Ben Rudiak-Gould wrote:
Spoonfed wrote:
The diagrams you have drawn show a Galilean Transformation, showing a
fairly small change in speed, less than ten percent of the speed of
light. This would cover the area within a billion light years of
Earth; within 10% of the radius of the universe.

First, that's the radius of the *visible* universe; nobody knows how big the
whole universe is. Second, in terms of comoving distance the radius of the
visible universe is about 47 billion light years, so one billion light years
is a lot less than 10%.

When we get outside that range, if Hubble's Law still holds true, we
need to use a Lorentz Transformation, as the Galilean transformation is
only an approximation.

As I've said before, the Galilean transformation is a better approximation
than the Lorentz transformation in this situation. More precisely, fix an
object O which is roughly stationary with respect to the CMBR, and choose
coordinates such that time is cosmological time and distance from the origin
is comoving distance from O. The coordinate systems so obtained, for
different objects O, are related by a coordinate transformation which is
similar to the Galilean transformation.

I know we've talked about this before, and I recall you said that you were
aware that your ideas were different from mainstream cosmology. If so, I
think you should tag your posts with "this is just my personal theory,
but...". And you should be aware that your model, if I understand it
correctly, is a special case of the standard big bang model with Omega ~ 0,
but Omega has been known to be about 1 for a long time. For as long as I can
remember, the only debate has been over whether it is slightly larger or
slightly smaller than one. Zero is way outside the error bars.

-- Ben

Actually, my idea is that k=0 and a(t)=1. These are terms from the
FLRW metric. As far as the cosmological constant goes, I don't even
know what it is, let alone what value it might have.

My personal theory, though it is a work in progress, is that the
universe started from (approximately) a point, and expanded outward
into space. Our local section of the universe underwent a huge
acceleration from the beginning. In the massive amount of energy
available in the beginning, Brownian motion caused the primordial
particles of our local universe to undergo immense acceleration.

By accelerating toward a receding object, but not matching pace with
it, we enter a frame of reference where the space between us and the
receding object is length uncontracted. It will be moving away more
slowly, but also more distant. In this way, the distance will be
greater than you would expect from its velocity. Likewise, the
faintness would be more than you would expect from its redshift.

Imagine at the dawn of the universe, we were being pushed HARD from
below by that hot part of the CMBR. Primordial Andromeda M31 galaxy
and and Fornax supercluster are right 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 to an ancient sphere (as old as it is big),
while we accelerate away from the very edge of that sphere, receding
right under our feet.

Millions of years pass by, and 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.

Because the area below us is length contracted, Hubble's constant
toward our feet, toward Virgo cluster, is a very tightly packed 55
km/sec/MPc. Meanwhile, overhead, in the length uncontracted region,
toward Fornax cluster, Hubble's constant is a much more loosely packed
80 km/sec/Mpc. These values for Hubble's constant have been argued,
but in my theory, they are both right.

Because of "uncontraction" all the stars overhead (toward Andromeda and
Fornax) are further away than they would be by the formula,
distance=rate * time. They are all dimmer than their redshifts would
indicate. But what about those below?

Our acceleration was right at the beginning of the universe... They
distances to them contracted at once, while the stars at our feet were
still nearby. An expansion of a little distance can get HUGE, but a
contraction of a little distance is still 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.

They may have even accelerated toward us after we stopped accelerating,
meaning they would have a higher average velocity away from us than
their current velocity away from us... So these stars should also be
slightly dimmer than their redshifts would indicate, although for
different reasons.

But where does that leave SN1997ff? It's a supernova that is much
brighter than it should be, as though it was staying close to us for a
long time, but then all of a sudden, it took off away from us.

Well, there's room in this model for mysteries. I'm guessing that
whatever caused it to go supernova also caused it to shoot downward
toward the near edge of the universe.

That's my theory, as it stands today, after spending most of the day
looking up Right Ascensions and Declinations for a bunch of those
objects. As far as whether it comes close to the standard model, I'm
pretty sure it doesn't, but I'm not 100% certain, because I've never
heard much about the standard model except that you can't understand it
without years of graduate level mathematics.

The neat thing about my explanation, though, is that it fits the data.

"Spoonfed" wrote in message
ups.com...
|
| Ben Rudiak-Gould wrote:
| Spoonfed wrote:
| The diagrams you have drawn show a Galilean Transformation,
showing a
| fairly small change in speed, less than ten percent of the speed
of
| light. This would cover the area within a billion light years of
| Earth; within 10% of the radius of the universe.
|
| First, that's the radius of the *visible* universe; nobody knows how
big the
| whole universe is. Second, in terms of comoving distance the radius
of the
| visible universe is about 47 billion light years, so one billion
light years
| is a lot less than 10%.


You have some evidence for this? Please cite the astronomer's name,
I'd be interested. 47 billion ly sounds rather a lot.



| When we get outside that range, if Hubble's Law still holds true,
we
| need to use a Lorentz Transformation, as the Galilean
transformation is
| only an approximation.
|
| As I've said before, the Galilean transformation is a better
approximation
| than the Lorentz transformation in this situation. More precisely,
fix an
| object O which is roughly stationary with respect to the CMBR, and
choose
| coordinates such that time is cosmological time and distance from
the origin
| is comoving distance from O. The coordinate systems so obtained, for
| different objects O, are related by a coordinate transformation
which is
| similar to the Galilean transformation.
|
| I know we've talked about this before, and I recall you said that
you were
| aware that your ideas were different from mainstream cosmology. If
so, I
| think you should tag your posts with "this is just my personal
theory,
| but...". And you should be aware that your model, if I understand it
| correctly, is a special case of the standard big bang model with
Omega ~ 0,
| but Omega has been known to be about 1 for a long time. For as long
as I can
| remember, the only debate has been over whether it is slightly
larger or
| slightly smaller than one. Zero is way outside the error bars.
|
| -- Ben
|
| Actually, my idea is that k=0 and a(t)=1. These are terms from the
| FLRW metric. As far as the cosmological constant goes, I don't even
| know what it is, let alone what value it might have.
|
| My personal theory,



Ahhh..... a personal theory... they abound. Got any evidence?


Androcles.




though it is a work in progress, is that the
| universe started from (approximately) a point, and expanded outward
| into space. Our local section of the universe underwent a huge
| acceleration from the beginning. In the massive amount of energy
| available in the beginning, Brownian motion caused the primordial
| particles of our local universe to undergo immense acceleration.
|
| By accelerating toward a receding object, but not matching pace with
| it, we enter a frame of reference where the space between us and the
| receding object is length uncontracted. It will be moving away more
| slowly, but also more distant. In this way, the distance will be
| greater than you would expect from its velocity. Likewise, the
| faintness would be more than you would expect from its redshift.
|
| Imagine at the dawn of the universe, we were being pushed HARD from
| below by that hot part of the CMBR. Primordial Andromeda M31 galaxy
| and and Fornax supercluster are right 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 to an ancient sphere (as old as it is
big),
| while we accelerate away from the very edge of that sphere, receding
| right under our feet.
|
| Millions of years pass by, and 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.
|
| Because the area below us is length contracted, Hubble's constant
| toward our feet, toward Virgo cluster, is a very tightly packed 55
| km/sec/MPc. Meanwhile, overhead, in the length uncontracted region,
| toward Fornax cluster, Hubble's constant is a much more loosely packed
| 80 km/sec/Mpc. These values for Hubble's constant have been argued,
| but in my theory, they are both right.
|
| Because of "uncontraction" all the stars overhead (toward Andromeda
and
| Fornax) are further away than they would be by the formula,
| distance=rate * time. They are all dimmer than their redshifts would
| indicate. But what about those below?
|
| Our acceleration was right at the beginning of the universe... They
| distances to them contracted at once, while the stars at our feet were
| still nearby. An expansion of a little distance can get HUGE, but a
| contraction of a little distance is still 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.
|
| They may have even accelerated toward us after we stopped
accelerating,
| meaning they would have a higher average velocity away from us than
| their current velocity away from us... So these stars should also be
| slightly dimmer than their redshifts would indicate, although for
| different reasons.
|
| But where does that leave SN1997ff? It's a supernova that is much
| brighter than it should be, as though it was staying close to us for a
| long time, but then all of a sudden, it took off away from us.
|
| Well, there's room in this model for mysteries. I'm guessing that
| whatever caused it to go supernova also caused it to shoot downward
| toward the near edge of the universe.
|
| That's my theory, as it stands today, after spending most of the day
| looking up Right Ascensions and Declinations for a bunch of those
| objects. As far as whether it comes close to the standard model, I'm
| pretty sure it doesn't, but I'm not 100% certain, because I've never
| heard much about the standard model except that you can't understand
it
| without years of graduate level mathematics.
|
| The neat thing about my explanation, though, is that it fits the data.
|

Ben Rudiak-Gould wrote:
Spoonfed wrote:
The diagrams you have drawn show a Galilean Transformation, showing a
fairly small change in speed, less than ten percent of the speed of
light. This would cover the area within a billion light years of
Earth; within 10% of the radius of the universe.

First, that's the radius of the *visible* universe; nobody knows how big the
whole universe is. Second, in terms of comoving distance the radius of the
visible universe is about 47 billion light years, so one billion light years
is a lot less than 10%.

When we get outside that range, if Hubble's Law still holds true, we
need to use a Lorentz Transformation, as the Galilean transformation is
only an approximation.

As I've said before, the Galilean transformation is a better approximation
than the Lorentz transformation in this situation. More precisely, fix an
object O which is roughly stationary with respect to the CMBR, and choose
coordinates such that time is cosmological time and distance from the origin
is comoving distance from O. The coordinate systems so obtained, for
different objects O, are related by a coordinate transformation which is
similar to the Galilean transformation.

I know we've talked about this before, and I recall you said that you were
aware that your ideas were different from mainstream cosmology. If so, I
think you should tag your posts with "this is just my personal theory,
but...".
And you should be aware that your model, if I understand it
correctly, is a special case of the standard big bang model with Omega ~ 0,
but Omega has been known to be about 1 for a long time. For as long as I can
remember, the only debate has been over whether it is slightly larger or
slightly smaller than one. Zero is way outside the error bars.

-- Ben

I am not really sure about the cosmological constant, I've written more
he

http://groups.google.com/group/sci.p...ac223f9?hl=en&

If I am not mistaken, bringing the cosmological constant up to 1
requires a whole lot of dark matter, or dark energy. I don't think
that dark matter or dark energy is necessary to explain what we see.
If saying "non-baryonic dark matter is unnecessary" is equivalent to
saying "the cosmological constant is zero" then, yes, I would say the
cosmological constant is zero, or near zero.

Inflation, surprising dimness of supernovas, CMBR and CMBR dipole,
asymmetric values of Hubble's Constant, can all be explained by
relativistic acceleration of our galaxy during the early universe, and
continued acceleration of early galaxies from the direction of the
virgo cluster.

Spoonfed wrote:
Actually, my idea is that k=0 and a(t)=1.

What does t denote here? Are you saying that a is a constant function of
time, i.e. that the universe is not expanding? Or are you saying that
a(now) = 1? The latter is not a hypothesis, just a normalization convention.

I'm not convinced that you "speak the language" yet; it looks like you're
just copying stuff from recent posts by Tom Roberts without understanding it.

These are terms from the FLRW metric.

I'd go farther than this and say that they only have meaning in the context
of the FLRW metric, i.e. in the context of big bang cosmology. If you're not
talking about the big bang theory, I don't even understand what you mean by
saying that k=0.

As far as the cosmological constant goes, I don't even
know what it is, let alone what value it might have.

You can ignore it for the time being; conceptually speaking, it's a detail.

By accelerating toward a receding object, but not matching pace with
it, we enter a frame of reference where the space between us and the
receding object is length uncontracted. It will be moving away more
slowly, but also more distant. In this way, the distance will be
greater than you would expect from its velocity. Likewise, the
faintness would be more than you would expect from its redshift.

I think you should stop talking about frames of reference and phrase things
in terms of what we can actually see, which is a 2D projection of our past
light cone. In particular, how should we define the distance of the
astronomical objects that we can see?

Here's an SR conceptual question which may be pertinent. At one end of Main
Street is a clock tower. Alice is running along Main Street toward the clock
tower at a relativistic speed. Bob is standing stationary on Main Street,
looking at the clock tower. At the moment Alice passes Bob, they compare the
times they see on the clock face. Does Alice see an earlier time, a later
time, or the same time?

Imagine at the dawn of the universe, we were being pushed HARD from
below by that hot part of the CMBR.

"Below"? Are you saying that the universe was not isotropic? What was the
distribution of matter? Is it still anisotropic in the present era?

Starting at around this point I can barely understand at all what you're
trying to say. I seriously have trouble distinguishing it from schizophrenic
raving, and I would dismiss it without a second glance if your relativity
tutorials didn't show obvious evidence of sanity. If you're going to make
this theory comprehensible to anybody, you're going to have to put a lot of
effort into clearing up the exposition. The first step in doing this is to
learn the current dominant theory of cosmology, and how to extract simple
predictions from it. Then you can describe how your theory differs from
that. For example, are you aware that the big bang theory predicts that
beyond a certain redshift, galaxies which are *farther* away will appear
*larger* in the sky? I assume your theory does not match this prediction.
This does not necessarily exclude your theory, because I don't know whether
this prediction of the big bang theory has been directly verified. If you
make clear predictions like this which differ from the big bang theory and
are not excluded by experiment, there is a chance that people might take you
seriously. At least they will understand what you're trying to say.

The neat thing about my explanation, though, is that it fits the data.

I'm sorry, but this is almost certainly just wishful thinking. It may fit
the data on whose basis you originally formulated it. But there is a lot
more data than you realize.

Read through Ned Wright's cosmology pages:

http://www.astro.ucla.edu/~wright/cosmolog.htm

They're full of charts showing the agreement of various cosmological
theories with the data. How confident are you that you can match all of
those data points?

Ned Wright's pages are, incidentally, the most accurate popular introduction
to big bang cosmology that I've ever seen. This is a great place to learn
more about thine enemy.

-- Ben

Ben Rudiak-Gould wrote:
Spoonfed wrote:
Actually, my idea is that k=0 and a(t)=1.

What does t denote here? Are you saying that a is a constant function of
time, i.e. that the universe is not expanding? Or are you saying that
a(now) = 1? The latter is not a hypothesis, just a normalization convention.

I'm not convinced that you "speak the language" yet; it looks like you're
just copying stuff from recent posts by Tom Roberts without understanding it.

These are terms from the FLRW metric.

I'd go farther than this and say that they only have meaning in the context
of the FLRW metric, i.e. in the context of big bang cosmology. If you're not
talking about the big bang theory, I don't even understand what you mean by
saying that k=0.

As far as the cosmological constant goes, I don't even
know what it is, let alone what value it might have.

You can ignore it for the time being; conceptually speaking, it's a detail.



Ben, you successfully identified my model as an Omega = 0 model. To
first order, it matches the diagram in Ned Wright's Cosmology page
http://www.astro.ucla.edu/~wright/cosmo_02.htm

I disagree with his definition of the word "now" as using the event of
a distant galaxy reaching 13.7 billion years as a definition of our
"now" is completely at odds with Einstein's methods of defining
simultaneous events in Special Relativity.

As far as the FLRW metric goes, I meant a(t)=1 as in, it is constant.
If I understood right, Tom told me that the FLRW metric was the family
of solution to some differential equation when you assumed that the
cosmological constant was zero. And yes, I repeated that back to him,
and I believe he confirmed it. So, in that respect, yes, I was simply
copying stuff from his posts without really understanding it. I
understood it well enough to answer your accusations that my model was
an Omega=0 model, to which my answer is guilty, as charged.

But it gets even simpler than that. Not only do I assume that Omega =
0, but I also assume that of the many possible solutions available in
the family of FLRW metrics, I am choosing the very most simple one.

If you take the FLRW general metric

ds^2 = dt^2 - a(t)(dr^2/(sqrt(1-k r^2)) + r^2
(d(theta)^2+sin^2(theta)d(phi)^2))

and set a(t)=1 and k=0, this becomes, (unless I've made a horrible
blunder)

ds^2 = dt^2 - dx^2 -dy^2 - dz^2

which is the definition of the differential space-time interval between
two differentially separated events.

Now, everything that I have been bringing up, which you have said I
should preface by saying "in my model" is based on this very simplest
possible metric which should, in my opinion, be the most well explored,
well known model of space-time. When I bring up my answers, the
experts should say, "Oh yes, of course, in the trivial, simple model
where Omega is equal to one and the scale factor is constant, and the
curvature, k is zero, of course that would be true, but we live in a
much more complex universe than that."

Ben, I congratulate you on doing just this. You came right out and
told me that I had an Omega=0 model, and after looking into it a little
deeper, I find that you are right.

By accelerating toward a receding object, but not matching pace with
it, we enter a frame of reference where the space between us and the
receding object is length uncontracted. It will be moving away more
slowly, but also more distant. In this way, the distance will be
greater than you would expect from its velocity. Likewise, the
faintness would be more than you would expect from its redshift.

I think you should stop talking about frames of reference and phrase things
in terms of what we can actually see, which is a 2D projection of our past
light cone. In particular, how should we define the distance of the
astronomical objects that we can see?

Here's an SR conceptual question which may be pertinent. At one end of Main
Street is a clock tower. Alice is running along Main Street toward the clock
tower at a relativistic speed. Bob is standing stationary on Main Street,
looking at the clock tower. At the moment Alice passes Bob, they compare the
times they see on the clock face. Does Alice see an earlier time, a later
time, or the same time?

Alice and Bob see the same moment on the clock face. However, Alice
sees the clock-face further away, and measures that the event happened
longer ago than Bob measures it to have occurred.

I attempted to make a demo of this phenomenon here
http://www.spoonfedrelativity.com/files/timetravel.swf

realized it was pretty badly written and tried to do it again he
http://www.spoonfedrelativity.com/files/newYears2.swf

Unfortunately it is still pretty bad, and probably doesn't get the
point across. The main issue is that the images are observed (by the
moving Speedy T and by the stationary Green Clark) at the centers of
the light spheres.

(Thanks for the set-up there, Ben. Really nice when somebody tosses a
question to me that I've got a demo for.)


Imagine at the dawn of the universe, we were being pushed HARD from
below by that hot part of the CMBR.

"Below"? Are you saying that the universe was not isotropic? What was the
distribution of matter? Is it still anisotropic in the present era?


On further research, I find that the direction I called "Below" is more
commonly known as Galactic North. Roughly 13 hours Right Ascension, 27
degrees, declination. Virgo Supercluster is 12 h 30 m, RA, 12 degrees
declination, where the Sandage Team measured Hubble's Constant at 57
km/sec/MPc. Ned Wright's page says there is a large excess of bright
galaxies in the "northern part of the sky" which I can only guess means
galactic north. This is the direction that I called "down" earlier.

And YES, my theory says the distribution of matter is anisotropic in
the present era--at least the parts of it we can see. The dark areas,
I believe, are still isotropic--undisturbed from the original
explosion.



Starting at around this point I can barely understand at all what you're
trying to say. I seriously have trouble distinguishing it from schizophrenic
raving, and I would dismiss it without a second glance if your relativity
tutorials didn't show obvious evidence of sanity. If you're going to make
this theory comprehensible to anybody, you're going to have to put a lot of
effort into clearing up the exposition. The first step in doing this is to
learn the current dominant theory of cosmology, and how to extract simple
predictions from it. Then you can describe how your theory differs from
that. For example, are you aware that the big bang theory predicts that
beyond a certain redshift, galaxies which are *farther* away will appear
*larger* in the sky? I assume your theory does not match this prediction.
This does not necessarily exclude your theory, because I don't know whether
this prediction of the big bang theory has been directly verified.

I really appreciate the extra time you are taking to give it a second
glance. I am developing it further, and perhaps it will become clearer
as I fill in more gaps, both in my explanation, and my understanding of
the standard model. For instance, I do see that gravitational lensing
actually happens, but I have my doubts that gravity can effect the
redshift of passing photons.

As my model does nothing to the scale factor of space, I would say that
distant galaxies should not appear larger than nearby ones.

If you
make clear predictions like this which differ from the big bang theory and
are not excluded by experiment, there is a chance that people might take you
seriously. At least they will understand what you're trying to say.

The simplest difference I know of is that I predict that a 600km/second
change in velocity would not significantly effect a measurement of the
CMBR dipole. This is very much at odds with the explanation for the
dipole given by NASA.


The neat thing about my explanation, though, is that it fits the data.

I'm sorry, but this is almost certainly just wishful thinking. It may fit
the data on whose basis you originally formulated it. But there is a lot
more data than you realize.

It's hopeful thinking. My prejudiced eyes see confirmation everywhere
I look.


Read through Ned Wright's cosmology pages:

http://www.astro.ucla.edu/~wright/cosmolog.htm

They're full of charts showing the agreement of various cosmological
theories with the data. How confident are you that you can match all of
those data points?

Ned Wright's pages are, incidentally, the most accurate popular introduction
to big bang cosmology that I've ever seen. This is a great place to learn
more about thine enemy.

-- Ben

Well, it's interesting, but I lose him when on page:

http://www.astro.ucla.edu/~wright/cosmo_02.htm

I lose him when he defines D_now as any event on the same hyperbola
instead of on the horizontal plane. That would be fine if he just said
"interesting idea" and moved on, but he appears to use it throughout
the rest of the tutorial as though it were the actual distance. Is he
correcting for this error in judgment when he introduces the scale
factor?

Jonathan Doolin

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.

it's - its
anomolies - anomalies
data suggests - data suggest
disentegrated - disintegrated

Spoonfed wrote:
Ben, you successfully identified my model as an Omega = 0 model. To
first order, it matches the diagram in Ned Wright's Cosmology page
http://www.astro.ucla.edu/~wright/cosmo_02.htm

I'm glad to hear this, since it means that I might just understand it after all.

I disagree with his definition of the word "now" as using the event of
a distant galaxy reaching 13.7 billion years as a definition of our
"now" is completely at odds with Einstein's methods of defining
simultaneous events in Special Relativity.

This is the crux of the matter right here. The only point of talking about
simultaneous events in special relativity is to relate it to the Newtonian
worldview, where simultaneity is taken for granted. There are not multiple
notions of distant simultaneity in relativity -- there is *no notion of
distant simultaneity at all*.

I come back to this several times below.

As far as the FLRW metric goes, I meant a(t)=1 as in, it is constant.
If I understood right, Tom told me that the FLRW metric was the family
of solution to some differential equation when you assumed that the
cosmological constant was zero.

Yes, your function a(t) has to satisfy the equations given here, which work
even when Lambda =/= 0:

http://en.wikipedia.org/wiki/Friedmann_equation

But it gets even simpler than that. Not only do I assume that Omega =
0, but I also assume that of the many possible solutions available in
the family of FLRW metrics, I am choosing the very most simple one.

Then you are definitely wrong! FLRW cosmology is well understood. It has a
few adjustible parameters, which are constrained by astronomical
observations. If your theory is a particular parameterized version of FLRW,
then there cannot be anything new about it. Either it's excluded by the
evidence, or it's identical with the currently accepted big bang model.

If you take the FLRW general metric

ds^2 = dt^2 - a(t)(dr^2/(sqrt(1-k r^2)) + r^2
(d(theta)^2+sin^2(theta)d(phi)^2))

and set a(t)=1 and k=0, this becomes, (unless I've made a horrible
blunder)

ds^2 = dt^2 - dx^2 -dy^2 - dz^2

which is the definition of the differential space-time interval between
two differentially separated events.

But your model then violates one of the assumptions behind the FLRW
solution, namely that rho and p depend only on t, not on x, y, or z. In your
model rho is nonzero inside an expanding sphere and zero outside it.

There's a second flat FLRW solution, which you get by taking k = -1 and a(t)
= t. It is a different coordinate cover of the same (flat, SR) spacetime.
With respect to those coordinates, your rho and p *do* only depend on t (if,
as always, I understand your idea correctly).

Here's an SR conceptual question which may be pertinent. At one end of Main
Street is a clock tower. Alice is running along Main Street toward the clock
tower at a relativistic speed. Bob is standing stationary on Main Street,
looking at the clock tower. At the moment Alice passes Bob, they compare the
times they see on the clock face. Does Alice see an earlier time, a later
time, or the same time?

Alice and Bob see the same moment on the clock face. However, Alice
sees the clock-face further away, and measures that the event happened
longer ago than Bob measures it to have occurred.

I agree with the first sentence, but the second is iffy. Again, this is the
crux of the matter. What you see is physically real, but these inferences
about distance and time are to a large extent arbitrary artifacts of one's
choice of coordinates. I don't think you understand this yet. I didn't
really understand it until I took GR.

It is not by a conspiracy of length contraction and time dilation that Alice
and Bob see the same moment on the clock face. It is simply because they are
both detecting photons *locally*; they are in the same place, so they
necessarily detect the same photons. Drawing conclusions about the origin of
those photons (e.g. reflection off a clock face) is a very complicated
business. Our innate sense of distance, which is based on binocular vision
and atmospheric scattering and the known size of familiar objects and other
such cues, does not work well in the relativistic domain.

Ned Wright's page says there is a large excess of bright
galaxies in the "northern part of the sky" which I can only guess means
galactic north. This is the direction that I called "down" earlier.

What he says is, "Hubble [...] found approximately the same number of faint
galaxies in all directions, even though there is a large excess of bright
galaxies in the Northern part of the sky." What this means is that the
universe is anisotropic on a small scale, but isotropic on a large scale.
The local anisotropy around the Milky Way is typical of the local anisotropy
one would see from anywhere else.

I'm not sure what he means by "northern", but it may well be terrestrial
north rather than galactic north.

And YES, my theory says the distribution of matter is anisotropic in
the present era--at least the parts of it we can see. The dark areas,
I believe, are still isotropic--undisturbed from the original
explosion.

Imagine for the moment that our present worldline pointed straight back to
the big bang. Would the universe then appear isotropic to us at large
scales, in your model? This is a physically meaningful question, so it
doesn't depend on coordinates -- you're free to analyze it with respect to
SR inertial coordinates. Your first impression might be that it won't appear
isotropic if we're near the edge of the expanding sphere, but if I
understand your theory, a careful analysis will show that the universe will
appear isotropic no matter where we are. Our motion with respect to the CMBR
cannot change this -- see below about the 600km/sec boost.

I do see that gravitational lensing
actually happens, but I have my doubts that gravity can effect the
redshift of passing photons.

Considering those two different coordinate covers of flat space may help. In
one, the redshift is explained by the SR formula. In the other, it's
explained by the change in the scale factor between emission and absorption.
This equivalence is a mathematical fact which doesn't depend on any
additional physical hypothesis. Einstein made the additional physical
hypothesis that every gravitational effect can be understood in the same
way, and he seems to have been right.

(I shouldn't really say this, because there is a coordinate-independent
sense in which gravitational fields do exist.)

As my model does nothing to the scale factor of space, I would say that
distant galaxies should not appear larger than nearby ones.

Actually I've changed my mind: I'm pretty sure I was wrong, and your theory
does predict that distant galaxies appear larger. :-) This is easier to see
if you use the FLRW coordinates, but since it's a physically real
prediction, you can in principle analyze it from SR inertial coordinates as
well.

The simplest difference I know of is that I predict that a 600km/second
change in velocity would not significantly effect a measurement of the
CMBR dipole. This is very much at odds with the explanation for the
dipole given by NASA.

But that's not even consistent with SR, let alone GR or the big bang theory.
A 600km/sec boost leads to Doppler shift and aberration *of your visual
field* which is completely independent of where that light originally came
from. The effect of a 600km/sec boost on the CMBR dipole is independent of
any cosmological assumptions. It only depends on local Lorentz symmetry.

http://www.astro.ucla.edu/~wright/cosmo_02.htm

I lose him when he defines D_now as any event on the same hyperbola
instead of on the horizontal plane. That would be fine if he just said
"interesting idea" and moved on, but he appears to use it throughout
the rest of the tutorial as though it were the actual distance. Is he
correcting for this error in judgment when he introduces the scale
factor?

Crux of the matter again. :-) It's not an error in judgment.

-- Ben

Ben Rudiak-Gould wrote:
Spoonfed wrote:
Ben, you successfully identified my model as an Omega = 0 model. To
first order, it matches the diagram in Ned Wright's Cosmology page
http://www.astro.ucla.edu/~wright/cosmo_02.htm

I'm glad to hear this, since it means that I might just understand it after all.

I disagree with his definition of the word "now" as using the event of
a distant galaxy reaching 13.7 billion years as a definition of our
"now" is completely at odds with Einstein's methods of defining
simultaneous events in Special Relativity.

This is the crux of the matter right here. The only point of talking about
simultaneous events in special relativity is to relate it to the Newtonian
worldview, where simultaneity is taken for granted. There are not multiple
notions of distant simultaneity in relativity -- there is *no notion of
distant simultaneity at all*.

I come back to this several times below.

As far as the FLRW metric goes, I meant a(t)=1 as in, it is constant.
If I understood right, Tom told me that the FLRW metric was the family
of solution to some differential equation when you assumed that the
cosmological constant was zero.

Yes, your function a(t) has to satisfy the equations given here, which work
even when Lambda =/= 0:

http://en.wikipedia.org/wiki/Friedmann_equation


I count TWO notions of time in relativity, and thus two notions of
distant simultaneity. You and Friedmann have been using "proper time"
and I've been using "coordinate time"

The equation given at http://en.wikipedia.org/wiki/Friedmann_equation
yields H^2 ~= 1/t^2 if we set a(t)=t and k=-1.

If we are talking about coordinate time, though, I believe the
space-time interval between differentially separated events is well
represented by

ds^2 = dt^2 - dx^2 - dy^2 -dz^2 which is achieved by using a(t)=1 and
k=0.

If we are talking about proper time, then we can use the a(t)=t and
k=-1.

I hope you are exaggerating when you say "there is no notion of distant
simultaneity at all" because I can see no way to have any discussion of
this topic at all without some notion of simultaneity.

But it gets even simpler than that. Not only do I assume that Omega =
0, but I also assume that of the many possible solutions available in
the family of FLRW metrics, I am choosing the very most simple one.

Then you are definitely wrong! FLRW cosmology is well understood. It has a
few adjustible parameters, which are constrained by astronomical
observations. If your theory is a particular parameterized version of FLRW,
then there cannot be anything new about it. Either it's excluded by the
evidence, or it's identical with the currently accepted big bang model.


That's what I'm trying to find out. I am holding out hope that it is
identical, but I've been told to preface every post with "This is my
own personal theory" which leads me to believe there must be some
difference. Most likely, it is just that "I don't speak the language
yet"

If you take the FLRW general metric

ds^2 = dt^2 - a(t)(dr^2/(sqrt(1-k r^2)) + r^2
(d(theta)^2+sin^2(theta)d(phi)^2))

and set a(t)=1 and k=0, this becomes, (unless I've made a horrible
blunder)

ds^2 = dt^2 - dx^2 -dy^2 - dz^2

which is the definition of the differential space-time interval between
two differentially separated events.

But your model then violates one of the assumptions behind the FLRW
solution, namely that rho and p depend only on t, not on x, y, or z. In your
model rho is nonzero inside an expanding sphere and zero outside it.


Basically correct. The density goes up toward infinity toward the edge
of the sphere, and is unknown outside it. But, I point out once again,
I am using coordinate time, and it appears to me that the FLRW solution
uses proper time.

There's a second flat FLRW solution, which you get by taking k = -1 and a(t)
= t. It is a different coordinate cover of the same (flat, SR) spacetime.
With respect to those coordinates, your rho and p *do* only depend on t (if,
as always, I understand your idea correctly).

Here's an SR conceptual question which may be pertinent. At one end of Main
Street is a clock tower. Alice is running along Main Street toward the clock
tower at a relativistic speed. Bob is standing stationary on Main Street,
looking at the clock tower. At the moment Alice passes Bob, they compare the
times they see on the clock face. Does Alice see an earlier time, a later
time, or the same time?

Alice and Bob see the same moment on the clock face. However, Alice
sees the clock-face further away, and measures that the event happened
longer ago than Bob measures it to have occurred.

I agree with the first sentence, but the second is iffy. Again, this is the
crux of the matter. What you see is physically real, but these inferences
about distance and time are to a large extent arbitrary artifacts of one's
choice of coordinates. I don't think you understand this yet. I didn't
really understand it until I took GR.


If you use proper time, I can see how inferences about distance and
time are to a large extent arbitrary.

We can really only tell the approximate distance to galaxies where they
were in our reference frame when they emitted the light that is
arriving now, then we can estimate where they are "now" and choose
between "coordinate time" now or "proper time" now.

To find their D_Now using "proper time" is to guess at where these
galaxies will appear to be when they reach a proper age of 13.7 billion
years. To find their coordinate time D_Now, take the observed
distance, divide by the speed of light, and multiply by their current
velocity and add to their observed distance.

(Coordinate time) D_Now = D_obs+(D_obs/c)*v_obs

Yes, this is based on our "choice" of coordinates, and in particular
our "choice" of reference frame. Our choice is not at all arbitrary,
however. It is extremely limited until we discover some method of
interstellar travel.

It is not by a conspiracy of length contraction and time dilation that Alice
and Bob see the same moment on the clock face. It is simply because they are
both detecting photons *locally*; they are in the same place, so they
necessarily detect the same photons. Drawing conclusions about the origin of
those photons (e.g. reflection off a clock face) is a very complicated
business. Our innate sense of distance, which is based on binocular vision
and atmospheric scattering and the known size of familiar objects and other
such cues, does not work well in the relativistic domain.


At first, it may seem like a conspiracy, but it is not a conspiracy.
The Lorentz transformation represents the only possibility that
maintains all lightcones, and all collisions while allowing changes in
velocity. And it does this very, very elegantly, taking care of all of
your "complicated business" of drawing conclusions about the origin of
those photons.

From all frames, events are seen to have happened at the center of the
light-cone produced by them. By predicting the space and time
coordinates of the event in the new reference frame, the Lorentz
transformation predicts the appropriate size, distance, and parellax
for binocular viewing.

[Snip]
-I was wrong about Bright stars in galactic north, sorry about that.
Yes, I think you are right. Polaris is nowhere near galactic north,
IIRC they are about 60 degrees apart. I feel a little foolish about
that. Especially since all of the data I've looked at since I wrote
that seems to indicate more acceleration in the opposite direction
(towards galactic north instead of galactic south)


And YES, my theory says the distribution of matter is anisotropic in
the present era--at least the parts of it we can see. The dark areas,
I believe, are still isotropic--undisturbed from the original
explosion.

Imagine for the moment that our present worldline pointed straight back to
the big bang. Would the universe then appear isotropic to us at large
scales, in your model? This is a physically meaningful question, so it
doesn't depend on coordinates -- you're free to analyze it with respect to
SR inertial coordinates. Your first impression might be that it won't appear
isotropic if we're near the edge of the expanding sphere, but if I
understand your theory, a careful analysis will show that the universe will
appear isotropic no matter where we are. Our motion with respect to the CMBR
cannot change this -- see below about the 600km/sec boost.


If our galaxy's worldline (tangent vector) points straight back to the
big bang event, then the universe should appear to be completely
isotropic. If we have accelerated a LOT since the big bang, then the
tangent vector would not point directly toward the big bang event and
we should be able to observe some form of anisotropy. In your view,
600km/sec is enough to account for this anisotropy. In my view, it is
not--(continued later)

I do see that gravitational lensing
actually happens, but I have my doubts that gravity can effect the
redshift of passing photons.

Considering those two different coordinate covers of flat space may help. In
one, the redshift is explained by the SR formula. In the other, it's
explained by the change in the scale factor between emission and absorption.
This equivalence is a mathematical fact which doesn't depend on any
additional physical hypothesis. Einstein made the additional physical
hypothesis that every gravitational effect can be understood in the same
way, and he seems to have been right.

(I shouldn't really say this, because there is a coordinate-independent
sense in which gravitational fields do exist.)


I don't understand this idea of equivalence. It seems to me redshift
must either be explained by the SR formula or the change in the scale<