological constant energy does not change
in a specific model of a massive object
according to the Einstein equation of Baez and
Bunn. We know the same is true for the
cosmological constant term. How can we
accommodate our need to maintain a constant
speed of light regardless of the velocity of an
observer if we have only constant terms for rest
mass density and cosmological constant? The
answer is simple. We can't.
We could speak in terms of “geodesics” and
“shortest paths”, and perhaps even provide
sufficient misunderstanding and apprehension
in the process to leave the reader feeling
desperate, though potentially entirely
unprepared, for a course in LaGrangian
mechanics. Instead, we'll consider the Einstein
equation.
On with the Show: The Einstein Equation of
Baez and Bunn
Baez' and Bunn's statement of Einstein's field
equation1 can be written as follows.
Evaluated solely at t = 0, with all elements of a
spherical system at rest with regard to each
other:
V t t
V
=−4π G( ρ+
1
c
2
( P x+P y+P z)).
7

Equation 1.0 – Baez' and Bunn's Version of
Einstein's Field Equation with Speed of
Light (“c”) NOT Normalized
where:
Vtt = the second derivative of the rate of change
of the volume of a sphere (of particles
representing the system of interest) in space-
time, (or the “acceleration” of the volume.)
V = the volume of the sphere (of particles
comprising the system of interest) in space-
time.
G = Newton's gravitational constant
= 6.67384 x 10-11 m 3 / (kg s 2). 21
ρ = the density of the matter in the spherical
system of interest.
Px = pressure component in the x axis direction
on the sphere's surface.
Py = pressure component in the y axis direction
Pz = pressure component in the z axis direction
c = speed of light = 300,000 km/s.
The result, as the left side of Equation 1.0
indicates, is in terms of per unit time (as a
change in a quantity “per unit time” divided by
the quantity).
Carroll (and Baez and Bunn) have published a
slightly different statement2 of the equation,
representing it alternately as:
Att
A
=
−4π G
3
( ρ+3 P ).
Equation 2.0 – A Different Presentation of
the Einstein Equation
where:
Att = the acceleration of a “scale factor”
associated with the universe (per the concept of
Hubble).
A = the scale factor associated with the
universe.
(Note: 58 A(t) = R(t) / R0 (or, here, A = R / R0),
where “R” is the radius being characterized,
and “R0” is the radius at a reference time (such
as the present age of the universe). Given this,
“A” is a unit-less or “per-unit” value expressed
as the multiple of a base value measured in
units of length, such as meters, light years, or
parsecs, and “Att” is in units of “per square
second” as a second derivative of a per-unit
value, although it too could be rendered entirely
unit-less through the imposition of a base value
of time, should it ever suit specific calculations.
The “per-unit” nature of scale factor serves to
explain the conversion necessary to compute a
volume for the universe (in cubic meters) when
we later apply the Friedmann-Walker-
Robertson (“F-W-R”) equation to evaluate the
results of a model that we will construct in a
computer program.)
G = gravitational constant.
ρ = the density of matter in the sphere
associated with the scale factor.
P = pressure in the spherical system of interest.
One fundamental difference between the two
Einstein equations (infinitesimal and Hubble
flow/scale factor based) now under
consideration is easily resolved if one
recognizes that Carroll states3 in his published
notes that setting the velocity of light equal to
one (normalization of “c”) is fundamental in the
analysis of Einstein's Equation. This is purely a
matter of units, but the effect must be
considered to produce equality between two
8

versions of the Einstein equation.
We should also note that according to a second
statement of the Einstein equation by Baez and
Bunn, the pressure term of our spherical,
Lemaître “cosmic egg” model in the first
instant of time must be multiplied by three
where the pressure in the system is the same
everywhere within the volume of interest, as it
would be (on the largest scale) for the universal
system we are considering.48 This is easily
explained based upon the concept of isotropy
and homogeneity, or a universe with properties
that are the same everywhere and in all
directions (in the context of a model of the
entire universe).
This renders fundamentally relevant the second
statement regarding the pressure terms by Baez
and Bunn in their paper presenting their version
of the Einstein equation, which asserts the need
to multiply the pressure in any system in which
the pressure is “the same everywhere” by a
factor of three. This is true because the
pressure must be “the same everywhere” within
our homogenous and isotropic model, whether
considered as a massive volume of space
comprising our modern universe, or a single,
infinitely small, geometric point particle
representing a Lemaître “cosmic egg”. Where
spatial expansion is the only basis for
increasing the size of the universe, and where
spatial expansion produces no particle motion,
there can also be no pressure variation in the
universe in our grand scale model. This renders
the basis for use of a multiple of three in the
pressure term clear with regard to the Einstein
equation of Baez and Bunn as compared to the
Einstein equation expressed in terms of “scale
factor”.
The use of a scale factor based upon radius and
applied to a sphere and the rate at which the
scale factor of the sphere changes (as the radius
at any time in the history of the universe
divided by the radius of the universe), per
Hubble, is based on the concept of the velocity
of expansion of the volume of the sphere and
the geometric link between volume and radius.
The tie between the two versions of the Einstein
equation already presented (one in terms of
volume, Equation 1.0, and the other focused
on scale factor, Equation 2.0) is less obvious
and is the focus of the first part of what follows.
Equivalence
Given:
A=
R
R0
.
At =
Rt
R0
.
At t=
Rtt
R0
.
The basis for the following relationships are
now clear:
R=R0 A .
Rt=R0 At
.
Rt t =R0 At t
.
Proof by Example
First, we'll consider a cube 100 centimeters on a
side. That cube is is 1,000,000 cubic
centimeters in volume. If we increase each side
by one percent in some time, the length of each
side is 101 centimeters, and the volume is then
1,030,301 cubic centimeters. The volume thus
increases by 3%.
As a result:
Vtt / V = 3 Rtt / R.
In our example,
9

Vtt / 1,030,301 = 3 Rtt / 101.
(1,030,301 – 1,000,000)/Δt2
/1,030,301
= 3 (101 - 100)/Δt2
/101.
30,301/1,030,301 = 3 (1)/101.
2.94% ≈ 2.97%.
For our purposes, this will suffice as proof of
equivalence.
Vtt and Rtt are calculated literally as
infinitesimals. In this computation, they are not
infinitesimals. This produces the error. (We
must wonder whether to divide the number by a
value on one end or the other of the volume. In
a true infinitesimal calculation, this would not
matter.)
Illustrating this Technique
Although spatial expansion associated with the
universe and the “big bang” theory does not
produce velocity for particles with mass, a
version of this “time jump” numerical
integration technique is later introduced and
presented in computer code in this discussion to
illustrate the numerical technique. The results
are compared to those of the Friedmann-
Walker-Robertson equation. (Use of a small
time step is no issue when one need merely
estimate the initial slope of a value before
initiating a “time jump”, and when the initial
slope does not substantially change even over
many intervals of a time step, or in a radical
manner even over the course of a well selected
interval for a “time jump”.)
We would expect the “time jump” numerical
method being described to underestimate the
reality when modeling spherical expansion with
an accelerating value of slope, which the
acceleration of the universe is said by
astronomers to possess, because the initial
estimate of the slope of acceleration of volume
in a brief interval at the beginning of any “time
jump” interval would tend to underestimate the
value of the actual slope over the majority of
each “time jump” interval. This is clearly
apparent, because the estimate of slope is
determined at the beginning of the “time jump”
interval, before acceleration relative to the
value of interest has occurred.
The results of the “time jump” technique will
clearly be affected by the accuracy of the
estimate produced at the beginning of each
“time jump”. The principle goal of the “time
jump” method just described in its use here
relative to the “big bang” and expanding space-
time is simply to illustrate the “time jump”
numerical technique and to evaluate the results
produced through application of the Einstein
equation of Baez and Bunn over the longest
conceivable time to the present without
substantial computational delay using a novel
method, to which we will now return.
Analysis: Applying the Einstein Equation of
Baez and Bunn to a Fundamental,
Cosmological Problem
At first sight Equation 1.0 may well appear to
offer only limited facility for application in the
most broadly conceivable realm of relativistic
applications, even if it is advertised by Baez
and Bunn as conveying the entire intent 1 of
Einstein's equation in terms of the relationship
between space and time in an instant. There are
a number of reasons for this.
1.Any spherical system analyzed must be
comprised of particles that have no initial
velocities relative to each other.
2.The Einstein equation presented by Baez and
Bunn is indicated by its developers to only be
valid in the first instant after time equal to zero
in a simulation (before particulate components
acquire unique velocities), which precludes any
detailed analysis of complex motion and
interactions over time due to the equation being
10

suitable only to a volume of particles within a
body in which all such particles are initially at
rest, and not in systems associated with many
particles in motion over time, which, in the
latter case, renders impossible the assurance
that all particles will not be moving at the
instant that the Einstein equation of Baez and
Bunn might be applied.
3.The Einstein equation of Baez and Bunn does
not consider such effects as gravitational
radiation. This is not particularly relevant to
consideration of the “big bang”, where
gravitational radiation is not an element of
focus (as it might be with massive objects, such
as black holes in some mutual orbit), but
gravitational radiation is a significant area of
application of the Einstein equation and general
relativity.
At this point, given the limitations that seem to
be imposed on the application of the equation,
and particularly if one has not read the related
paper by Baez and Bunn1, one might begin to
feel suspicious of the range of facility of the
referenced variation on the Einstein equation,
so it is worth considering whether this version
of the Einstein field equation can be applied
meaningfully to some small but not entirely
insignificant problem without further delay to
test the level of insight that the Einstein
equation of Baez and Bunn might have the
power to bestow. One could select from a
number of gravitational alternatives in the
context of specific systems, some of which are
discussed in a most insightful manner
elsewhere in the paper by Baez and Bunn in a
highly logical development and with impressive
results.
Rather than attempting to retrace such familiar
ground in what would inevitably prove to be an
inferior effort, a single, modest problem in the
field of gravitation and cosmology will be
considered to help to allay any apprehension
that may have arisen regarding the importance
of the Einstein equation of Baez and Bunn due
to its limitations. The problem of interest here
can be simply stated.
Can the Einstein equation of Baez and Bunn be
used to produce a valid, order of magnitude
estimate of the scale of the modern universe
approximately fifteen billion years after the
“big bang”? We shall make an effort to apply
the Einstein equation of Baez and Bunn at the
moment of the “big bang”, and hope for the
best (with some help from modern astronomy)
in estimating the resulting expansion of the
universe. Having agreed to pursue this
common goal, if we succeed, how could anyone
still be inclined to chaff when considering an
equation in a form that easily offers useful,
cosmological insight into the fundamental
origin and future history of everything in the
cosmos after billions of years of evolution?
A Simplified Analysis of the Cosmological
Evolution of the Universe Via the Einstein
Equation of Baez and Bunn
A Cosmological Equation
We should recall that Einstein's field equation is
essentially a classical equation. In fact, its
classical nature, disconnected from quantum
physics, is recognized to pose substantial
challenges for physicists working on problems
where powerful gravitational forces interact
with the world of the very small.19
Because of our need for observational guidance
in th proposed calculations, we suddenly find
ourselves requiring astronomical data and
related, published insights relevant to our
computational aspirations. What has the
universe been doing over the past fifteen billion
years (or so)?
The Universe with a Cosmological Constant
The most likely statistical range for the
cosmological constant has been considered
elsewhere.14, 15 Recently published data
11

suggests that “dark energy” (commonly
employed today as a synonym seeking to
conceptually flesh-out “cosmological
constant”) may comprise seventy percent of our
universe.16 Present day models of the universe
assert that long ago the universe “slipped” the
bonds of gravity and is now expanding
endlessly, cooling and growing less dense over
billions of years.35
As we'll see later in the Friedman-Walker-
Robertson (“F-W-R”) simulation, “slipped”
may be too subtle a word. Gravity's capacity to
restrict the expansion of the universe based
upon modern data relative to its component
energy densities seems to have always been
tenuous, a fact concealed only by observational
limitations and what may have simply been bias
in favor of a perfectly balanced, and, perhaps,
intellectually (or philosophically) satisfying,
“Einstein universe”, possessing a very unique,
steady-state future. (WMAP analysis suggests
that we may be closer to this “Einstein
universe” result in terms of the balance between
matter and cosmological constant energies than
modern headlines might be taken to suggest,
but a graph of the expansion of the universe
from the “big bang” to the present that results
from the asserted energy balance and the
Friedmann-Walker-Robertson equation seems
more determined to support consistent
expansion.)
Einstein's original basis for the cosmological
constant was founded in an attempt to produce
a non-expansive, steady-state universe in which
the gravitational impact of mass and positive
energy on space-time curvature would be
balanced by cosmological constant.40,43 His
tenuously balanced, “steady-state” (or “Einstein
universe”) model corresponds to an expansion
that slowly decelerates after a “big bang” origin
to achieve a constant rather than an ever
expanding or subsequently imploding volume.
(As mentioned previously, some have presented
this simply as a by-product of the limits of
astronomical observations at the time and not
an attempt by prominent scientists to introduce
unique characteristics to the universe occupied
by man.)
Such an “Einstein” or “steady-state” universe
would never renew itself with a “big crunch” or
suffer endless “evaporation” via re-accelerated
expansion after slipping past the steady-state,
matter density limit fueled by “cosmological
constant” roughly seven billion years ago47 in
the manner consistent with astronomers' current
(as of 199859) model for our universe. 22, 35, 50
The “Einstein universe” model is the first
approach to modeling the universe over its
entire, approximately fifteen billion year life to
(the present time) that we will consider using
the Einstein equation of Baez and Bunn (via the
first pass at developing the “EXP” model). (A
present estimate of the age of the universe is
thirteen point four billion years. It is rounded
to the nearest five billion years in the math that
is performed here. A minor adjustment to one
of the “F-W-R” model's parameters is made to
compensate for the approximation and produce
the thirteen point four billion year estimate of
volume in fifteen billion years.)
It is interesting to observe that Einstein may
have been somewhat uncomfortable with the
notion that one need merely add a cosmological
constant to his equation for general relativity to
produce an equation that would consider an
“anti-gravity” force at work in the universe.
The notion that so pervasive a force as “anti-
gravity”, particularly given present day model's
that grant cosmological constant claim to
seventy percent of the energy of the universe,
would affect only one term of the Einstein
equation expressed merely as an addend may
still empower a sense of subtle surprise or
unease.
Modern observations have produced
widespread belief in an exponentially
expanding universe35,9, which appears to have
had enough energy associated with a
cosmological constant to overcome the effect of
12

gravity, as suggested in Figure 3.0. This is, of
course, mentioned here only to establish why,
in the following development of a
computational method by which to estimate the
size of the universe roughly fifteen billion years
after the “big bang” (again, approximately
thirteen point four billion years later, for
sticklers), the decision is made at first to very
crudely approximate the universe (in the
context of pursuit of an order of magnitude
estimate of its volume) as having stopped in its
most recent, near steady-state inducing
deceleration via the “Einstein universe” model
and ignore the rapid, outward acceleration
proclaimed observationally by astronomers in
the 1990s and affirmed in the first decade of the
20th century.
Our purpose in seeking a model based upon this
initial, immensely “crude” perspective is not
cognizant denial, merely the simplification of
the mathematical modeling approach used here
in a manner that is taken to an extreme and
designed to produce a speedy, ball-park, order
of magnitude estimate of the volume of the
universe in the present day. If the order of
magnitude predicted through this course of
action is absurd, the concepts associated with
an exponential model and the Einstein equation
of Baez and Bunn should guide a quick
“refinement” or correction to improve the result
based upon astronomers' modern perspectives
regarding the history of the expansion of the
universe.
If we err in representing some constant or the
equation for some other key value, we won't
have spent our time incorporating the error into
a complex mathematical relationship designed
to represent a final, more complex model that
we find, in the end, also will not work in the
context of producing a valid, order of
magnitude estimate of the current volume of the
universe. Simply producing a more complex
looking equation does not necessarily assure
better results.
This “crude” initial modeling approach helps to
break the development into smaller pieces that
can be easily evaluated to determine if an error
exists within them. If nothing else, subsequent
improvement, if necessary, given our order of
magnitude goal, should help to demonstrate the
underlying concept of making a “crude”,
mathematical approximation then refining it
further if it is not sufficiently accurate for a
particular model's purpose.
The Einstein Equation of Baez and Bunn at
the Instant of Creation
Figure 3.0 illustrates the curve most relevant to
the mathematical modeling approach that we
have selected for our first attempt at producing
an order of magnitude estimate of the present
volume of the universe consistent with modern
astronomical theory by using an “Einstein
universe” model. If we accept that it is
reasonably valid in the context of the universe
expanding exponentially9 toward a final volume
near the end of fifteen billion years (with a five
billion year time constant) in the context of
imposition of a “steady-state” conclusion to that
expansion in our first pass at developing a
solution, then we need only follow a
straightforward mathematical approach based
upon the acceleration predicted by the equation
of Baez and Bunn at the moment of creation to
apply the exponential curve of the first half of
Figure 1.0 (or all of the curve of Figure 3.0) to
the time that followed creation (“the big bang”).
Perhaps we should evaluate the basic nature of
what astronomers have proposed before we
proceed to any mathematical model in order to
insure that we have shed potential mis-
conceptions. The “Einstein universe” (or
steady-state universe) model is not the preferred
image of the evolution of the universe
according to modern cosmologists. The
relevant picture changed abruptly in the late
nineteen nineties and what had been presented
with a great deal of uncertainty began to seem
clear with some help from the Hubble Space
13

Telescope.
The “Einstein Universe” and the “Big Bang”
as Perceived Today
Most astronomers and cosmologists today
embrace the conclusion that our universe has
too much cosmological constant to be closed.
Such an assertion is contrary to modern
astronomical observations. That means that our
“Einstein universe” model is known to us to be
wrong from the very beginning.. It is important
to realize that no argument is made here to
counter that fact. We're simply seeking a
simple mathematical model to produce an order
of magnitude estimate of the current volume of
the universe, and a steady-state universe model
produced using an exponential equation seems
to offer some potential in this regard.
Cosmologists predict a curve associated with
the size of our universe over time that may be
classified via some variation on the plot of
Figure 1.0. The “variations” are generated
using different combinations of mass and
cosmological energy in the Einstein equation.
No argument is made here to resist that
perspective, but we are seeking to apply the
equation of Baez and Bunn to produce a
reasonable estimate over the entire age of the
universe to the present day, and with that
equation limited merely to the first instant of
time (if one is intent on producing accurate
results per the limitations of the Ricci tensor, as
previously outlined), we must find another
mathematical equation to “fill in the blanks”
relative to the shape of the curve for the volume
of the universe from the “big bang” to the
present day.
(Note: The path toward acceptance of the
concept of a cosmological constant over time
by astronomers is discussed in greater detail in
an exposition on the subject of modern,
cosmological theory by an award winning
author that may be read or downloaded via the
internet.35 NASA also offers discussions of the
topic.50)
Figure 1.0 – Conceptualization of One
Perspective of the Expansion of the
Universe50
Figure 1.0 is meant to initially describe an
increase in the volume of the universe to a
magnitude that occurs over one era during a
period of slowing expansion, followed by an
endless period in which universal expansion
accelerates. What is proposed here in the
context of an “Einstein universe” as a first pass
model is to very crudely ignore the later, re-
acceleration of the expansionary phase (that
began some seven billion years ago) and simply
seek an order of magnitude estimate for the
present volume of the universe based upon a
steady-state conclusion to expansion as
reflected in the plot of Figure 3.0.
A Diversion to Consider Expansion and the
Nature of our Exponential Model's Guiding,
Astronomical Data
We might presume that analytical computations
based upon redshift serve to identify the rate of
expansion of our universe at various times in
the past from data culled from increasingly
distant sources providing ever more ancient
light and related insight into the rates of
expansion in the past, but if we accept
astronomers' claims of faster than light
expansion in the most distant regions of the
14
Volume
Time
Initial, Exponential
Increase in Size
A region of
slowed growth
Accelerating
Expansion

universe, some of the universe's newest light
and the sources that emitted it are now
permanently beyond our view in the context of
the events they are experiencing this very
moment. Light travels at only one speed, and
only the stuff of space-time can expand fast
enough to make it impossible for light to
traverse a distance over a vast region of FTL
(faster than light) space-time expansion
sufficient to prevent that light from falling
behind the rate of expansion of the space-time
in the direction of its path, thus hiding the light
from distant viewers on the opposite side of the
faster than light expanding vacuum (as
illustrated in Figure 2.0).
Astronomers have always gazed into the past.
This exposed them to light from objects that
were at shorter distances from us long ago in a
universe in which the rate of expansion was
much smaller for the most distant objects.
Using light that departed from those
astronomical objects billions of years ago in a
past in which the universe was a smaller, less
rapidly expanding place empowers modern
perceptions of depth of field.
Astronomers plot acceleration curves with
distance and look for patterns pointing to
increasing velocities of recession with time and
related distance (after they have identified a
standard source of light that will produce a set
of reference wavelengths at the distances of
interest to use to evaluate redshifts for all
sources). Ancient light (“radiation”) from
distant sources traveling through expanding
space-time also suffers a redshift both due to
the recession of the source and the expansion of
the space-time through which the light travels.
Expanding space-time significantly reduces the
energy associated with radiation subject to such
effects over cosmologically relevant time
intervals.49 This reduces the impact of radiation
on space-time and cosmological expansion over
time.
If the universe seems to be spherical, a rate of
expansion via a red shift curve over distance
(and time) consistent with faster than light
expansion for the most distant objects would
present the effect described in Figure 2.0.
(When considering astronomical data we must
always keep in mind that astronomers today use
ancient light to photograph distant objects, even
though the light that such objects may be
emitting at this very moment may never reach
the earth, stopped by its passage through too
rapidly expanding space-time.)
Figure 2.0 – Observable Limit Due to Faster
than Light Expansion at Some Distance
From Earth if FTL Expansion of the
Universe at Some Distance Is Valid (Not to
Any Scale)
Back to An Extremely Crude Assumption, our
“Einstein Universe”, a “First-Pass” Model
We have chosen to presume that Figure 3.0 is a
vaguely reasonable candidate for a model of the
expansion of the universe to the present day in
the context of the universe starting with rapid,
slowing, exponential9 expansion, and
aggressively ignoring the later initiation of a
more rapid expansion.23,35,40,43,44 We've also
15
Faster
Than
Light
Expansion
Radial Observational
Limit
Earth
Space-time
Surrounding Earth
Expanding at a
Constant Rate
at Any Instant in Time
"The
Universe"

chosen to extend the period of decelerating
initial expansion that we presume to associate
with the early universe to the present in an
“Einstein universe” model. Any exponential
equation representing such behavior has an
interesting property, as shown in Figure 3.0.
Figure 3.0 shows two things. As the plot
marked by squares, the exponential plot,
increases toward its final value of one for the
function shown (with the magnitude of 1.0
representing 100% of the final value), most of
the increase in magnitude occurs, as measured
in thirds of the time required to reach ninety-
nine percent of its final value, in the first third
of the time required to reach that final value. In
fact, in any natural phenomenon that can be
Figure 3.0 – How an Increasing Exponential
Equation Changes on the Way to a Final
Value
described by the exponential plot shown in
Figure 3.0, one would be reasonably close to
the final value if one considered only the
magnitude of the plot after the first third of the
time required to reach ninety-nine percent of
the final value. The plotted result would have
achieved roughly two-thirds (63%) of the final
value after one third of the time (one “time
constant”) required to very nearly achieve that
final value.
The graph in Figure 3.0 shows that the
exponential plot can be approximated by two
lines, which share the value of the exponential
plot at one third of the time required for the
exponential plot to reach its final value. The
lower time valued linear plot has a slope less
than the initial slope of the exponential plot, so
if we plotted a line using the initial slope of the
exponential curve, we'd overshoot our final
value (of 1.0, here) considerably. (This is why
we won't linearize our model given only data in
the first instant of the “big bang”.)
Figure 3.0 suggests that we might at least
attempt to make a crude, order of magnitude
estimate of the final volume if we were to apply
these concepts and use the initial rate of change
of a process presumed to be exponentially
varying (to which the acceleration of the
expansion of the universe could be grossly
assigned over much of its past 9) to estimate the
final value using a time change equal to one
third of the time required to reach 99% of the
final value as the time constant.
The prescribed model is admittedly imperfect
and likely to substantially underestimate the
modern scale of the universe given that
astronomers believe that the universe has
moved past its phase of slowing expansion and
shifted into a new era of accelerating growth (as
suggested by Figure 3.0). The goal here is
simply to employ the inherent simplicity of an
exponentially slowing expansion model to
advantage for an order of magnitude
calculation. We are, after all, “thinking crude”.
Next we must consider how this model will be
built?
Building a Mathematical Model for the
Expansion of the Universe Based on the
Einstein Equation of Baez and Bunn
The Einstein equation of Baez and Bunn is
capable of producing an estimate for the
acceleration of the expansion of the universe
only at the first instant of time. It has no power
to establish a time constant for an exponential
expansion over time, because the expansion of
a presumably spherical universe places particles
16
1 2 3 4 5 6 7 8 910
0
0.5
1
1.5
Exponential Plot
Column B
Column C
Time (t)
1-e^(-t/c)

within it in different reference frames
associated with their individual velocities after
the first moment of time, and we can only
consider a system using the Einstein equation
of Baez and Bunn if we are assured that there is
no initial velocity of expansion (to remain
consistent with concepts of general relativity).
We have to rely upon astronomical data for the
time constant for an exponential model.
For the order of magnitude computation in
which we hope to engage, we could conclude
that the acceleration will occur over fifteen
billion years in an exponential manner like that
shown in Figure 3.0 to produce the current
volume of the universe (knowing that we are
ignoring the later, re-acceleration phase of
expansion described by astronomers in this first
pass model). If we have some vague belief in a
universe that behaved like an exponentially
slowing process relative to expansion before the
expansion began to speed back up, and if we
aren't too particular with regard to accuracy,
that may not be entirely unrealistic, given that
we are only seeking an order of magnitude
estimate of volume. (We will characterize it
here simply as a crudely computed, minimum,
“ball park” estimate for the present size of the
universe.)
We've shown that exponentially slowing
processes moving toward steady-state (Figure
3.0) reach (roughly) two thirds of their final
value after roughly one third of the total time
required to approximate that final value. (The
one third of total time to approximate final
value interval is the time constant for the
exponential process.) For our purposes, one
third of the total time to the present size of the
universe is one third of approximately fifteen
billion years, or five billion years.
One could, at this point, assert the wisdom of
employing an exponentially slowing equation
for outward acceleration (Vtt) based on the first
moment of acceleration as:
V t t (t )=V t t i ni t (e
(−t / δ )
) .
where:
δ = time constant.
Vtt(t) = the acceleration of the volume at any
point in time.
Vttinit = the initial acceleration of volume per the
equation of Baez and Bunn.
After the first integration of this equation, the
velocity of volume equation is given by:
V t (t )=−δ V t t i n i t e
(−t /δ )
+V t 0
.
Since the velocity of expansion must be zero at
time equals zero (Vt(0) = 0), the value of Vt0 is
prescribed to render the right side of the
preceding equation zero at time equals zero:
V t (t )=−δ V t t i n i t e
(−t /δ )
+δV t t i n i t
.
A second integration of the preceding produces
the following equation for volume, which must
be evaluated between two points in time:
V (t )=δ
2
V t t i n i t e
(−t /δ )
+δ t V t t i n i t
.
Equation 3.0 – Volume for Universe, Precise
With the closed form of this integral over time
always ranging between time equal to zero and
three time constants (“3δ”, or three times five
billion years for our universe), or, equivalently,
the present, we can produce a closed form
version of the preceding equation to the present
age of the universe:
V u =δ
2
V t t i n i t e
(−3)
+3 δ
2
V t t i n i t −( δ
2
)V t t i n i t
.
The first term on the left of the right side of this
equation can be eliminated within a reasonable
approximation, since it is much smaller than the
17

middle term, and the result is (approximately):
V u ≈2 (δ
2
)V t t i ni t
.
Equation 4.0 – Approximating the Volume of
the Universe After Three Time Constants
(roughly to the present, with “δ”, the time
constant, equal to five billion years) for an
“Einstein Universe” Crude Model Based on
Acceleration at First Instant of “Big Bang”
with Time Constant Given by One Third the
Present Age of the Universe.
The approximation of Equation 4.0 will be
counted as sufficient for use here to determine
if we have finished completing our exponential
model, as the “first step” in the process. The
selection of a universe with an initial size of
one cubic meter in volume at time equal to zero
may trouble some fond of the concept of a
quantum point (or “singularity”) at the “big
bang” expanding into a massive universe, but a
smaller initial volume renders the mathematics
related to volume less straightforward, because
quantum point particles have no volume.
(We've discussed this in the context of a
geometric argument favoring zero initial
velocity of expansion for the Einstein equation
of Baez and Bunn in terms of volume if it is to
be equivalent to the Einstein equation in terms
of scale factor.)
An atomic or smaller initial volume for the
universe would increase the density of
cosmological constant in the computation, and
increase the initial acceleration of volume (as
suggested in Table 1.0). A larger initial volume
for the universe slows the expansion rate for
time equal to zero with less concentrated
cosmological constant energy. Since the goal
here is to compute the size of the universe
fifteen billion years after the “big bang”, one
cubic meter of initial size for the universe at
time equal to zero is a convenient measure for
use in Equation 4.0, and is certainly not so
large as to likely introduce a significant
temporal error over the interval of interest due
to the time associated with the expansion from
a quantum scale object to an object one cubic
meter in volume in the context of a “big bang”
origin based on the present intent of that origin
theory.
Getting Pressure Data from Density and
Velocity
In a model of the expansion of the universe due
to the expansion of space-time we don't need to
consider the velocity of mass particles.
Particles have no velocity in a universe in
which only space-time is expanding. If we
consider the motion of mass particles in a
spherical volume under the influence of a
gravitational field, we do need a means of
considering the “flux of momentum” of the
particles, which in the case of the Einstein
equation of Baez and Bunn, requires a pressure
term. That is the purpose of the development
that immediately follows, although adjunct to
our needs.
Newton assures us that:
F = ma,
Equation 5.0 – Newtons Law of Force
where:
F = force.
m = mass.
a = acceleration of the mass due to force.
To work with the Einstein equation of Baez and
Bunn we must consider the impact of the
spherical volume in which we are interested
expanding into a field of test particles with
density, “ρ”. These test particles, imagined as
the “flux of momentum” induced by velocity of
mass particles, strike the surface of our
spherical, mass volume and induce a pressure,
“P” on that surface as it expands. We begin
18

with:
A = 4πR2,
where,
A = surface area of spherical Volume
representing the mass in which we are
interested (that of the universe).
R = radius of the cross-sectional area of the
spherical volume in which we are interested
(the radius of the universe).
If we take the field of test particles striking the
surface of our sphere of particles under
gravitational influence to have a mass density,
“ρ”, and the volume that is being struck by the
moving cloud of external test particles to move
with a radially oriented velocity given by
“vradial”, the velocity of a point on the sphere's
surface expanding outward orthogonally to the
surface, then the encompassing test particle
mass that is coming into contact with the
surface area of this spherical system at any time
interval, “Δt”, is:
m = ρ 4πR2 vradial Δt,
with all variables as previously defined.
Notice that this equation has two components
that make its relevance here apparent. The
“4πR2” component is the surface area of the
sphere. The component represented by “vradial
Δt” is the distance traveled by the surface of the
sphere in the context of outward expansion
perpendicular to the surface of the sphere at
every point. It is the increase in the radius of
the sphere due to expansion in the time “Δt”.
This means that the product of the “4πR2” and
“vradial Δt” terms is the additional volume
acquired by the sphere due to expansion in the
time “Δt”, described in the MKS system in
terms of cubic meters. If this volume is
multiplied by a mass density, the result is the
mass of the test particles encountered by the
surface of the sphere, with the test particles
assigned the mass density of the sphere.
What is the acceleration given to each of these
particles?
a = Δvradial / Δt.
This makes it possible to compute the force due
to the test particles on the area of the surface of
the universe/sphere perpendicular to the
direction of motion. We know the equation for
force according to Newton in terms of mass and
acceleration:
F = ma.
We can thus simply expand the equation by
substituting known terms for mass and
acceleration.
Given that the test particle mass being
encountered by the surface of the sphere, under
the assumption that “Δt” is infinitely small, so
that there is no significant change in the density
associated with the spherical system during that
time interval, is given by:
m = ρ 4πR2vradial Δt,
with:
ρ = density of test particles (or mass particles in
spherical system).
R = radius of sphere
Vradial = radial velocity of point on surface of
sphere.
Δt = time for variation in volume of sphere.
We define acceleration as before:
a = Δvradial / Δt.
19

We can now substitute these two components
into the equation for force.
F = (ρ 4πR2 vradial Δt)(Δvradial / Δt).
The preceding equation for force simplifies to:
F = ρ 4πR2 vradial(Δvradial).
If the radius of the object at the moment of
interest is fixed, so “R” does not vary (as with a
planet), the force is purely a function of the
change in velocity, so the cumulative force is
given by:
F =∫ ρ 4π R
2
vr a d i a l d vr ad i al
F = 2ρπR2 (vradial)2 .
The surface area of the entire spherical volume
is given by:
Asphere = 4πR2..
If pressure is force divided by area, the average
pressure on one hemisphere of the sphere in the
direction of motion is:
P = F/A = 2ρπR2 vradial
2 / (2πR2).
P = (ρ vradial
2).
We could directly substitute the result obtained
for pressure into Equation 1.0 as Equation
5.5:
V tt
V
=−4π G( ρ+
3 ρ v
2
c
2
−2 Λ) .
Equation 5.5 – Einstein Equation of Baez
and Bunn with Pressure Term Associated
with Mass with Velocity Directly Inserted
Is An Expanding Universe the Same as A
Spherical Mass Expanding Due to Internal
Forces Not Related to Space-Time
Expansion?
We need to consider the model that is
represented by Equation 5.5. The development
of Equation 5.5 considered a sphere expanding
into space-time. We might think of a cartoon
model of a round ball with a fuse that has been
lit. If the bomb goes off, in an ideal fashion,
the mass elements of the device will expand
radially outward assuming the explosive is
spherically packed with constant density.
This has absolutely nothing to do with the
expansion of space-time over fifteen billion
years since the “big bang”. A bomb explodes as
the result of a (typically) chemical reaction.
Equation 5.5 and its development would NOT
apply to that situation. Any moving mass
expanding outward from a sphere that had just
exploded would acquire real velocity. No
velocity is obtained here. Space-time, in the
context of expansion of the universe, is actually
increasing in volume in between the bits and
pieces of the universe driven by cosmological
constant (or “dark energy”).
If we presume that mass does not acquire
velocity or “flux of momentum” due to
expansion of the universe because space-time
in between massive objects is the only element
that is expanding, so the mass is not generating
a “flux of momentum” in space-time due to its
own motion, how might we write an equation
for the expansion of the universe based upon
Equation 5.5 while incorporating the
cosmological constant? We might consider
Equation 5.6 to be a logical conclusion:
V t t
V
=−4π G( ρ−2 Λ).
Equation 5.6 – Einstein Equation Assuming
No Pressure Term is Required Because
Expansion of the Universe Does not Produce
Any Real Velocity of Objects in the Universe
Due the Motion of Objects Because of
Cosmological Constant's Effect Inducing
Expansion of the Fabric of Space-Time
Between Such Objects
20

Taking the Exponential Model Using a Basic
Einstein Equation (with no pressure term) to
Compute Initial Acceleration at the “Big
Bang” for a Test Drive
Accumulating Data for Computations – Not
Entirely Trivial in an Era of Rapidly
Expanding Astronomical Knowledge
We need to try out Equation 4.0 (developed for
our exponential model) and Equation 5.6 to
determine if the results that they produce when
they are employed together are credible. We
need some readily accessible data to do that.
The value of the speed of light, “c”, is
299,792,458 m / s. The gravitational constant,
“G”, is 6.674 × 10-11 m3 kg-1 s -2.
The value of the cosmological constant is not as
consistently established as that of the
gravitational constant or the speed of light in
scientific literature of the past several decades.
Astronomical observations provide a basis for
its determination.
Warning! - The Only Data of Interest With
Regard to the Mass of the Universe When
Applying an Einstein Equation are “Density”
and “Volume”
Density, not my density, or your density, but the
density of the universe, is one of the two most
critical pieces of information required for any
Einstein universe model. The other information
that must be relevant to determining the mass of
the universe is its volume. One is unwise to
assume that the “mass of the universe” posted
on what appears to be a relevant internet web
site in a respectable entry is necessarily always
suitable for use with a specific volume of the
universe.
As astronomers peer ever further into space, the
known volume of the universe changes. If the
volume of the universe changes, and the typical
density estimate remains the same (as one
might expect), then the total mass of the
universe must expand, or Einstein's equation
won't work. The reason is straightforward. If
the total mass data is not correct, the total
cosmological constant data won't be right as a
multiple of that mass, and if both are based
upon a volume that understates the latest
estimates for the latest “volume of the
universe” (or “radius of the universe”), the
results achieved using any version of the
Einstein equation will be wrong. It's as simple
as that.
Some Mass Data
At least one internet site proclaims the total
mass of the Universe to be 1 x 1053 kilograms.32
One should not rely upon such an estimate in
terms merely of kilograms with no
corresponding volume data for the universe.
Mass data is intrinsically based upon an
estimate of the size of the universe. Neither
may be up-to-date, and may not correlate. One
must correlate the volume of the universe, the
mass density of the universe, and the mass of
the universe before applying this information in
any combination.
Recent (Relative to This Writing), but Not
Absolute Data Regarding the Mass and
Volume of the Universe
One should take all of the following data with a
grain of salt, and confirm the mass density and
volume of the universe information that you
employ before proceeding to apply it to an
Einstein equation, or you may find that you've
gotten the proverbial cart before the horse, and
your results may be disappointing. Data on the
volume or radius of the universe tends to
change with time. In the past decade and a half,
it has changed quite rapidly.
The density of the universe is given at a
university web site (from the U.S.) as 3 x 10-30
g / cm3.80 This is 3 x 10-27 kg/m3. Should we
trust this value? We can find an estimate for
the total energy density of the universe on a
21

NASA web site, and, recognizing that the “total
energy density” data is based on a summation
of both mass energy and cosmological constant
energy, without regard to sign in the Einstein
equation, conclude that somewhat less than one
third of the WMAP estimate84 (as supported by
the pie chart at the cited source) is fairly close
to the data just cited from the university web
site. We now need a credible estimate of the
volume of the universe, and we can compute
the mass of the universe directly.
We might have encountered a 2004 news story
focused on data derived from the Wilkinson
Microwave Anisotropy Probe (WMAP) and
data it gathered from the cosmic microwave
background radiation. The result proclaims the
radius of the universe to be seventy-eight
billion light years.81 That corresponds to a
volume of 1.69 x 1081 cubic meters.
If we were in a hurry, we might simply multiply
this seemingly modern volume estimate by the
density estimate and compute a mass for the
universe. For better or worse, this is not the
eighteen hundreds, and astronomy has been
moving much faster than it did in that era, so
we should probably strive to find some
information that is a little more recent.
If we were professional astronomers, we might
have seen something in the monthly notices of
the Royal Astronomical Society from 201182
that would have given us some rapid insight
into the matter of the most recent estimate of
the volume of the universe in a paper from
2010, which also is based upon WMAP data.83
The 2010 estimate asserts the radius of the
universe to be 27.9 gigaparsecs in a flat space-
time model. This correlates to a volume for the
universe of 2.67 x 1081 cubic meters. (That's a
change of approximately 1 x 1081 cubic meters,
corresponding to more than half of the 2004
estimate, only six years later.)
We'll use the larger estimate, since that seems to
be the prevalent direction in which estimates of
the volume of the universe tend to be going.
With the 2010 estimate of the volume of the
universe, the mass of the universe, based on the
density data previously cited, is roughly 8 x
1054 kilograms. This is the estimate that we'll
use in this discussion for the mass of the
universe. Don't use it in any future analysis
until you've researched the latest estimate for
the density of the universe and its volume (or
radius) and performed the simple calculation to
produce a result.
Cosmological Constant and Mass Based on
the Most Recent Ratio
If the most recent astronomical data is taken to
establish that mass is thirty percent of the
universe and cosmological constant is seventy
percent of the universe (with radiation taken to
have no relevant presence compared to these
other two quantities effect), then the
cosmological constant mass density would be
two and one third times the magnitude of the
normal mass density, or, based on the number
just given for the mass of the universe,
cosmological constant would have an
equivalent mass of 18.67 x 1054 kilograms.
(As recently discussed, we could have easily
arrived at the wrong conclusion regarding the
magnitude of the mass and cosmological
constant of the universe, because it is not
difficult to find estimates of the volume of the
universe on the internet that include 3.38 x 1080
cubic meters 28 and 1.2 x 1079 cubic meters29,
suggesting one source that is out of date and
another based only on the distance traveled by
light since the estimated beginning of the
universe, ignoring the expansion of space-time
as required for consistency with an Einstein
equation. Note that it is also not difficult to
encounter misleading data regarding the mass
density of the universe in the context of a 3x10-
28 kilogram per cubic meter estimate that was
inconsistent with the WMAP published result
by a factor of ten.32)
22

The initial volume of the universe at the first
instant of the “big bang” will be estimated to be
an easily scalable one cubic meter (which is
purely for computational convenience and is
not meant to suggest that our theoretical
assumption of a “cosmic egg” is incorrect).
This one cubic meter initial volume exists at the
“big bang” and will, of course, be associated
with the spherical model we have developed.
With the volume of the modern universe
crudely approximated to be 2.67 x 1081 cubic
meters, the density of the universe is then given
by:
ρ=8 x10
54
k g/2.67 x10
81
c ubi c me t e r s.
ρ=3 x10
−27
k g pe r c ubi c me t e r .
If the current matter and cosmological constant
densities (30% and 70%, respectively) of our
universe are estimated here, for an order of
magnitude, crude approximation, to have
originally been within a spherical region of one
cubic meter volume at the “big bang”, the
original density of the positive mass in the
universe and the original density of the
cosmological constant at the “big bang” would
have been:
ρ=3 x10−27
Λ=7 x10
−27
These results are consistent with a “seventy-
thirty” ratio of mass to cosmological constant.
The Result
The resulting rate of initial expansion
acceleration, based upon Equation 5.6, is:
V t t
V
=−4π G( ρ−2 Λ).
where, at the “big bang”:
V = 1 cubic meter.
G is 6.674 × 10-11 m3 kg-1 s -2.
ρ = 8 x 1054 kg/m3 .
Λ = 7ρ/3 = 7/3(8 x 1054 kg/m3).
Λ = 17.66 x 1054 kg/m3.
The acceleration of the volume at the “big
bang” is then given by:
V t t
V
=−4π G( ρ−2 Λ).
V t t=−( 4π )6.674×10
−11
x(8 x10
54
−(2)17.66 x 10
54
)V .
V t t=2.29 x10
46
cubic meters per square
second.
Assuming a 15 billion year present age for the
universe, with one time constant, “δ”, given by
five billion years (or 5,000,000,000 years x 365
¼ days/year x 24 hours/day x 60 minutes/hour
x 60 seconds/minute = 157.788 x 1015 seconds),
the current size of the universe would be on the
order of 1.7 x 1079 cubic meters, produced via
Equation 4.0:
V u≈2 (δ
2
)V t t i ni t
m3.
V u ≈2(157.788 x 10
15
)
2
2.29 x10
46 m3.
2 x 2.29 x 1046 m3 /s2 x (157.788 x 1015 s)2 =
1.1 x 1081 m3.
This result is somewhat smaller than the
approximately 2.67 x 1081 cubic meter volume
estimate for our current universe produced by
astronomers. Still, this conclusion is not bad
for a crude estimate based only upon
acceleration during the very first moment of
creation computed via the Einstein equation of
23

Baez and Bunn, and, though low, is within an
order of magnitude of current, published
estimates.
We might question whether our one cubic meter
initial size for the universe produced a lucky
result. Of course, the initial size of the universe
affects the initial density of the mass and
cosmological constant. As has already been
discussed, if we make the size of the universe at
the first instant smaller, we compress the
cosmological constant energy density further.
This accelerates the expansion, a factor
countered by the accelerated expansion
affecting a smaller initial volume, which can be
understood via Equation 4.0.
Table 1.0 shows the effect of different initial
sizes of the universe in the context of the
predicted current volume of our universe for
various initial sizes of the universe. Use of
various initial sizes for the universe at the “big
bang” in Table 1.0 permits us to examine our
“lucky guess” concern regarding the initial
volume that we chose of one cubic meter.
Table 1.0 – Variation in Size of Predicted
Universe After Fifteen Billion Years Based
Upon Initial Size of Universe at Big Bang
Using Equation 4.0 and Equation 5.6
The “lucky guess” hypothesis relative to the use
of one cubic meter as the initial size of universe
at the “big bang” in the initial computation
resented here is established to not be valid
based on the results of Table 1.0. The one
cubic meter scale was simply appealing because
it forced Equation 4.0 to produce results in
terms of cubic meters per second squared,
which made it a computationally convenient
choice. It has been shown in Table 1.0 to not
be necessary to select a one cubic meter initial
volume for the universe at the “big bang” given
the same equations employed in order to
produce the same mathematical outcome.
The only conversion that was necessary to
produce Table 1.0 was multiplication of the
mass density and cosmological constant density
used for one cubic meter at the “big bang” by
the multiplier produced by one cubic meter
divided by the whole or fractional cubic meter
volume indicated in the table. The result
produces more or less concentrated levels of
cosmological constant, which correctly
modifies the initial acceleration of the
expansion to produce the same, final volume
for any initial volume choice.
There is no change in the results of Table 1.0
for different initial volumes at the “big bang”,
as long as the initial matter and cosmological
constant densities are suitably adjusted for the
new, initial containment volume. Of course, we
haven't justified the level of cosmological
constant used beyond the assumption that what
is present in the universe today was present at
the “big bang”, which isn't a perspective that is
easily defended on its own.
“Zero point energy” is the lowest energy state
of the vacuum and thus must demonstrate an
increase in its net value in the universe with
space-time volume expansion. As a result, net
cosmological constant energy should increase
with the expansion of the universe. We've
modeled the mass and cosmological constant
energy that is present today as being present at
the “big bang”. The same is true for the
Friedmann-Walker-Robertson equation based
estimate that comes later.
A “Good” Result?
24
Initial Modern
Initial Expansion Universe
Size Acceleration rho Lambda Pred. Size
(m^3) (per sec^2) (kg/m^3) (kg/m^3) (m^3)
0.01 2.29E+048 8.00E+056 1.76E+057 1.1E+081
0.1 2.29E+047 8.00E+055 1.76E+056 1.1E+081
1 2.29E+046 8.00E+054 1.76E+055 1.1E+081
10 2.29E+045 8.00E+053 1.76E+054 1.1E+081
100 2.29E+044 8.00E+052 1.76E+053 1.1E+081
1000 2.29E+043 8.00E+051 1.76E+052 1.1E+081

The most straightforward (though erroneous)
conceptualization of the “big bang” (as some
form of explosion that distributed all of matter
and energy through space and time by means of
its physical force) may support the belief that
expanding space-time isn't needed to produce a
universe that approximates the estimate that has
been used for the volume of the universe of
1080 cubic meters. Astronomers correct that
erroneous viewpoint by asserting that
propulsion of matter by a physical explosion is
not a valid paradigm for the action of the “big
bang” and does not explain observed FTL
data35 for the most distant observable objects'
recessional velocities from us.
We are told that all that we require to explain
the expansion of the universe is an environment
in which cosmological constant energy
produced by the “big bang” can generate an
expanding volume of space-time that
establishes a universe that is big enough to
conform to astronomical observations through
the expansion of the space-time in which matter
is trapped as space-time expands to increase the
separation between stars and galaxies. (Any
particles traveling at light speed could traverse
1.4 x 1026 meters in fifteen billion years. If we
use this value as the radius of a sphere, the
universe would have a maximum, possible
volume of 1.2 x 1079 cubic meters based upon
light speed limited expansion, which explains
the basis for one of the estimates of the volume
of the universe previously cited.)
A few points associated with our results seem
hard to deny. The “Einstein universe” model
we employed produces a result that is roughly
one order of magnitude less than the correct
value for the volume of our universe after
fifteen billion years. The time required to
produce a one cubic meter volume universe
from a quantum scale universe in “big bang”
models is not a significant fraction of fifteen
billion years (so we were not wrong to leave
out the time of the related expansion phase to
one cubic meter from a "cosmic egg" given the
very little time required for the universe to
expand to one cubic meter according to
common, cosmological theories relative to the
fifteen billion year age of our universe and our
own estimated value for the initial rate of
expansion of the universe via the Einstein
equation of Baez and Bunn).
The results achieved here, given the crude
nature of the mathematical approach employed
to produce them, although low, as one would
expect for an expansion model that produces a
steady-state outcome, seem surprisingly close
to modern, scientific estimates of the volume of
the universe relative to the “order of
magnitude” nature of the exercise being
attempted with an exponential approximation to
an “Einstein universe” model and the
acceleration of volume computed only at the
first instant in time. In short, the result
generated is not horrible in that it approaches
the right order of magnitude for the volume of
the universe of the present day in the right
amount of time from a relatively small volume
of space-time, but not so “good” in its too crude
conceptualization and low, ball-park estimate,
combined with the steady-state, rather than
expanding nature of the result.
Some may find the one meter initial scale of the
universe to be particularly troubling. To delve
down to the Planck scale “big bang” universe,
we'd need to understand how the universe was
changing, and how those changes might
influence the physical model that we would
employ. That is well beyond the wildest
dreams of this discussion, which is only meant
to demonstrate how the Einstein equation of
Baez and Bunn might be used to approximate
the expansion of the universe from the time of
the “big bang” to the present. As long as we
accept that the time required for the universe to
expand from a "cosmic egg" to one cubic meter
was not significant relative to fifteen billion
years, the approach is reasonable.
The results of Table 1.0 are consistent with an
25

acceleration rate that slowly drops to zero after
fifteen billion years in the “Einstein universe”
model, relative to a real universe in which
acceleration of volume is believed to have
begun to increase some seven billion years47
ago. We are thus now in a position to question
the relevance of an extremely “crude” model
that assumes exponential deceleration of
expansion to the present. We should reconsider
our initial model but seek to modify it in a
straightforward attempt at improvement in a
“second pass” at the problem.
The Second Pass - State-Space Equation
Model
We can expand the model of Equation 5.0 to
allow for re-acceleration of expansion after
seven and a half billion years using Equation
6.0. With the time interval being considered
lasting fifteen billion years, it is a small matter
to create a temporally bifurcated model in
which time is divided into two intervals, each
lasting seven and a half billion years, in which
Equation 4.0 dominates over the first seven
and a half billion years, and Equation 6.0, with
re-acceleration, dominates over the second
seven and a half billion years. The time
constant of either model becomes two and half
billion years.
V u ≈2 (δ
2
)V t t i ni t e
3 .
Equation 6.0 – Exponential Model's Re-
acceleration Equation for the Second Half of
the Expansion of Our Universe (from Seven
and a Half Billion Years to the Present)
Equation 4.0 and Equation 6.0 represent the
volumes produced by expansion rates in the
universe over two different eras. Since we have
crudely modeled the initial expansion to have
lasted 7.5 billion years, and the re-accelerating
expansionary era to have extended over the
same, subsequent time interval, we can present
the relevant time frames for the two equations
modified to add data points based on the known
behavior of the exponential models. (The data
points of the model correlate with equations
described below.)
With “δ” crudely taken to be equal to 2.5 billion
years for our universe in both the initial
expansion and second, re-acceleration phase of
expansion, and with the assumption that for an
exponential equation approaching a final value
(during the initial expansion), the magnitude is
sixty-three percent of the final value after one
time constant, while for an exponential
equation increasing exponentially to infinity,
the value after two time constants is sixty-three
percent of the value after three time constants,
so we can compute a series of data points for a
plot of the volume of the universe from the “big
bang” to the present.
For t = 2.5 billion years (one time constant after
the “big bang”):
V u≈(0.63 )2(δ2
)V t t i n i t=1.26( δ2
)V t t i ni t
For t = 7.5 billion years:
V u≈2 (δ
2
)V t t i ni t
Now we compute values after the second
exponential:
For t = 12.5 billion years:
Vu≈0.63( 2(δ
2
)Vt tin it )e
3
=1.26(δ
2
Vt ti nit )e
3
(Equal to the rough 63% of final value after two
time constants, plus the final value from the
first equation.)
For t = 15 billion years:
V u ≈( 2(δ
2
)V t t i ni t )e
3 .
(Equal to final value, plus the final value from
the first equation.)
The initial value of the rate of expansion of the
universe at the “big bang”, represented as
“Vttinit”, is given by Equation 5.6, rewritten as
26

follows:
Vt tinit=−4π G( ρ−2 Λ)Vinit
,
where:
Vttinit = initial acceleration rate of volume of the
universe.
Vinit = initial volume assumed for the universe
at the time of the “big bang” (per volume data
in preceding tables).
All other variables are as previously defined.
The preceding equations neatly divide the
universe into two time periods (a bifurcated,
state-space model) of seven and a half billion
years each. The first is characterized as a
decelerating, exponential expansion that ends at
approximately seven and a half billion years (in
this model) with the universe at the predicted
volume. The second is a re-accelerating
expansion to the present, fifteen billion year age
of the universe, which leads to the
approximation of the present volume of the
universe of Table 2.0.
We summarize the expansion values over time
and plot the resulting expansion of the universe
using the 2.67 x 1081 cubic meter estimate of
the current size of the universe to compute
matter and cosmological constant densities at
the “big bang” as shown in Table 2.0 and
Figure 4.0 using a simple, exponential (“EXP”)
model:
Table 2.0 – Results of Attempt to Produce a
Better Estimate of Expansion Using
Equation 4.0 and Equation 6.0 (Data is
Plotted in Figure 4.0.)
Note that the results of Table 1.0, for an
Einstein universe with a five billion year time
constant, comes numerically closest (in terms
of the exponential model results of Table 1.0
and Table 2.0) to the estimate of the current
volume of the universe used here based on
comparison of the final values after fifteen
billion years of the pure Einstein universe
exponential model and the temporally
bifurcated exponential model with a two and
half billion year time constant.
Figure 4.0 – Result of Attempt to Improve
the Approximation of the Expansion of the
Universe Using the Exponential Assumption
Over Two Time Intervals.
The order of magnitude of the predicted volume
of the universe at present in Table 2.0 is 5.7 x
1081 cubic meters via the bifurcated “EXP”
(exponential) model, which is on the same
order of magnitude as astronomer's estimate of
2.7 x 1081 cubic meters 28 and is certainly
reasonable as an order of magnitude
approximation. The shape of the curve in
Figure 4.0, controlled by the dual interval,
state-space equations (one a decelerating
exponential to seven and a half billion years,
the other an accelerating exponential from
seven and a half billion years to fifteen billion
years) and the exponential assumptions that
guided their development present a logical and
27
0.00E+000 1.00E+010 2.00E+010
0.0E+000
1.0E+081
2.0E+081
3.0E+081
4.0E+081
5.0E+081
6.0E+081
7.0E+081
"EXP" Model of Expansion of the Universe
Volume (m^3)
Time, Years
Volume,CubicMeters
Time (Yrs.) Volume (m^3)
0.00E+000 1.0E+000
2.50E+009 1.8E+080
7.50E+009 2.9E+080
1.25E+010 3.6E+081
1.50E+010 5.7E+081

relatively smooth variation in volume.
Cosmological Fuel and the Initial,
Decelerating Expansion
If we take cosmological constant to be literally
proportional to the volume of space-time, then
its power to drive expansion vanishes with it in
the vanishingly small volume of our current
universe's space-time at the “big bang”.
Equation 4.0 will not produce an adequate
“bang” without cosmological fuel. We assume
some source of energy existed to produce the
“bang”.
Use of substantially less than the value of
cosmological constant energy employed in our
computation of the initial acceleration of the
volume of the universe is not reasonable based
on our attempt to reproduce astronomers'
estimates of the current volume of the universe
in the context of a decelerating, exponential
expansion over roughly the first seven and a
half billion years followed by a re-accelerating
expansion over the subsequent seven and a half
billion years.
Inflaton Theories and Colliding Membranes
It is likely that most astronomers inclined to
take experimental data seriously, even where
some level of uncertainty may be inherent due
to the use of extremely distant objects to obtain
the data, would see in the “big bang” an
opportunity to seek to explain the expansion of
space-time via modern theories of quantum
physics. (This dates to Lemaître's “cosmic
egg”.44)
That goal might be motivated by the simple
question of how a “big bang” could drive
expansion to produce a universe with its
present, estimated volume and still retain the
details produced through astronomical studies,
such as cosmic background radiation with its
very limited variation in temperature given low
levels of “zero point energy” measured by
scientists and our “flat” space-time
environment. Such theories support
cosmological perspectives that do not rely upon
simply stuffing all of the energy of a
cosmological constant after fifteen billion years
into a space vastly smaller than that of the
modern universe and computing the expansion
that the Einstein equation we derived from that
of Baez and Bunn predicts using an exponential
modeling assumption.
Advanced, modern cosmological theories may
invoke an “inflaton” field that produces
immense expansion very rapidly in a manner
not consistent with an exponential model for
expansion with a two and a half billion year
time constant. Subsequent expansion to the
present is on a much smaller scale. Past
concepts in this regard have included
invocation of a “super-heated Higg's field”6 as
the basis for an “inflaton” field. Add to these
insights offered by string theorists regarding
colliding membranes as the origin of the “bang”
in the “big bang”, and we may begin to wonder
regarding both the nature of the forces driving
the expansion initially and the scale of the
initial expansion.
Simple application of “cosmological constant”
within the Einstein equation in the context of a
“big bang” and a rapid, exponential
expansion20,30,9 to the present may seem less
insightful upon encountering concepts such as
the “inflaton” field and colliding membrane
theories42 if the simple assertion of a correlation
between cosmological constant levels and the
volume of space-time in the universe is not
sufficiently daunting when considering a
quantum scale singularity as the origin of the
universe.
The extent of the variation in modern “big
bang” theories from a quantum scale singularity
out of which all of matter, energy, and space-
time develop41 (as envisioned by Lemaître44) to
“banging” membranes that occasionally collide
and create great bursts of energy amid massive
28

Instant of Relativity Revised Copyright 2012 to 2016 by Del John Ventruella

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