Galaxies expanding with space? The Space Stretch
Nick
or space is stretching out inbetween them.
What does the space stretch do to the
space's geometry?
Space stretch geometry?
-- Light Falls --
George
its not stretching , rather expanding,so geometry remains as it is
T Wake
"Nick" <macro...@yahoo.com> wrote in message
news:1123653046.0...@o13g2000cwo.googlegroups.com...
> Either the galaxies are moving through space
> or space is stretching out inbetween them.
Didnt you like the answers you had last time.
> What does the space stretch do to the
> space's geometry?
Ask a new question.
> Space stretch geometry?
Why do you think this is wrong?
Nick
Growing inbetween?
Looks like stretching to me!!!
T Wake
> Expanding inbetween?
>
> Growing inbetween?
>
> Looks like stretching to me!!!
>
Can you see it? Which way is it stretching?
Nick
T Wake
> what is the geometry of stretching space?
>
What do you mean by "stretching space?"
IIRC the expansion of space was not a stretch, more an increase in the
distance between large scale structures.
Puppet_Sock
And, once more, in order to understand what the
current best answers are to such questions, you
would need to understand the basics of general
relativity. Have you done this?
Socks
Nick
Just put up *the answer* or shut up!!
What's the stretch?
Geometry and dimension...
Puppet_Sock
And, still again, in order to understand the answer,
you would have to understand the basics of general
relativity. Have you done this?
See, I'm not about to try to teach you what typically
takes a full eight months of university effort, usually
at the fourth year level. There really is no point in
talking to you about cosmological solutions to the
general relativity equations if you don't know what
the GR equations are, what solutions to them mean, or
what a cosmological solution is.
Do you even know what a metric is? Or what a curvature
tensor is? Or what a geodesic path is? Really, it's
hopeless to try to answer your questions if you don't.
You are like a little child demanding to understand
how the television works, but can't accept that it
requires a great deal of understanding of electronics.
Well, if you want more understanding than is available
to a little child, you have to learn more than that
little child.
But I'm not holding out much hope.
Socks
Nick
They don't add
If space inbetween galaxies is getting larger
that space is stretching out.
What is the geometry of stretching space?
xx...@bellsouth.net
xxein: That is your accepted belief. Do I have to explain what a
belief is? Does an aspirin cure a headache or does it just alleviate
the sensation? Show me a metric.
What is voodoo? What is a psychological profile? What is a relative
anything? Show me a metric.
Go to a movie. Did you like it? Show me a metric.
A metric, as mathematical as is you may want it to be, is just a belief
that you will accept. Nothing more.
2+2 is a metric. Is your wife a metric also?
Is something you do not understand a metric? Is everything a you
decide to examine a metric?
Please tell me 'what is a metric'?
George
in-between of "what"?
Puppet_Sock
What is your knowledge of geometry? Geeze, are
you in fact this dense or just a troll?
Look, you are asking about things that require
differential geometry to answer. What do you
know about differential geometry? If you have
had the basics, we can talk. Otherwise, I'm not
going to try to teach you this stuff over a
news group. It would be tough enough to do with
a web page, especially since you don't seem to
think the effort is required.
Socks
T Wake
"Puppet_Sock" <puppe...@hotmail.com> wrote in message
news:1123767071.9...@o13g2000cwo.googlegroups.com...
> Nick wrote:
>> How do dimensions grow?
>> They don't add
>>
>> If space inbetween galaxies is getting larger
>> that space is stretching out.
>>
>> What is the geometry of stretching space?
>
> What is your knowledge of geometry? Geeze, are
> you in fact this dense or just a troll?
He seems to be a bit of both really.
T Wake
That said, Nick, lots of people have asked you questions and you don't seem
to answer - yet you post your... question... repeatedly.
Jim Black
> Nick wrote:
> > How do dimensions grow?
> > They don't add
> >
> > If space inbetween galaxies is getting larger
> > that space is stretching out.
> >
> > What is the geometry of stretching space?
>
> What is your knowledge of geometry? Geeze, are
> you in fact this dense or just a troll?
See:
http://groups.google.com/groups?q=light-speed-acceleration&start=0&scoring=d&hl=en&
Nick
T Wake
> What's the answer to the space stretch socks?
>
Is there any point answering?
Nick
What's the answer to the space stretch?
Spoonfed
Depending on who you talk to. If you talk to me, that's what I would
tell you. I would also tell you that "space stretching" is a misnomer,
but a lot of people don't know that. They use dots on a polka-dot
balloon to show that the center of an expanding two dimensional surface
is not necessarily on the surface, then go on to say that it is the
same with our universe.
The analogy is very vague, and it must break down at some point. Does
the universe wrap around on itself like the surface of the balloon?
Does the center of the universe lie on a vector perpendicular to all
spatial directions? Do the dots on the surface of the balloon expand,
or do they shrink relative to the surface of the balloon, being held
together by gravity and molecular forces? If the dots on the surface
expand, then of course, it would not seem like the universe were
expanding at all, since our meter sticks would expand at the same rate.
Finally there is no cause and effect related to this balloon model. We
don't see some evidence pointing to the balloon and only the balloon.
All we see is that objects in the universe are moving away from us. We
also see that there is a dim all-pervasive black-body radiation called
the CMB. The CMB has been carefully mapped, and resembles an acoustic
signature of a roughly spherical object (for instance, a balloon)... As
long as you don't look to close. An acoustic signature would, of
course, involve molecules bouncing off each other and exchanging energy
across the surface of the sphere, whereas the surface of this sphere
has a radius of at least 13 billion light years.
However, to a number-cruncher, an acoustic signature is an acoustic
signature, and currently the money (whatever tiny amount there is), and
the interest (also pretty small, I think), is in looking at the
anomolies in the acoustic signature, and trying to find local causes
for the lowest frequency elements of the acoustical signature to be
missing.
The issue of these missing frequencies, according to the August issue
of Scientific American "could send us back to the drawing board about
the early universe." I've not been able to understand what is on the
current drawing board, but I do know one thing that is missing: A
proper appreciation for the relativistic effects due to the momentum of
the receding objects.
Why are people ignoring these effects? I gather that people believe
"The galaxies aren't really receding. The space is expanding between
the galaxies making them appear to recede." Which returns us to Nick's
question. What is the geometry of stretching space?
The clearest answer is what T Wake has given. Space is not stretching.
The objects within it are moving apart. However, this is in direct
conflict with the belief that the galaxies are not really receding.
Finally, another issue obfuscates this further. There is more than one
way to define simultaneity in cosmology.
The first way is what I would think is most obvious, to simply take a
photograph, and anything on a surface equidistant from the lens that
shows up in the photo should be considered simultaneous, making some
minor corrections for differences in gravitational potential of the
source image. *totally obvious, right? actually it sounds a lot more
complicated than I thought.*
The second way to define simultaneity is by measurement of any
particular object's proper age, that is the time it would measure
itself aging since, for instance the big bang. It is in this model,
where if you plot positions versus time, you actually get a universe
which is infinite in extent, completely homogeneous, and expanding over
time. This seems to me to be an odd way of defining simultaneity, but
it does make some of the other often repeated statements about
cosmology make more sense.
So there are a lot of arguments caused because there is more than one
context in which to describe the phenomena, analogies are useful in one
context and not in another, and the mathematics are sometimes suggested
without reference to specific problems. The answers are vague, and the
questions are even more vague, and the more people know about the
topic, the less they want to say, because they are more aware of what
they don't know.
This is just the way it seems to me this morning.
Nick
universe where there is no boundary expanding at
the speed of light. Instead the space inbetween
the galaxies is stretching.
To answer my question, the space stretch can be
represented in Riemanian geometry as the expanding
surface of a hypersphere. Curvature would go down
inbetween the galaxies as more distance is created.
Gravity will only get weaker inbetween galaxies.
Spoonfed
B. F. Schutz (Schuetz?) book assumes from the beginning, (if I read it
correctly) that the universe is infinite, homogeneous, and galaxies
appear to be moving apart. Also, Einstein suggested a mathematical
problem making the same assumption, which later Friedmann, Lemaitre,
Robertson, Walker all solved independently with the same solution.
Of course it is possible to develop this mathematics to fit this
assumption. The problem is with the assumption. There is no reason to
assume that the universe should be infinite and homogeneous.
The universe may or may not have an infinite amount of matter, but this
matter should be distributed (roughly) by an equipartition of momentum
model, which results in a Lobachevskian geometry (made familiar from
Escher's Circle Limit art). A relativistically expanding sphere that
becomes (possibly) infinitely dense at the edges.
I simply do not agree that space is stretching, nor that we have any
measurements to suggest that it is. You will find a large list of texts
which disagree with me, and you will also find a rare gem which does
agree with me, but which side are you on?
T Wake
> Space stretching can be represented by a closed
> universe where there is no boundary expanding at
> the speed of light. Instead the space inbetween
> the galaxies is stretching.
You are wrong, and space stretching is your interpretation not mine.
Nick
Did you originate this thread?
Did I mention your interpretation?
I say the surface of a hypersphere.
T Wake
However it is not possible to get others to explain or justify confusion
generated by "your" (lack of) understanding.
macro...@internetcds.com
T Wake
<macro...@internetCDS.com> wrote in message
news:1124092530.9...@g14g2000cwa.googlegroups.com...
> And you've been brainwashed
>
No where near as much as you have.
macro...@internetcds.com
T Wake
<macro...@internetCDS.com> wrote in message
news:1124093267.4...@g43g2000cwa.googlegroups.com...
> But I haven't went to school
>
Well that is fairly obvious.
Jim Black
> I say the surface of a hypersphere.
It'll work for this problem, as long as you bear in mind that it only
represents one of the three possibilities, the others being an infinite
flat universe and an infinite saddle-shaped universe. If you want to
get an idea of what each looks like, you can build a model. You can
approximate a sphere nicely with pentagons and hexagons in the pattern
of a soccer ball. Substitute hexagons for the pentagons, and you get
flat space. Substitute seven-sided heptagons, and you get the
saddle-shaped possibility.
The next step in what it seems you are trying to do is to introduce
time. Special relativity treats time as a dimension, just like length,
width, and height. The main difference between time and space is that
for a right triangle where the legs are distances, the hypotenuse has a
length of sqrt(x^2 + y^2), but if one of the legs is a time, the
hypotenuse's length is sqrt(x^2 - c^2 t^2).
macro...@internetcds.com
Closed universe obeying No boundary proposal
That universe is the surface of a hypersphere.
Jim Black
The others don't have boundaries either, because they're infinite.
If you make your hypersphere big enough, it gets closer and closer to
an infinite flat universe. If you accept a closed universe as a
possibility, you should be okay with the flat case, unless you have
some objection to an infinite universe.
What objection could that be?
macro...@internetcds.com
It had a beining and is expanding at a finite rate.
You can't get infinity out of that.
The only infinity is the future.
T Wake
<macro...@internetCDS.com> wrote in message
news:1124427847....@f14g2000cwb.googlegroups.com...
Do you understand what gibberish this is.
If you had a piece of string, of infinite length, it would have a start
point and as you followed it, its length would expand at a finite rate.
If the universe is finite - where is the boundary?
macro...@internetcds.com
it was orignally intended twake:
If a closed universe is expanding there is no edge.
Instead the space inbetween the galaxies is stretching.
What is interesting is that in cosmology the light
transversing this space is also stretched. It gets
longer and less energetic. This redshift is how we
determine their distances.
If there is no boundary the universe can be seen to
be the surface of a hypersphere.
T Wake
<macro...@internetCDS.com> wrote in message
news:1124504038....@g44g2000cwa.googlegroups.com...
> Let me take the oportunity to use this post for which
> it was orignally intended twake:
Ok, a novel approach for you.....
> If a closed universe is expanding there is no edge.
Well, it depends how you use closed. The general cosmological use for a
"Closed Universe" is one which has a finite amount of expansion possible. It
is still infinite in size.
If the universe is anything but infinite in size, it has an edge.
> Instead the space inbetween the galaxies is stretching.
No. Stretching is a bad analogy as it implies things which aren't so.
The balloon model is not the real thing. It is an analogy to help people
understand some of the concepts.
> What is interesting is that in cosmology the light
> transversing this space is also stretched. It gets
> longer and less energetic. This redshift is how we
> determine their distances.
Where does the energy go?
>
> If there is no boundary the universe can be seen to
> be the surface of a hypersphere.
>
Nope.
macro...@internetcds.com
What is your explanation?
John Sefton
macro...@internetCDS.com wrote:
> Space stretches and this is the cause of the light stretch.
>
> What is your explanation?
>
Uniform stretch everywhere doesn't work.
This is why shells grow in spirals.
Take 3 towns,; A, B, and C.
B is 10 km north of A. C is 10 km
north of c.
Now double all the distances
in one unit time.
B is now 20 km north of A.
C is now 20 km north of B.
So what?
Well, B moved 10 klicks.
How far did C move? (Hint: B is now
where C used to be.)
Well, B moved 10 klicks. How far
did C move in equal time? What about
D, E, F.....all originally at 10 klick
intervals? Get the picture?
John
Nick
that it only gets weaker at its center. The center of
gravity of more than one galaxy would be somewhere
in the space inbetween and the space inbetween
is stretching.
As the space stretches so do the light waves.
Jim Black
macro...@internetCDS.com wrote:
> Infinite universe Jim?
> It had a beining and is expanding at a finite rate.
> You can't get infinity out of that.
It would have to be infinite in the beginning.
Jim Black
That's why the velocity of galaxies away from us is proportional to
their distance (until relativistic effects kick in, and their relative
velocity compared to us becomes ambiguous).
Jim Black
Mitchell is right here about a finite universe not necessarily having a
boundary. This is the case if space has a positive curvature (assuming
it is isotropic and homogenous).
See, for example:
http://map.gsfc.nasa.gov/m_uni/uni_101bb2.html
Such a space is generally referred to as "closed." The
cold-dark-matter model of cosmology implied that if and only if there
was enough matter in space to make it closed, there was enough to make
it recollapse. However, the discovery of the accelerating expansion of
the universe indicated that there is more to the behavior of space-time
than the gravitational influence of ordinary and cold dark matter.
Thus, the statement that a closed universe must necessarily collapse is
no longer true, if you use "closed" in the usual way.
John Sefton
Which is fine if we are the center.
John
Jim Black
Watch what happens if we simply change reference frames:
time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------> position
time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------> position
time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------> position
time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------> position
Every galaxy sees the other galaxies moving away from it at a rate
proportional to their distances.
donsto...@hotmail.com
doing? Does NASA have a mission planned to go visit them? Mommy, will
Humanity ever live in peace?????????
- Donsky Oatsky, The Nut (Pecan) Ranch
Spoonfed
True. In fact, Hubble's law is nothing more than Distance = Rate *
Time: You can verify this by checking the units.
Hubble's Constant * Distance = Velocity
50 km/second per MegaParsec is equal to somewhere around 1/13 billion
years.
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.
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.
But the d=r*t law remains true not only in the Galilean transformations
you have shown, but is also true with Lorentz Transformations. Observe
the following animation, showing a Lorentz Transformation, similar to
the Galilean Transformation you have shown.
http://casa.colorado.edu/~ajsh/sr/wheel.html#spacetime_wheel
Notice, when the lines are nearly vertical, they are fairly similar to
those you've drawn with ASCII above. The lines at the edges, on the
other hand, are squeesed in extremely tightly.
Though the lines get squeezed in together, they do not change their
linear quality. They are straight lines, indicating a linear
relationship, preserving Distance=Rate*Time.
Spoonfed
So it would seem, B moved 10, C moved 30, D moved 50, E 70, F 90, to
infinity, so of course it looks like somewhere along the way something
must be moving at faster than the speed of light.
However, if you know your Special Relativity, you know that F is both
time dilated and length contracted. G is more so, H is more still, I,
more still, etc. Until you get out to Y which is moving 99.999% of the
speed of light, has experienced, for all intents and purposes, no time
at all, and is still adjacent to Z.
Yes, one second has passed for you, but one second has not passed for
"the universe"
The universe looks like this:
http://www.spoonfedrelativity.com/files/rel-big-bang.gif
Not like this
http://www.spoonfedrelativity.com/files/galileanreal.gif
Spoonfed
Jim Black wrote:
>
> That's why the velocity of galaxies away from us is proportional to
> their distance (until relativistic effects kick in, and their relative
> velocity compared to us becomes ambiguous).
Relative velocity does not become ambiguous when relativistic effects
kick in. There might be a bit of extra work involved in establishing
precisely when and where the relative velocity happened or how long it
lasted, it is all very definable and not ambiguous at all.
Events can be described in space and time very precisely according to
an agreed upon reference frame, just as we on earth all describe time
on earth according to GMT.
Nick
only works if the space stretch inbetween the galaxies
is equivalent to them moving away *through* space.
Space stretch stretches light just like velocity does.
Mitch -- Light Falls --
Spoonfed
I believe you may be confused.
Jim Black
If we want to make a meaningful statement about the relative velocity
between us and very distant galaxies, we must specify the path one
taken in going from one object to the other, and how much time is spent
on each part of the path. Otherwise, the statement is ambiguous, not
because of special relativity, but because of general relativity. The
idea that one gets different answers for different ways of getting from
one object to the other is at the very core of general relativity.
Consider the situation in which we want to compare the velocity of the
center of the earth (A) at a certain time t1 with the velocity of an
object (B) falling towards the earth at some later time t2. Suppose
that if we compare the velocity of A and B at time t1, that we find
that they are at rest with respect to each other. If we then wait at
object B until time t2, we will detect no velocity change, since the
object is freely falling. We would conclude that the relative velocity
between A at time t1 and B at time t2 was zero. If on the other hand,
we begin by waiting at point A until time t2, and then make the
comparison with object B, we will detect a velocity difference.
Ben Rudiak-Gould
>Jim Black wrote:
>
>>That's why the velocity of galaxies away from us is proportional to
>>their distance (until relativistic effects kick in, and their relative
>>velocity compared to us becomes ambiguous).
>
> Relative velocity does not become ambiguous when relativistic effects
> kick in.
I think the crux of the disagreement is that he's talking about general
relativistic effects, while you're talking about special relativistic effects.
He is correct. You would be correct if the universe were accurately
described by special relativity at cosmological scales, but it's not.
-- Ben
Ben Rudiak-Gould
> 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
Spoonfed
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.
Androcles
"Spoonfed" <jonatha...@spoonfedrelativity.com> wrote in message
news:1124931583....@g43g2000cwa.googlegroups.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.
Ahhh..... a personal theory... they abound. Got any evidence?
Androcles.
Spoonfed
I am not really sure about the cosmological constant, I've written more
here:
http://groups.google.com/group/sci.physics.relativity/msg/f5b0705d3ac223f9?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.
Ben Rudiak-Gould
> 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
Spoonfed
> 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 here:
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
Spoonfed
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/files/rel-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.
Autymn D. C.
anomolies -> anomalies
data suggests -> data suggest
disentegrated -> disintegrated
Ben Rudiak-Gould
>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
Spoonfed
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/physics/RelativisticRedshift.html
Except that I would stop at equation (2)
z=sqrt((1+beta)/(1-beta))-1
since the assumption of v/c<<1 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
Spoonfed
Posted September 7, 2005.
Ben Rudiak-Gould
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.physics/msg/c62984b2c8511d91
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
Spoonfed
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.
Ben Rudiak-Gould
> 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 are:
* 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
Spoonfed
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 are:
>
> * 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.