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

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
factor. If putting the redshift effect into the scale factor makes the
math easier, this should be described explicitly as a mathematical
shortcut for calculation purposes.

I should add that in my concept, the redhift of distant galaxies is
almost 100% accounted for by their recession velocities. (There would
also be a slight redshift due to the gravitational potential difference
between the surface of the star and the surface of the earth.)

http://scienceworld.wolfram.com/phys...cRedshift.html

Except that I would stop at equation (2)

z=sqrt((1+beta)/(1-beta))-1

since the assumption of v/c1 comes from NOWHERE!


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.


What phenomenon are you expecting to make distant objects seem larger?

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

Redshift is not independent of the speed of its source.

z=sqrt((1+beta)/(1-beta))-1

I cannot speak for the standard model, but in my model, the Cosmic
Background Radiation is coming from an almost solid receding wall of
plasma--specifically, the light of matter as it gets cool enough where
electrons have a low enough energy to form electrical bonds with
protons and form atoms.

After this point in time, the atoms will not glow again unless they are
pushed or pulled together to create stars.

The light, having come from very similar events across the universe,
would be at the same temperature due to the nature of the event. Since
we see a difference in the temperature, from galactic north to galactic
south, it indicates (1) that one side is closer to us than the other or
(2) that one side is moving faster than the other or (3) both.

I am going with (1) now because it is easier to explain than (2): The
times here are just to explain a concept--not taken from any actual
data.

Imagine that we are in a sphere that is slowly cooling as time goes by.
One side of this sphere is 6 billion light years away, and the other
is say, 16 billion light years away, (just pulling this number out of a
hat)

Six billion years later the light from the near side reaches us. It is
very hot. Another ten billion years pass, and the light from the far
side reaches us. It is much hotter than the light from the far side
which has aged another 10 billion years.



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

So what I saw as a lack of good judgment is simply a lack of clarity in
his definition of universal time.

The crux of the matter is whether we decide to define simultaneity in
terms of proper time, in which case, "there is *no notion of distant
simultaneity at all*." or we can use coordinate time, which does
contain a fairly rigid definition of distant simultaneity for any
particular observer at any given event.

As for these other issues, I am not entirely clear on what the standard
model says about them. If the standard model agrees with mine on these
issues, then it is the same theory, and it was only a matter of
miscommunication.

(1) The CBR coming from a nearly solid receding wall of plasma.

(2) Redshift of distant galaxies is determined by the equation
z=sqrt((1+beta)/(1-beta))-1 where beta=v/c


Thanks,
Jonathan Doolin

Search for article: "Spoonfed Big Bang Cosmology Model, Take 2"
Posted September 7, 2005.

Sorry for the late reply. I've been very busy.

Spoonfed wrote:
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"

No, the t in a(t) is a coordinate time; they're just different coordinates.

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.

There's no physically meaningful notion of distant simultaneity. If you
choose a coordinate system, you get a notion of distant simultaneity (equal
coordinate time), but it's physically meaningless.

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.

You simply must unlearn this stuff about the Lorentz transformation in order
to understand general relativity. It is because you have this idea of the
Lorentz transformation operating globally on the universe that you're having
trouble with GR, and with abandoning distant simultaneity. See below about
Alice and Bob.

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
factor. If putting the redshift effect into the scale factor makes the
math easier, this should be described explicitly as a mathematical
shortcut for calculation purposes.

That would be similar to describing the primed coordinates in the Lorentz
transformation as just a mathematical shortcut to make the calculation more
convenient, which is what Lorentz did. It's better to treat the unprimed and
primed coordinates as equally valid. That's also true of other coordinate
systems that aren't related by a global Lorentz transformation.

What phenomenon are you expecting to make distant objects seem larger?

See this message:

http://groups.google.com/group/sci.p...2984b2c8511d91

This is for a(t)=t and k=0, but I think it applies to your model also. (But
keep in mind that I'm less sure of this than of the other stuff in this
discussion.)

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.

Redshift is not independent of the speed of its source.

Indeed not, but what I said is correct. If Alice and Bob are at the same
place at a particular moment, both moving inertially, and you know what
Alice sees, and Alice and Bob's relative velocities, that is enough to
determine what Bob sees. Information about the objects that originally
produced the light is unnecessary. This is true in SR -- nothing to do with
GR -- but it remains true in GR, and I think that once you understand it you
will understand GR better.

Restricting ourselves to SR, consider the following two ways of working out
what Bob sees, in terms of what Alice sees:

* In terms of Bob's rest frame: the whole universe is
Lorentz-transformed from Alice's rest frame; distances, times,
redshifts, etc. change, so Bob sees different things.

* In terms of Alice's rest frame: the universe is not
Lorentz-transformed, but Bob is Lorentz-transformed from his
own rest frame: his eyes are distorted, causing him to see
different things than Alice sees.

You should be able to see that these two approaches yield equivalent
predictions: they are, after all, Lorentz transformations of each other.

Most SR courses only teach the first (global) approach. But only the second
(local) approach works in GR.

-- Ben

Ben Rudiak-Gould wrote:
Sorry for the late reply. I've been very busy.

Spoonfed wrote:
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"

No, the t in a(t) is a coordinate time; they're just different coordinates.


First off, I will need to know where to find a derivation of this:
http://en.wikipedia.org/wiki/Friedmann_equation

Second,
If you have sound and a good bandwidth on your computer, have a look at
this
http://www.spoonfedrelativity.com/movies/Plot6to9.htm

If you mapped a region of constant "space-time interval" as I call it
in the demo, or constant "proper time" as I've been calling it here,
you would get a hyperbola. If you mapped a region of constant
coordinate time, you would get a straight line.

I used the terms "proper time" vs. "coordinate time" as defined in
Lewis Carroll Epstein's Relativity Visualized. Maybe they are not in
common usage.

Coordinate time goes along with coordinate space, with all the
(x,y,z,t) defined events that go along with them. Proper time goes
along with individual particles describing how much they have aged.

If you looked at the universe from the perspective of a given particle
at a given age, it should look the same as the universe from the
prespective of any other particle at the SAME age.

Specifically, each particle should observe itself to be at the center
of a sphere that looks (in cross-section) like this:
http://www.spoonfedrelativity.com/files/250%20plus.JPG

The circle should have a radius proportional to the age of the particle
(since the initial event), and all of the particles should be moving
outward at a speed of v=H*d, where H=1/s where s is the age of the
particle, AKA the proper time of the particle, AKA the spacetime
interval.

By this logic, the universe looks the same FROM all places in the
universe. Expanding at an equal speed. But we don't look at the
universe FROM all places, we look at the universe from HERE. And from
HERE, the universe appears to increase in density towards infinity
toward the edges. Just as it does FROM everywhere else.

Anyway, I just want to see the derivation of the Freidmann equation,
and I thought you might have a good reference.

Spoonfed wrote:
First off, I will need to know where to find a derivation of this:
http://en.wikipedia.org/wiki/Friedmann_equation

I imagine it's in most textbooks. I'm not sure about online. I can outline
it here.

The assumptions leading to the Friedmann equations a

* Spacetime is described by the FRW metric.

* The stress-energy tensor is that of a homogeneous isotropic perfect
fluid, which I seem to recall is given in FRW coordinates by

/rho 0 0 0\
| 0 p 0 0|
| 0 0 p 0|
\ 0 0 0 p/

where rho is the energy density and p is the pressure. (Both are
functions of the FRW coordinate t.)

* The GR field equations hold.

The Friedmann equations are the extra constraints needed to make all of
these assumptions consistent. You get them by working out the Einstein
tensor G_uv from the FRW metric (which is straightforward but tedious --
just tons of differentiation), and setting it equal to 8 pi G times the
stress-energy tensor given above.

Second, If you have sound and a good bandwidth on your computer, have a
look at this
http://www.spoonfedrelativity.com/movies/Plot6to9.htm

Yes, I think it's neat. Keep up the good work. But in order to understand GR
you have to realize that the Lorentz transformation is valid only locally.

I used the terms "proper time" vs. "coordinate time" as defined in
Lewis Carroll Epstein's Relativity Visualized. Maybe they are not in
common usage.

Well, his approach to special relativity is quite different from everyone
else's, as I'm sure you know. I'm pretty sure his definition of proper time
is the same as mine. When I say "coordinate time" I simply mean the value of
the time coordinate of some event, with respect to some agreed-upon
coordinate system.

Coordinate time goes along with coordinate space, with all the
(x,y,z,t) defined events that go along with them. Proper time goes
along with individual particles describing how much they have aged.

Right.

If you looked at the universe from the perspective of a given particle
at a given age, it should look the same as the universe from the
prespective of any other particle at the SAME age.

Under certain symmetry assumptions, yes.

Specifically, each particle should observe itself to be at the center
of a sphere that looks (in cross-section) like this:
http://www.spoonfedrelativity.com/files/250%20plus.JPG

I understand where this picture comes from, mathematically. But you are
attaching to it far more physical significance than it deserves.

(Pseudo-)Riemannian manifolds are not like (pseudo-)metric spaces. Metric
spaces are nonlocal: you plug any two points in the space into the distance
function, and wham, you get a distance between them. Riemannian manifolds
are local: From a point, you can only figure out the distance to nearby
points. You can't jump from here to there; you have to move continuously
from here to there. Your notion of spacetime interval is s^2 = t^2 - x^2,
but there's no such thing on a Riemannian manifold. There's only ds^2 = dt^2
- dx^2.

-- Ben

Ben Rudiak-Gould wrote:
Spoonfed wrote:
First off, I will need to know where to find a derivation of this:
http://en.wikipedia.org/wiki/Friedmann_equation

I imagine it's in most textbooks. I'm not sure about online. I can outline
it here.

The assumptions leading to the Friedmann equations a

* Spacetime is described by the FRW metric.

* The stress-energy tensor is that of a homogeneous isotropic perfect
fluid, which I seem to recall is given in FRW coordinates by

/rho 0 0 0\
| 0 p 0 0|
| 0 0 p 0|
\ 0 0 0 p/

where rho is the energy density and p is the pressure. (Both are
functions of the FRW coordinate t.)

* The GR field equations hold.

The Friedmann equations are the extra constraints needed to make all of
these assumptions consistent. You get them by working out the Einstein
tensor G_uv from the FRW metric (which is straightforward but tedious --
just tons of differentiation), and setting it equal to 8 pi G times the
stress-energy tensor given above.


It seems like there was some derivation along these lines in Schuetz's
book. I will try to go back to it again to look. He presented a model
of "dust" representing a infinite, static and homogeneous distribution
of particles throughout space, but I couldn't remember him actually
making the assumption that it represented the universe. To me, it
seemed like one possible distribution, and I skimmed through the rest
of the book looking around for a more believable non-static
distribution, but it seemed like he wasn't going to get around to
presenting another example.

Perhaps I'll have a chance to go review it again, now that I have some
better idea how the static homogeneous distribution in the FRW metric
with k=-1, a(t)=t is the same as a relativistically expanding
lobachevskian distribution in Euclidian space.

(Maybe it will be back at the library next time I check.)

Second, If you have sound and a good bandwidth on your computer, have a
look at this
http://www.spoonfedrelativity.com/movies/Plot6to9.htm

Yes, I think it's neat. Keep up the good work. But in order to understand GR
you have to realize that the Lorentz transformation is valid only locally.

I used the terms "proper time" vs. "coordinate time" as defined in
Lewis Carroll Epstein's Relativity Visualized. Maybe they are not in
common usage.

Well, his approach to special relativity is quite different from everyone
else's, as I'm sure you know. I'm pretty sure his definition of proper time
is the same as mine. When I say "coordinate time" I simply mean the value of
the time coordinate of some event, with respect to some agreed-upon
coordinate system.


That's the same as my definition.

I think Epstien gets into trouble for calling the business about curved
space "hocus-pocus." To me, his exile from the ranks of respected
authors on Relativity appears to be more a matter of politics than
substance. Here are a couple quotes from Relativity Visualized.

"Proper means the measure of a thing as perceived by an agent not in
motion relative to the thing being measured. If you ride on a ship,
you measure its proper length. If you measure the ship's length as it
flies past you, you don't measure it's proper length."

And another quote from Epstein:
"The spacetime diagram in this book represents the speed of light as a
horizontal line. The spacetime diagram in many other books represents
the speed of light as a sloped 45 degree line. How come? Because the
diagram in this book plots proper time against space. The diagram in
the other books plots coordinate time against space. Which is right?
Both are. They are different views of the same thing"

Coordinate time goes along with coordinate space, with all the
(x,y,z,t) defined events that go along with them. Proper time goes
along with individual particles describing how much they have aged.

Right.

If you looked at the universe from the perspective of a given particle
at a given age, it should look the same as the universe from the
prespective of any other particle at the SAME age.

Under certain symmetry assumptions, yes.


Good point.

Specifically, each particle should observe itself to be at the center
of a sphere that looks (in cross-section) like this:
http://www.spoonfedrelativity.com/files/250%20plus.JPG

I understand where this picture comes from, mathematically. But you are
attaching to it far more physical significance than it deserves.

(Pseudo-)Riemannian manifolds are not like (pseudo-)metric spaces. Metric
spaces are nonlocal: you plug any two points in the space into the distance
function, and wham, you get a distance between them. Riemannian manifolds
are local: From a point, you can only figure out the distance to nearby
points. You can't jump from here to there; you have to move continuously
from here to there. Your notion of spacetime interval is s^2 = t^2 - x^2,
but there's no such thing on a Riemannian manifold. There's only ds^2 = dt^2
- dx^2.

-- Ben

Also a good point; I am considering events which are considerably more
than differentially separated. I am assuming that for a first order
approximation, gravitational effects can be neglected, and that we have
several choices in how to perform a path integral to calculate s.

You can choose to take the path integral either along hyperbolic-curved
lines, between differentially separated events of the same proper-time,
which is what you and Baez appear (to me) to be doing. Epstein does
this at well, but he makes it a lot more explicit.

.... or you can take the path integral along horizontal straight lines,
between differentially separated events of the same coordinate time
(for a particular observer), which is what I have been suggesting.

A third, most appropriate option, given our observational limitations,
would be to take the path integral along 45-degree-downward straight
lines representing differentially separated events along the
past-light-cone. This would represent the locus of events in the
universe which we currently see, because the light would be currently
reaching us.