Cosmological Principle
--------------------
All viable models of the universe start out by asserting the
'cosmological principle'. This posits that the universe is homogeneous
and isotropic everywhere at the same time.
'Homogeneous' means that the prospect on the universe from any
point within it is essentially the same as that from any other point.
We can not distinguish any one point from any other by the scenery
presented by them.
This concept is called the Copernicus principle because he removed
the Earth from its privileged site at the center of the universe and
reduced it to a mere one of the planets in orbit about the Sun.
'Isotropic' means the view is essentially the same in all
directions from any point.We can not single out any one direction
from any other by the view presented by them. Many astronomers treat
isotropy as the fundamental premise with homogeneity being a corollary
of it. They argue that if a universe is isotropic from all points it
must necessarily be homogeneous.
This is false. Two points can have an isotropic prospect of the
universe, but a different overall scene. And two points may have a
homogeneous view, but not isotropic ones. We could else select the
point with a such-&-such prospect as special over the other.
Homogeneous and isotropic are separate concepts.
There is yet no positive proof of the cosmological principle. We
do apply it in our studies and so far have not encountered any serious
violation.
Asserting the cosmological principle works immense simplification
in the maths involved in the models. On the other hand as yet there
is no agreed mathematical description of a universe lacking isotropy
and homogeneity. In this treatise we take the cosmological principle
as a given premise.
Newton versus Einstein Physics
----------------------------
Modern cosmology derives from Einstein physics. The description of
gravity and matter in that discipline lead directly to the many extant
models of the universe. Being that fluency in Einstein physics is rare
among home astronomers, treatments of cosmology for them are often
qualitative, superficial, or even juvenile. But it is feasible to
develop a competent understanding of cosmology banked off of Newton
physics, already embedded in home astronomers. This was elaborated by
Milne & McCrea about 1935.
On the grand scale of the cosmos only gravity governs the behavior
of the universe. Any electric and magnetic forces cancel out, being
that there are two opposite charges for each. Electric fields have
positive and negative charges; magnetic fields, north and south poles.
Within a cosmic-size volume, these tend to be of equal numbers,
netting their force to zero.
Gravity has only one kind of charge, which we call 'matter' or
'mass'. This accumulates ever more so as we embrace larger and larger
quarters of the universe. The net force is always attractive and grows
to arbitrarily huge strength with increasing enclosing volume.
In a universe operating under the cosmological principle any
spherical volume within it, tho arbitrarily huge, does in deed obey
Newton physics. The cosmological principle allows any such sphere to
be comprehended and therefore the actual size doesn't matter. We can
interpret the 'radius' of the sphere as a dimension line scribed
inside the sphere and drop all concerns about the true radius and even
the sphere's surface.
The Lambda Force
--------------
Cosmology derived from Einstein physics incorporates the lambda
force and the pressure of matter. Both are absent from a cosmology
developed from Newton physics. While both features are today mostly
omitted them from Einstein-based cosmologies, they are under study for
possible cause of 'dark energy'. We have no good idea what dark energy
is, but ir seems to distort the classical all-gravity models of the
universe. .
The lambda force is a historical accident. In Einstein's early
days galaxies were believed to be nebulae within the Milky Way. Their
disposition thruout space beyond the Milky Way was not yet realized.
Even after the nature of galaxies was known, the objects were presumed
to sit still in deep space Way and the universe was thought to be
quite static.
Einstein's initial forays into cosmology leaded to a collapsing or
infalling of space, which did not match the then-known behavior of the
universe. He introduced a new force LAMBDA that preserved a static
space.
After the Hubble expansion redshift of galaxies was confirmed
Einstein revoked the lambda force -- the universe actually does
expand. The lambda force was since then called Einstein's greatest
blunder in science. It is not sure if Einstein personally described it
as such. . We may today look at LAMBDA as a historical episode.
The pressure raised up by matter is a feature of Einstein, but not
Newton, physics. Tho demonstrably valid, in any accumulation of mass
in the universe it is vanishingly small compared to the gravity of
this same mass. So it is routinely neglected by most astronomers.
The Acceleration of the Universe
------------------------------
If we examine a spherical volume of space, with interior mass m
and radius r, the gravity field strength, or acceleration, at the
surface is
2der(r,t) = -gamma * m / (r^2)
= -gamma * 4 * pi * r^3 * rho / (3 * r^2)
= -4 * pi * gamma * rho * r / 3
+-------------------------------------------+
| NEWTON PHYSICS OF THE MODEL SPHERE |
| |
| 2der(r,t) = -4 * pi * gamma * rho * r / 3 |
+-------------------------------------------+
gamma is the Newton constant 6.672E-11 n.m^2/kg^2. pi is
3.1415... . rho is the density of the mass within the sphere.
Only the mass within the sphere contributes to the acceleration at
the surface. The acceleration due to the exterior mass nets to zero.
The sphere accelerates as a unit such that the surface comoves with
the mass particles. There is no motion of matter across or thru the
surface and the total mass within the sphere remains constant.
To avoid having to deal with specific sizes of sphere the actual
radius r is expressed in terms of a scalefactor R. R can be regarded
as a dimension mark inscribed in the space within the sphere and which
partakes in the changes of the actual radius. The r itself may be
unknown and even really unknowable.
The scalefactor in the present epoch R0 is set to unity, altho on
occasion we do explicitly write it out; R0 = 1.
r = r0 * R / R0
= r0 * R
rho = rho0 * (R0 / R)^3
= rho0 / (R^3)
Then
2der(r,t) = -4 * pi * gamma * rho * r / 3
= -4 * pi * gamma * rho * r0 * R / 3
= r0 * (-4 * pi * gamma * rho * R / 3)
= r0 * 2der(R,t)
2der(R,t) = -4 * pi * gamma * rho * R / 3
Next convert the rho
2der(R,t) = -4 * pi * gamma * (rho0 / R^3) * R / 3
= -4 * pi * gamma * rho0 / (3 * R^2)
+-----------------------------------------------+
| ACCELERATION EQUATION - |
| |
| 2der(R,t) = -4 * pi * gamma * rho * R / 3 |
| = -4 * pi * gamma * rho0 / (3 * R^2) |
+------------------------------------------------+
The acceleration can be expressed two ways, with rho or rho0.
Both are equivalent and interchangeable. Some treatments of cosmology
miss out the nice distinction between rho0 (the constant) and rho (the
time variable). This can cause rank confusion up and down the track.
Either equation is used as better suits the task at hand.
With the use of R in the stead of r we can identify the model
sphere with the universe at large and R is the scalefactor of the
universe. Some treatments say that R is the 'radius' of the universe
as tho the universe has a 'surface' and a 'center'. It in fact has no
surface and center in any usual sense of these words.
Energy of the Universe
--------------------
Integrate the acceleration equation. Use the form with rho0 to
avoid the extra function in time, rho = rho(t).
2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2)
1der(R,t) * 2der(R,t) = (-4 * pi * gamma * rho0
/ (3 * R^2)) * 1der(R,t)
= (-4*pi*gamma*rho0/3)*(R^-2)*1der(R,t)
intl(1der(R,t) * 2der(R,t),t) = (-4 * pi * gamma * rho0 / 3)
* intl((R^-2) * 1der(R,t),t)
(1 / 2) * 1der(R,t)^2 = (-4 * pi * gamma * rho0 / 3) * -R^ - 1 + C
= 4 * pi * gamma * rho0 / (3 * R) + C
C [large-c] is the integration constant. For reasons soon evident
this is cast as C = -k*c^2, c [small-c] being the speed of light. The
important point at the moment is that k is, too, a constant, so the
entire -k*c^2 is itself merely a constant that replaces the generic
integration constant C.
+------------------------------------------------------------------+
| ENERGY EQUATION |
| |
| (1/2) * 1der(R,t)^2 = 4 * pi * gamma * rho0 / (3 * R) - (k *c^2) |
+---------------------------------------------------------------- +
This is the energy equation for the universe. (1/2)*1der(R,t)^2 is
the kinetic energy; 4*pi*gamma*rho0/(3*R) is the potential energy. The
-k*c^2 is the sum of these two, the total energy of the universe. All
of these are normalized for unit mass, m = 1kg. It's easier to see
that -k*c^2 is the sum of the kinetic and potential energies by
rearranging the energy equation thusly
-k * c^2 = (1/2) * 1der(R,t)^2 - 4* p i* g amma * rho0 / (3 * R)
which resembles the familiar expression from ordinary mechanics
(total energy) = (1/2) * V^2 - (gamma * M / R)
Friedmann Model
-------------
The acceleration and the energy equations are one particular
solution of the general equations of Einstein physics that describe
the geometry and dynamics of the universe. They were derived first by
Friedmann in 1922 and are called the Friedmann equations of cosmology.
The cosmology based on them is the Friedmann cosmology, which we
explore in depth here.
There are slates of other, mathematicly proper, solutions not
discussed here. They, tho of intense historical importance, are
generally not believed to represent the observable universe.
+---------------------------------------------------------------+
| FRIEDMANN EQUATIONS OF COSMOLOGY |
| |
| 2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2) |
| |
| 1der(R,t)^2 / 2 = 4 * pi * gamma * rho0 / (3 * R) -( k * c^2) |
+-------------------------------------------- -+
-k*c^2 looks like m*c^2,which would gives k the units of mass.
Being that the energy equation is normalized for unit mass, k can
have only the dimensionless integer values +1, 0, or -1, by the nature
of a normalized parameter.
There are three scenarios for the Friedmann universe. For k = -1,
-k*c^2 = c^2 and the universe has net positive energy. Such a universe
is characterized by an 'open' and 'hyperbolic' geometry.
When k = 0, -k*c^2 = 0, indicating an 'open' and 'flat'
geometry.
Finally, k = +1 makes -k*c^2 = -c^2 for a negative energy. This
universe has a 'closed' and 'spherical' geometry.
There is no theory to choose among the three possibilities; all
are valid cases of Friedmann cosmology. We here study the case of k =
0 from mathematical simplicity and emotional nicety.
The Standard or Classical Model
-----------------------------
The Friedmann equations with k = 0 constitute the basis of the
'standard' or 'classical'[!] model of the universe. This model is also
called the Einstein-deSitter model after the two workers who presented
it in 1932. Keeping in mind that this model is only one of many
possible models, treatments of cosmology typicly are founded on this
standard model. This mainly is because it is mathematicly one of the
simpler ones that seem to agree with observed behavior of the
universe.
+---------------------------------------------------+
| EQUATIONS OF THE STANDARD MODEL |
| |
| 2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2) |
| |
| 1der(R,t)^2 / 2 = 4 *pi * gamma * rho0 / (3 * R) |
+---------------------------------------------------+
Hubble Parameter
--------------
In the present epoch 1der(R,t) is positive, toward increasing R;
the universe is seen to expand. The rate of this expansion as an
increment of the instant R is 1der(R,t)/R and is exactly the quantity
measured by Hubble when he discovered the expansion phaenomenon. This
1der(R,t)/R is the very Hubble parameter H. Its value now, H0, is
uncertain and is the subject of repeated assessment. Nowadays H0
ranges from 50 to 75 km/Mpc.s or 15 to 23 km/Mly.s. These are the
astronomer's units and they merely state that the expansion rate is so
many km/Mpc or km/Mly every second.
H = 1der(R,t) / R
= ((8 * pi * gamma * rho0 / (3*R)) / R)^(1/2)
= (8*pi*gamma*rho0/(3*R^2))^(1/2)
H0 = (8 * pi * gamma * rho0 / (3 * R0^2))^(1/2)
= (8 * pi * gamma * rho0 / 3)^(1/2)
+-----------------------------------------------+
| EXPANSION OR HUBBLE PARAMETER |
| |
| H = 1der(R,t) / R |
| = (8 * pi * gamma * rho0 / (3 * R^2))^(1/2) |
| |
| H0 = ( 8 * pi * gamma * rho0 / 3)^(1/2) |
+-----------------------------------------------+
Note carefully that H0 and rho0 in this equation are ganged
together. Asserting a value for the one fixes the value for the
other. But H0 and rho0 are the present epoch values of H and rho AS
ASSESSED BY OBSERVATION. Their values COULD be discordant with this
equation. More on this later.
By the 1990s, spaceprobes COBE and WMAP confidently measured H0
to be 71km/Mpc.s. There seems to be no evidence for a value much lower
than this,not so low as 50km/Mpc.s. We keep the obsolete value to show
how the universe could behave under various assessed H0.
Many treatments call H and H0 the 'Hubble constant' as tho it were
an other one of the physical constants of nature. It is not.It is a
quantity derived from the behavior of the universe. Hubble assessed H0
at about 500km/Mpc.s based on the scanty data he had. Since then it
has been continually modified by continuing observations. In light of
this character of H0 we cease to regard it as a constant and refer to
it as the Hubble parameter or Hubble factor.
One megaparsec equals 3.086E19 kilometers and one megalightyear
equals 9.46E18 kilometers. These allow the alternate -- and more
relevant -- citation of H0.
+-----------------------------------------+
| RANGE OF ASSESSMENTS OF H0 |
| |
| H0[50;75]km/Mpc.s = [1.62;2.43]E-18/s |
| |
| H0[15;23]km/Mly.s = [1.58;2.43]E-18/s |
| |
| H0[lo] = 1.6E-18/s, H0[hi] = 2.4E-18/s |
+-----------------------------------------+
These simply state that the scalefactor of the universe increases
by, to pick one of the values, 1.6E-18 part of its instant value per
second.
Hubble Time
---------
If the value of H0 prevailed thruout all past time, then there is
a moment when R = 0. This is the notion of the bigbang creation of the
universe. The time elapsed since R = 0, T0, under H = H0 is
H = 1der(R,t) / R
= delR /( delT * R)
H0 = delR / (delT * R0)
= delR / delT
= (R0 - 0) / (T0 - 0)
= R0 / T0
= 1 / T0
T0 = 1 / H0
+---------------------------------------+
| HUBBLE TIME |
| |
| T0 = 1/H0 |
| |
| T0[lo] = 6.25E17s T0[hi] = 4.17E17s |
| = 1.98E10y = 1.32E10y |
| = 19.8Gy = 13.2Gy |
+---------------------------------------+
T0 is the Hubble time of the universe, the time elapsed since the
bigbang if H = H0 all during the expansion process. It is NOT the
actual age or the time since the bigbang because there is good cause
for H to vary with time. H is greater in past time and is now
declining, according as the particular geometry and dynamics we work
with. The Hubble time is a figure-of-merit for comparing various
models of universe answering to 'age'.
The Redshift in Galactic Spectra
------------------------------
The importance of redshifts in the spectra of galaxies took a long
time to appreciate. In 1905 Slipher called attention to a most curious
feature of spiral nebula spectra: They showed strong redshifts. At
that time the nature of spiral nebulae was unknown and the shifts were
assumed caused by recession velocity of the nebulae. The fact that
very few spiral nebulae exhibited a blueshift, for accession, strongly
heightened the mystery of these objects. By 1920 redshifts for about
40 spiral nebulae were established, mainly as a byproduct of other
work on these objects.
Early cosmologists ignored the spiral nebulae for ignorance of any
possible use for them in the large scheme of the universe.Spiral
nebulae were thought to reside within our Milky Way and the cosmos
beyond was empty space. Einstein ignored the nebulae in his first
model of the universe in 1917, which stated that the universe is
static with no special motion.
deSitter in 1917 worked up a model that incorporated a vast and
colossal expansion of empty space. He said there was no way to
document this expansion because there was nothing in deep space to act
as markers. He, too, ignored the spiral nebulae. However, he noted
that if there were sprinkled in the empty space a cosmic confetti the
expansion could be monitored by the redshifts in its spectrum.
deSitter -- and not as some treatments tell Hubble -- was the first to
demonstrate an expanding universe AND TO PREDICT THE REDSHIFT CAUSED
BY IT. The expansion of space was for many years actually referred to
as the deSitter effect.
In the ongoing debate about these nebulae, a few astronomers
argued that they were far off systems of stars like the Milky Way.
By 1925 evidence accumulated to declare that the spiral nebulae are
really other Milky Ways, other galaxies, in deep space beyond the
Milky Way. Astronomers then tried to assess their distances by various
means. Among these were delta Cephei stars, texture on photographs,
brilliance of supernovae, and angular size of star clouds and
clusters. Thru all this period Slipher continued to collect redshift
data.
Gradually there built up a body of galaxy distances and redshifts.
Only after the extragalactic character of these nebulae was realized,
after the galaxy distances were assessed, after the principles of
Einstein physics perculated thru astronomy then the time was ripe for
Hubble's work.
Hubble was NOT the first to discover the distance-redshift
relation! In 1922 Wirtz, before the galactic nature and distances of
the spiral nebulae were known, found that the Slipher redshifts
increase inversely as angular size and apparent brilliance of the
nebulae. He figured that the size and brightness were indices of
remoteness and suggested that the relation was really one of redshift
versus distance. Furthermore Wirtz suggested that the relation was
consonant with the deSitter model of the cosmos -- the universe seems
to be ever expanding.
In 1928 Robertson studied the newly assessed distances and
addiurnated redshifts of the nebulae. By now these were known to be
actual other Milky Ways populating deep space. He noted that the
redshift varied linearly as the galaxy distance and thus correlated
well with the deSitter concept of an expanding universe.
Hubble announced his discovery in 1929 with input from his own,
and Humason's, new data. Humason & Hubble enjoyed free run of the
Mount Wilson 2.5m telescope, the largest and best outfitted in the
world. No other observatory could capture the galaxy data as well as
this one could. Curiously, while other astronomers soon associated his
redshift relation with an expanding universe, Hubble did not go along
with them. He held out for some kind of conventional explanation for
the recession of the galaxies. Not until 1937 did Hubble come around
to accept his discovery as an effect of Einstein physics.
Expansion or Hubble Redshift
--------------------------
The raw observation of the remote targets reveals an increase of
the wavelength of their spectrum lines. A line identified as having a
certain wavelength today, as determined on Earth now, is seen to be of
longer wavelength in the remote's spectrum. The cause of this increase
is nothing more or less than the increase in the scalefactor between
the emission of the light and its reception.
The light takes so long to reach us that the universe had time to
expand noticeably. We may think of the wavelength as a dimension mark
on the fabric of space, like the very R itself. It is stretched by the
scalefactor expansion along with space itself and serves as an
indicator of the expansion.
We start with the very discovery of the expansion by means of the
redshift in the spectra of remote galaxies. Because along the
lightpath from the remote to us the speed of light c is the same, the
wavelength of the light relative to the instant scalefactor is the
same for all points on the lightpath. Placing the one point at the
remote site and the other at us we have
c * delt0 / R0 = c * delt / R
c*delt can be set to the wavelength of the light
lambda = c * delt
lambda0 / R0 = lambda / R
lambda0 / lambda = R0 / R
The ratio of the wavelength at us to that at the remote source is
the ratio of the scalefactor at us to that at the remote site. This is
manifested as a redshift in the remote's spectrum because R0 is larger
than R or R0/R > 1.
For historical reasons the ratio of wavelengths is names Z+1.
+------------------------------+
| EXPANSION OR HUBBLE REDSHIFT |
| |
| Z+1 = lambda0 / lambda |
| = R0 / R |
| = delt0 / delt |
+------------------------------+
By long tradition from spectrometry the redshift Z is defined as
(lambda0-lambda)/lambda. But
Z = (lamba0 - lambda) / lambda
= lambda0 / lambda - lambda / lambda
= (lambda0 / lambda) -1
Z+1 = lambda0 / lambda
In fact and deed this is all the expansion redshift tells us. It
is merely the scalefactor ratio from the emission to the reception of
the light. And that's really all there is to it. This was first
announced by leMaitre & Robertson about 1925.
This cause of a redshift is totally different from the Doppler-
Fizeau effect. The Doppler-Fizeau effect involves a source in motion
relative to us within the same coordinate system. The redshift is the
speed of the source compared to that of light.
Time Dilation
-----------
The increase in wavelength is accompanied by a decrease in
frequency nu because lambda*nu = c by definition. Hence
R0 / R = (c / lambda) / (c / lambda0)
= nu / nu0
= Z+1
nu / nu0 = (1 / delt) / (1 / delt0)
= delt0 / delt
= Z+1
The decrease of frequency with the scalefactor increase is not
limited to light or even to radiation. It applies to all processes and
activities! Think of frequency as a clock. The clock's rate, the number
of ticks per second, is the frequency. The clock ticks slower at large
Z. Hearts beat slower. People age slower. At very large Z actions
procede ever slower. At Z = 8 a process we observe to take one day
actually occupies but 3 hours at the source. At Z = 1000 that same
one-day event occupies not quite 1-1/2 minutes at the source. In the
limit at Z = infinity all activity halts.
We must factor in the time dilation when interpreting observations
of very remote targets. A delta Cephei star, allowing it can be seen
at high Z+1, seems to take much longer for each cycle compared to a
similar one close to us. A supernova takes much longer to go thru its
decay than a similar close one.
Hubble Expansion vs Recessional Velocity
--------------------------------------
Note well that nothing is said about the 'speed of recession' of
the galaxies and quasars. H0 in no way tells 'how fast' these bodies
are rushing and streaming away from us. The units of H0 are NOT
[speed/distance]. They are [deldistance]/[distance*/deltime].
Arithmeticly these look the same but in nature they are entirely
distinct.
Note also that the expansion redshift says nothing about the
distance between us and the remote. To extract a distance from the
redshift we must declare the model of universe to use. A common, tho
wrong, model is that with H = H0 and R0/R = r0/r = t0/t for all time.
Then
H0 = delR / (delt * R)
= delr / (del t *r)
v = delr / delt
= H0 * r
= c * Z
r = c * Z / H0
Astronomers nowadays are turning away from converting Z into
speed or distance and are just citing Z itself.
When the expansion redshift was first discovered the Doppler-
Fizeau effect was the only credible means in astronomy of producing
redshufts. The redshifts were interpreted as ordinary recession from
us, resulting in quotations of some very high speeds. (leMaitre &
Robertson were unknown to mainstream astronomers.) Hubble & Humason,
discerning huge redshifts in the galaxies, laid them to huge recession
speeds. They expressed them in ordinary [oldstyle] units and plotted
them against the galaxy distances.
These latter they assessed from delta Cephei stars, star counts,
cluster resolution, and so on. As these speeds became large fractions
of c, many astronomers applied the Einstein version of the Doppler-
Fizeau formula, without recognizing that the redshift of expansion has
nothing to do with material movement away from us.
The remotes are not at all careering thru space away from us at
super-high speeds. They are relative to us more or less at rest. They
move only locally under tidal action from neighboring remotes with
speeds only up to a few thousand kilometer/second.
It is astounding how many treatments, even authoritative ones, mix
this up! They use the analogy of acoustic waves from trains or cars,
even tho their mechanism is utterly irrelevant to Hubble redshifts.
They speak of galaxies and quasars fleeing pell-mell away from us at
super-speeds, all nitidly cited in kilometers/second.
Being that Hubble was a dean of astronomy, his work was pretty
much gospel. And so his Doppler-Fizeau interpretation was blithely
handed down to our day in the ordinary textbooks.
What's Going on Here?
-------------------
We derived the Friedmann formulae of the universe and we will
explore it extensa mente in this paper. However, we employed only
regular Newton physics, not Einstein physics. Any orthodox text on
cosmology derives the exact same Friedmann equations from the theory
of relativity. To do so some agility in maths is required as well as a
thoro grounding in relativity theory. Yet we did the same via simple
maths within a familiar envelope. What's going on here?
In 1934 Milne & McCrea discovered that the Friedmann formulae in
fact can be derived from Newton precepts. The results are the same as
those obtained from the full blown relativity derivation. They called
their derivation the Newton cosmology but this is a poor name. The
Milne-McCrea scheme is not a model of the universe; it is a method of
demonstrating the Friedmann model. Furthermore the term 'Newton
cosmology' has two uses in astronomy. The one refers to the Milne-
McCrea method. The other refers to the world pictured by Newton and
his school in the 17th to 19th centuries. Cosmology works use the term
in both senses; please keep awake. Here we mean only Newton's own
description of the universe. The modern derivation of the Friedmann
equations is the Milne-McCrea method.
Milne & McCrea's Trick
--------------------
Starting with the assumption that the universe is every where
uniform and even, homogeneous and isotropic, any small part of the
whole world behaves just like any other part. Homogeneity and isotropy
are almsot as axioms in cosmology today, altho there are challenges
against them from time to time.
We also start with the observed phaenomenon of Hubble expansion.
If we rope in a given spherical volume with a comoving envelope, the
sphere expands against its own self-gravitation. The matter undergoes
its own expansion, even within similarly defined spheres that enclose
or overlap the instant one.
Note very well that there is no unique center from which the
expansion issues, nor any edge encasing all the matter of the world
within this sphere.
Now we must understand that Einstein physics does not overthrow
Newton's; it is a superset of Newton physics. In the limiting case of
small volume and slow speed and weak gravity the two physics converge
to yield the same results to various experiments. This is why with the
several generations of experience with relativity theory we can still
employ Newton's laws in so very many situations in astronomy. The bulk
of cases simply do not approach the threshold for switching over to
relativity laws.
The downside is that this very feature of Milne & MccCrea
dissuaded early adaoption of Einstein physics by mainstream
astronomers. The situations of astronomy then known were thoroly
handled thru Newton. Einstein didn't intrude into them. Only when we
discovered phaenomena and objects with conditions too extreme for
newton, mostly via astrophysiccal satellites, did astronomers take up
relativity studies.
Hence if we take our surrogate sphere small enough the mass
internal to it raises up a weak gravity, the geometry is Euclid, and
the expansion speed is slow. Under these conditions we can describe
the behavior of the sphere using Newton physics of the sort familiar
to home astronomers and not entangle with Einstein physics. Since the
end result must still be the same which ever physics is used, the one
and same Friedmann equations fall out from both derivations. We
arrived at them via the Milne-McCrea method for the simplicity of the
maths and physics.
17th Century Cosmology?
---------------------
Some works speculate on Newton's failure to discover the Friedmann
model and launch cosmology in the 17th century. Certainly he was
maths-wise capable of doing so and the Milne-McCrea method banks off
of Newton's own work. The answer seems to be flat out no. For starts
there are two concepts that Newton could never have appreciated.
We displaced the actual size of the expanding sphere by switching
to a scalefactor R in the stead of the absolute radius r. And we
introduced the comoving coordinate system such that within it matter
has no flow. In the extreme the coords of all particles remain fixed
thruout the expansion phase.
Neither concept was within the grasp of Newton and his school. He
worked with an absolute external coordinate frame whose existence
transcended any distribution of mass within it. And he held that
gravitational bodies of spherical symmetry have a definite center and
radius.
The more fundamental reason Newton could not discover the
Friedmann universe by the Milne-McCrea derivation is that this
derivation is really a retrosolution from the Friedmann model to a
Newton world. That is, the answer was already at hand. Furthermore the
Hubble expansion was already observationally established. Newton could
know of neither the Friedmann solution nor of the Hubble phaenomenon.
He worked in a static and quiet universe from the word go.
Misinterpretations
----------------
The Milne-McCrea method as simple as it is can be oversimplified
with disastrous consequences. By far the most common botchup with the
method is the notion of a closed and bounded sphere. The surrogate
sphere contains all the matter in the universe -- it is the 'cosmic
ball' -- and expands from a center into some empty space that engulfs
it. This leads to a border where the world is neither isotropic nor
homogeneous. There arises the issue of what happens when this edge is
crossed. There is in this picture an 'outside' for the universe!
Yet in so many popular treatments the universe is shown in a film
or video as a tiny dot in otherwise empty space. The dot blows up into
this space. Globs spew out, turn into quasars, condense into galaxies,
drop off planets. All this stuff flies past the viewer and off the
edge of the screen, as tho we were watching the episode from outside
the universe. We saw that this is not only erroneous but is
nonsensical.
A corollary to the cosmic ball picture is the barrier of
lightspeed. This is usually carefully sidestepped in popular works. As
the material flows out in absolute space in real motion it obeys the
Hubble relation. At some great distance from the center the Hubble
expansion approaches the speed of light. The material would begin to
pile up at this speed barrier some definite radius from the bigbang
point, thus creating a throroly not isotropic or homogeneous region..
In time all the material would collect along this outer shell and the
universe comes to an end state with everything moving outward at near
lightspeed.