The model I describe below is a model of matter expanding into
pre-existing space. This can be pictured as being similar to a nuclear
bomb in space, viewed from the distance. As such, it is possible for
energy from outside the universe to enter. Toward the end of the
argument, I will mention "unknown objects" pushing through the
universe, creating and accelerating galaxies. These objects would have
started from the edge of the universe, disintegrating on impact with
the expanding shell, with momentum imparted to a finite number of the
outer particles to send them hurtling through the inner universe.
Part I: The Single-Particle Non-Accelerated Universe
Consider an instant in time and space containing only one type of
particle, but an infinite number of them. Each of these particles
occupy the same point in time and space, but do not share the same
momentum, thus the Pauli Exclusion Principle is not violated.
The Pauli Exclusion Principle states that no two fermions can occupy
the exact same set of quantum numbers. Quantum numbers are used to
denote linear momentum, angular momentum, spin, oscillations, and other
modes of motion and/or "energy storage." As long as each of our
particles has a DIFFERENT linear momentum, it should be possible for
them to occupy the same point in time and space for a single instant.
In this single place in space and time, all of the particles at this
point form what can be described as a fermi gas. In a fermi gas, there
are a certain number of particles, and a certain amount of energy
available. There is either just enough energy for the particles to
occupy their different momenta, (also known as modes of motion) or
there is more than enough inergy for them to occupy their different
momenta.
If there is MORE THAN ENOUGH energy for the particles, then it is
difficult to make predictions about a pattern. If there is JUST ENOUGH
energy to supply each particle with a different momentum, then these
different momenta should form a fairly regular pattern.
As an analogy, imagine an astronaut filling a round jar with BB's. If
the jar is much bigger than the volume of the BB's, he cannot predict
where the BB's will locate themselves. However, if he fills the jar,
completely, the BB's will arrange themselves, more-or-less in a
lattice, and more specifically, viewed from certain angles, this
lattice will have planes of BB's arranged in a hexagonal pattern.
The pattern of these BB's are of course, positional, whereas the
pattern we want in our particles is in their momenta. However, just as
the positions of the BB's can be mapped by vectors from the position of
an arbitrarily selected BB, the momenta of the particles can be mapped
by vectors from the momentum of an arbitrarily selected particle.
If we assume there is just enough energy to put each of the particles
in a unique momentum state, we should find the pattern of momenta to be
just as mathematically predictable as the locations of BB's in a packed
jar.
For now, instead of focusing on the whole three dimensional structure,
I will only address one plane, along which the BB's would arrange
themselves in a regular hexagonal pattern. Also, I presume that this
regular hexagonal pattern continues out to infinity in all directions,
which means I am assuming there is "just enough energy" to put each of
an infinite number of particles in a unique momentum state.
1. Map the momenta of the particles
As I am working in only one plane, it is much easier to work with flat
circles instead of spheres. In order to find the vectors of available
momenta, I started with one arbitrary penny, and set a zero-momentum
origin at its center. Then, I took one finger, counting the nearest
{l=1} six pennies {t=0,1,2,3,4,5}. Then I took two fingers {n=0,1} and
identified a pattern to define the coordinates for the next nearest
{l=1} concentric hexagon {t=0,1,2,3,4,5}. By repeating this pattern
with three fingers, four fingers, etc. I found that I could explicitly
locate an individual penny in an infinite plane of pennies with a set
of three numbers {l, n, t}.
Then by doing a little geometry and trigonometry, I found the x,y
coordinates of these pennies, in units of penny lengths.
(1) x(t,l,n)=l*Cos[t*Pi/3]+n*Cos[(t-2)Pi/3]
y(t,l,n)=l*Sin[t*Pi/3]+n*Sin[(t-2)Pi/3]
t:{0,5} represents the six initial directions
l:{1,Infinity}: Represents the integral distance in each of the six
directions
n:{0, 1, ..., l} Represents the offsets of extra pennies
These equations generate an infinite set of regular hexagonal
coordinates, spaced at unit length apart.
2. Find the velocity of the particles
In relativity, of course, there is a speed of light limit. However,
this does not limit the momentum of a particle in any way.
The momentum of a particle is equal to mass*velocity*gamma where
gamma=1/sqrt(1-(v/c)^2). In this model, by choosing units that the
speed of light is 1 (whether that be 1 light year per year, or 1 light
second per second, or just a little less than 1 foot per nanosecond).
Since there is only one type of particle, we can say it's mass is 1
particle mass.
(2) Then p = v/sqrt(1-v^2)
where p = momentum; v=velocity in units such that v=1 represents the
speed of light.
The units of momentum is mass*velocity, and in this case, those units
are the mass of the particle times the speed of light. Since this is a
new particle and a new unit, as far as I know, I will make up my own
name for it--the Umph.
To get a feel for momentum in these units, first we can find the
inverse function of (2) which is
(3) v= p / sqrt(1+p^2)
So a momentum of 1 Umph, corresponds to a velocity of .707c. 2 Umphs
Corresponds to a velocity of .894c, 3 Umphs to velocity .949c.
Recall, above, that I assumed the particles would take every available
momentum state, and by doing so, would form a regularly spaced pattern.
This means, along any straight line from the origin, between any two
evently spaced pair of momenta, there should be the same number of
particles. So if there are a billion particles moving straight north
between 0 and 1 Umph, there should also be a billion particles between
1 and 2 Umphs, 2 and 3 Umphs, 3 and 4 Umphs, 1001 and 1002 Umphs, etc.
The number of particles moving with momenta between two successive
momenta is called the linear momentum density. Because of the even
spacing of particles between each momentum, we can say that I've
assumed a constant linear momentum density. Though the linear momentum
density remains constant, the velocity density increases more and more
rapidly as we approach the speed of light. This will be more apparent
in the animation presented below.
Let us call this linear momentum density, s. So s= the number of
particles between rapidity 0 and rapidity 1 along one of the six
straight lines from the central particle. We need this, because
although we have already have v as a function of p, we don't yet have v
as a function of our variables t, l and n.
(4) p_x(t,l,n)=(l*Cos[t*Pi/3]+n*Cos[(t-2)Pi/3])/s
p_y(t,l,n)=(l*Sin[t*Pi/3]+n*Sin[(t-2)Pi/3])/s
This is similar to equation 1, but by introducing the variable, s, we
have scaled the pennies, until s pennies can fit side-to-side in a row
of length 1.
(5) p^2 = p_x^2 + p_y^2
= (n^2 - n*l + l^2)/s^2 (Note...This step takes several trig
identities)
We need p^2 to plug into equation 3. I made several starts on this
problem myself, before I got it right. It is really good practice in
trigonometry.
(6) v(t,l,n) =
{(l*Cos[t*Pi/3]+n*Cos[(t-2)Pi/3]),(l*Sin[t*Pi/3]+n*Sin[(t-2)Pi/3])}
-----------------------------------------------------------------------
sqrt(1+(n^2-n*l+l^2)/s^2)
This equation is a repetition of equation 3, but with terms replaced
with solutions from equations 4 and 5.
After generating these velocities, the actual locations of the
particles can be generated by multiplying the velocity vectors by time.
If we have a finite number of particles, arranged uniformly around the
center, then they will be accelerated back toward the center. If there
are an infinite number, each particle will see itself in the center of
the sphere, and thus have no preferential direction for acceleration.
For the sake of this simple model, I assume that the number of
particles is infinite, and there is exactly the amount of energy needed
to give each of the particles a unique linear momentum state. Thus,
there is no gravitational acceleration for any of the particles in any
direction.
The result, letting s=25, and only going out to l=80, looks something
like this:
http://www.spoonfedrelativity.com/fi...l-big-bang.gif
(Note with l=80, s=25, The momentum of the outermost shown particles
are p_outermost= l/s = 3.2, and thus, by equation (3) v=.95. The
circle continues to get more dense as you go out the last 5% of the
radius, but this detail is NOT shown in the animation, because of the
exponential growth in processing time needed to plot those points.)
Even though this is a two-dimensional case, it gives a very good idea
what the particle distribution of a perfectly homogeneous univers
should look like. It is a primitive model, which does not take into
account any sort of particle forces, yet it very clearly predicts a
dense outer shell, which would, (seemingly paradoxically) be from the
moment of the big bang, early in the universe from when it was still
small, and yet at the same time, be surrounding the older, more
expanded universe.
In fact, since all of the momentum is presumed to be linear, from the
very beginning, this model does not profess to describe the heat of the
big bang. It does not examine the electro-magnetic fields, and thus
does not show how the light from the hot dense region around the edges
is redshifted from the perspective of an observer at the center.
All this model does, is that it points out that an infinite, isotropic
and homogeneous distribution of linear momenta will, given time, result
in a fairly well defined pattern of positions describing a perfect
circle, with an outer shell of infinite density. I have no doubt that
if we did the same with a three dimensional Hexagonal Close Packed or
Face Centered Cubic distribution of linear momenta, we would similarly
find a sphere with an outer shell of infinite density.
In my model of the actual universe, the matter is similarly
distributed, and when we see the CMBR, we are actually seeing the inner
side of this infinitely dense shell. Because of the doppler effect;
both the normal doppler effect and the transverse doppler effect, this
shell is redshifted by a factor of several thousand.
I leave open the possibility that the number of particles in the
universe is not truly infinite, but very very large. In which case,
there will be a constant pull in a certain direction. But in this
model, that pull would be in a very specific direction, and may yet be
determined.
Among the most important things this model should explain, though, is
#1 Why there appear to be galaxies in the universe which are OLDER than
the Milky Way.
As I have described it thus far, every particle is moving with constant
speed. If you determine the proper age of any particle moving away
from an observer, the moving particle always ages slower than the
observer. Thus it would seem that our galaxy should be the oldest
galaxy, and all others should be younger, as we look out in the
distance.
#2 Why there are a predominance of bright galaxies toward the galactic
north.
#3 Why the CMBR is "hotter" in the galactic North
#4 Why Hubble's constant has been measured to have a smaller value
towards galactic north than it does toward galactic south.
#5 How the measured radius of the universe is closer to 25 billion
light years instead of 13.7 billion light years, though it is only 13.7
billion years old.
#6 Account for the era of Inflation during the first microsecond of the
universe, which is used to explain this by the standard model.
#7 Account for patterns of polarization in the light from the CMBR
While I have not settled down with ALL of the data, devoting sixty
hours a week to poring over every single thing, and doing the very
difficult work of mapping out every coordinate, I AM devoting more time
than I can really afford in simply presenting the distant view of the
model that should eventually be found to answer many of these questions
with one single phenomenon.
A Lorentz Transformation, performed on any event after the initial
event, mapping the coordinates from the first reference frame to the
second reference frame will
1) cause one side of the universe to expand much faster than the speed
of light, instantly pushing it out to an unlimited distance.
2) Cause us to enter a new reference frame where the objects in the
region which we accelerated toward to be much much older.
3) Cause Lorentz Contraction effects on the undisturbed portions of the
lattice in our local region, which might result in polarization of
light from the CMBR.
4) Cause one section of the CMBR to be much closer and younger than
another, and the other section of the CMBR to be much further away and
older.
5) Cause the universe to be closer and flatter in one direction than
the other, resulting in brighter galaxies and smaller measurements of
the Hubble constant.
==========================
Part II: Lorentz Transformation from an event in a Single Particle
Universe
To predict inflation, simply take the toy model universe as given, and
perform a Lorentz Transformation on the entire set. For instance, take
a particle that is moving at .99c and decelerate it down to zero. I
don't know when I'll be able to get around to doing this, myself, but I
know that the end result would create a result qualititatively similar
to our own universe, with asymmetries in Hubble's Constant, a CMBR
dipole, and a universe larger than could be accounted for by AGE*Speed
of Light.
You might be able to get an idea of how to do this transformation from
http://www.spoonfedrelativity.com/worldRegions.html
In the following, I use Above and Below to describe opposite
directions--Below is galactic North, while Above is Galactic South. At
the dawn of the universe there would have been no galaxy with which to
reference direction. The only thing you could use is that the
direction from which you were being pushed would seem like DOWN, and
the direction you were being pushed would seem like UP. Thus I use
these directions to describe the initial acceleration.
===========================
Part III How it Happened:
This section describes a few of the anomolies of the standard model,
and how they can be accounted for by assuming an immense acceleration
immediately after the big bang.
The explanation involves both time dilation and length contraction, and
more importantly, length "uncontraction." The key is a huge
acceleration of the local matter, near the beginning of the Big Bang.
Imagine at the dawn of the universe, we were being pushed HARD from
below by the hot part of the CMBR. By checking the Right Ascension and
Declination of these objects, you can verify that primordial Andromeda
M31 galaxy and and Fornax supercluster are over our heads, and
SN1997ff, M87, and Virgo are at our feet.
We are forced up, accelerating, and with each change in velocity, the
universe under us is scrunched by length contraction, while overhead,
distances to receding particles are Lorentz "uncontracted" until we
match pace with them... but there are always more particles outpacing
us, so as we continue to accelerate, the region above us expands.
This expansion is not limited by the speed of light. This is the
process of entering, or getting closer to the reference frame of the
receding object. As we enter this frame, that object gets much older,
and much further away, as can be calculated from the Lorentz
Transformation, and finding the intersection of that object's worldline
with our plane of simultaneity (or world-region).
So, though our galaxy barely aged during this time, the rest of the
universe expanded to an ancient sphere (as old as it is big)
The region above us has expanded, but the region below us, has become
more length contracted. After we are through with this acceleration
(inflation) period, we find ourselves at the very edge of an ancient
spherical universe, though we are still at the dawn of time.
Toward the end of our acceleration era, we match pace with Andromeda
galaxy, and start to overtake it, so it starts falling "down" towards
us. (If we were Andromedans, we'd see that at just after we finished
accelerating, the Milky Way started to overtake us.)
Because the area below us was length contracted during that
acceleration phase, Hubble's constant toward our feet, toward Virgo
cluster, is a very tightly packed 55 km/sec/MPc. Since that initial
era, the edge close to our feet has been expanding at the speed of
light, just like the edge far over our head.
Meanwhile, overhead, in the length "uncontracted" region, toward Fornax
cluster, Hubble's constant is a much more loosely packed 80 km/sec/Mpc.
You can check the directions and findings for the Fornax team and the
Virgo team, who used Cepheids to find Hubble's Constant. Virgo cluster
is almost precisely lined up with the hot dipole of the CMBR, while
Fornax is near the cold dipole.
Because of "uncontraction" all the supernovas overhead (toward
Andromeda and Fornax) are further away than they would be by the
formula, distance=rate * time. Their distance expanded by length
"uncontraction" so their velocities are not high enough to account for
their distance. Thus, they are all dimmer than their redshifts would
indicate. This dimness, is often used, inexplicably, to suggest that
the universe is "accelerating." You can ask a proponent of the
standard model about that.
But what about the supernovas below? With only a few exceptions in the
galactic north (under our feet), all of the Supernovas are dimmer than
astronomers expect.
For explanation, consider this: our acceleration was right at the
beginning of the universe... The distances to those Supernova
contracted at once, while the stars at our feet were still nearby. The
immediate expansion of the little distance over our head made it HUGE,
but the immediate contraction of the little distance under our feet
couldn't go less than little. These stars may have been delayed a
couple million years in taking off away from us, but still, they should
be very close to matching the distance=rate*time.
Many of the supernovas in the galactic north are slightly brighter than
we expect them to be considering their redshift, most notably SN1997ff.
This fits with the shorter Hubble constant in that direction, and the
closer universal edge.
SN1997ff lies well outside the redshift/luminosity curve. This
Supernova lies directly under our feet. It's a supernova that is much
brighter than it should be--much MUCH closer than would be indicated by
its redshift. The data suggests to me that it was staying close to us
for a long time, but then all of a sudden, it took off away from us.
I'm guessing that whatever caused it to go supernova also caused it to
shoot downward toward the near edge of the universe.
Part IV: Distortions From Outside
Finally, the weblike pattern of superclusters throughout the visible
universe has an explanation in my model. The most likely is that at
some time in the early universe, energy came from OUTSIDE the sphere
and passed through the region, disrupting the regularly spaced pattern
of particles. This energy was most likely in the form of other
particles, planets, stars, galaxies or universes which were
disentegrated by the outer edge of the expanding sphere of our
universe. The change in momentum of a large but finite number of
particles passed back through the universe, smashing particles together
as they flowed, resulting in both the formation of superclusters, and
the sudden instant of acceleration which I have been describing.