The Infinite Cosmic Lattice

[BurtJordaan]

One of more interesting ways of picturing the cosmos is by way of Escher's Infinite Lattice, which I have recreated here in color. The red cubes represent the 'particles' of the perfectly flat, isotropic and homogeneous universe, while the blue bars are just distance markers (one could just as well have taken them away, but they help somewhat with the visualization).

A lot of cosmology insight can be gained by simply looking at and contemplating this lattice. Here are the first five of my top ten interesting insights:

1. Say each blue bar is now 1 km long, but we somehow 'grow' the length of each bar (not the cubes) at a steady rate of one meter per year. The distance between any two red cubes will then grow at n meters per year, where n is the number of bars between them. This is equivalent to Hubble's law - the apparent recession speed of a distant galaxy is proportional to its distance from us. The 'Hubble constant' (H-zero, or Ho) of the lattice would be one meter per year per km distance. Take note that it is a distance divided by a time and a distance, so the unit of measure is essentially one over time. If we divide Ho into 1000, we get the Hubble time of this lattice as T_H = 1000/1 = 1000 years. The factor 1000 is just the number of meters in a km.

2. An observer somewhere near any red cube (say at the viewpoint in the diagram) will get the impression that she is at the center of a large expanding system of cubes, with all cubes moving away from her at speeds depending upon their relative distance. To her, the color of distant red cubes will appear redder than nearby cubes (light with longer or 'stretched' wavelengths), from which she will be able to deduce that more distant cubes recede faster than nearby ones. Whether she attributes the reddening to Doppler shift or to stretching of the wavelengths does not matter at this point, but it is equivalent to the distance/redshift relationship that astronomers use.

3. If our observer extrapolates backwards in time, she will conclude that if the individual growth rates of bars have always been one meter per year, the red blocks must have been touching each other at a certain time in the past. With all bars one km long and a stretch rate of at one meter per year, this must have happened 1000 years ago. So, our astronomer may conclude that expansion of the lattice have probably started 1000 years ago. This is not necessarily the age of the lattice, since she does not quite know what may have happened before the time when the cubes (presumably) touched each other. The ‘age’ of our universe is determined in a similar way.

4. Since we are working with a (scaled down) toy model here, let us say the speed of light in this lattice is an extremely pedestrian one km per year, so that in 1000 years, light would have traveled only 1000 km. Later we will replace kilometers with light-years, but for now, please bear with me. If our cube-astronomer should use a telescope to look as far as she possibly can, what will she see? Due to the finite speed of light, she must be looking back in time - she would be seeing every successive cube at an earlier time, i.e. at the time when the light left it. At some distance, she must be looking back to the time when all the cubes were touching each other. Unless the cubes were all transparent, she could not observe anything more distant, or to an earlier time, for that matter. This is equivalent to our present observation of the cosmic microwave background (CMB) radiation, the most ancient observable light.

5. Shortly after the bars started to expand, light could however move through the lattice. Suppose our lattice astronomer knows the natural color (and hence the wavelength of emissions) of the ancient cubes, but presently she detects them at a wavelength that is a thousand times longer. She could deduce that the length of the (now one km long) blue bars must then have then been only one meter. Because of her assumption of a steady growth in the length of one meter per year for each bar, she can also estimate the time when the light has left the most distant visible cubes: about one year after the expansion started, or roughly 999 years ago. This is equivalent to our present observation of the CMB radiation, which is stretched by a factor of 1088, coming from approximately half a million years after expansion started.

The black background that you see when you ‘look down the shaft’ at the bottom-left of the lattice represents the CMB of the toy-model. The cubes do not stop there; you can just not see the ones farther out, towards infinity…

I think this is more than a mouthful already. I will proceed with the last five of my top ten ‘lattice insights’ in a follow-on post. Any feedback would be welcomed.

[BurtJordaan]

In part 1, we discussed the first five of my top ten insights gained from the infinite lattice. Recall that our resident lattice astronomer was last time observing the cosmic microwave background (CMB), which is the oldest light that we can observe. It is visible as the black background when you look down the 'shaft' at the bottom-left. Insights 6 to 10 are all about cosmic distances and their meanings. For ease of reference, here is the lattice again.

Infinite lattice (c) Relativity-4-Engineers

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6. Co-moving distance. This is the most common distance that astronomers and cosmologists use. Fortunately, it is also the simplest one in the lattice model: you simply count the number of blue bars between two cubes. The reason for "counting bars" is that the distance between cubes are then not influenced by the length of bars, which may change over time. Recall that we got the recession speed of a cube by multiplying the growth rate of each bar by the number of bars – so, it is a useful measure of cosmic distances. In real life, astronomers use 'bars' of one parsec (pc) long, which translates to 3.26 light-years.^[1]

7. Proper distance. Another good measure of distance is the proper distance, which changes over time, as the bars grow in length. It is like taking a 'freeze frame' photo of the lattice at any specific time and then measuring the distance between cubes on the photo, using a present day blue bar as a meter stick. Suppose that from a specific cube, our astronomer receives light that has doubled in wavelength (stretch factor S = 2). Since the blue bars are now 1 km long, it means that they must each have been 500 meters long when the light left that cube. Also, as they grew at only 1 meter per year, it must have taken 500 years to grow to 1 km length. This must then also be the time that light took to reach her telescope. It is called the 'lookback time' and is sometimes (incorrectly) stated as "the distance of the object in light-years". This is not the proper distance of the object. We will now turn to how our astronomer would determine proper distances.

8. D_now and D_then. In our lattice, proper distance requires a simple, though not a quick calculation, because she has to work bar by bar, reasoning as follows: one year ago the bars must have been 999 meters long and the year before that, 998 meters, until 500 years ago when they were all 500 meters long. Light always traveled at 1000 meters per year in our lattice, so in its first year of the journey, it crossed almost 2 bars (i.e. 1000/501 ~ 1.996 bars). This way the number of blue bars that the light must have crossed can be approximated as: 1000/501 + 1000/502 + ... + 1000/999 + 1000/1000, i.e. 500 terms, one for each year.^[2] Using her computer, she arrives at the answer: 694 blue bars were crossed in the 500 years. Since the bars are now 1 meter long, that cube must now be about D_now = 694 km away. Another question: how far was the cube in question from the ‘home cube’ when the light was emitted? Fortunately, this is quite easy – we have seen that all bars were then 0.5 km long, so the cube was then D_then = 694 x 0.5 = 347 km away. This is similar to how cosmologists determine actual distances in the universe.

9. Hubble radius. The Hubble radius is defined as the distance that light can travel in one Hubble time. Recall from part 1 that the Hubble time is essentially the inverse of the Hubble constant (1/Ho), so the Hubble radius is the speed of light divided by Ho. Since our Ho = 1 meter per year and our 'speed of light' is 1 km per year, the present Hubble time of the lattice is 1000 years and the corresponding Hubble radius is 1000 km. The 'Hubble constant' of our lattice (Ho = 1 meter per year per km) does not remain constant over time; 999 years ago, bars were only 0.001 km in length, but their lengths still changed by 1 meter per year, so the Hubble rate was 1/0.001 = 1000 meters per year per km distance. It then decreased to the present Ho = 1 and in the far future it will eventually approach zero for this toy-model. The Hubble radius, being the inverse of Ho, will start from very small and will grow to a very large value as H goes to zero in the far future.

10. Cosmic event horizon. A final interesting question: given the values that we have, could a distant cube recede at a speed greater than the speed of light? Say it is presently 1000 km from the home cube; at Ho = 1 meter per year per km distance, its present recession speed will be 1000 meters per year, which is equal to the toy model’s speed of light. This is known as the cosmic event horizon (CEH_now) distance of the model.^[4] All cubes farther than 1000 km will presently be receding at rates faster than the speed of light and they are outside of our observer's CEH. Any observer, irrespective of where in the lattice, will have a CEH around them.^[4] In the real universe, CEHs also exist for all observers, but for that we need to consider a more realistic model.

In a follow-on article, we will look at such a more realistic model, still based on the infinite lattice.

Notes:

[1] Parsec stands for "parallax-second". It is essentially the distance to an imaginary nearby star that would show a parallax of plus and minus one arc-second as Earth moves around the Sun. The fact that it is directly measurable for many stars makes it preferable to the (more commonly used) light-year, which is not directly measurable.
[2] For more accuracy, she could have taken the average length of the bars in every year, i.e. 1000/500.5 + 1000/501.5 + ... + 1000/998.5 + 1000/999.5, but for an example, it not necessary to complicate thing like that.
[3] Remember that our lattice has infinite size, so there are an infinite number of cubes outside of any home cube's CEH. Because of this model's declining Hubble constant, the distance to the CEH will increase without limit, so that future lattice astronomers will eventually be able to observe ever more cubes. In our real accelerating expansion cosmos, this will not be the case.
[4] In this "Dinky Toy" model, the Hubble radius and the CEH has the same value, but this is not quite true for a realistic model, as will be discussed next time.

[Marshall]

So far this is the best elementary explanation of expansion cosmology basics that I can remember seeing.
You do it without equations, and still make it *quantitative*. Not merely verbal.

Distance expansion is the simplest kind of geometry-change and it is infact quantitative, not qualitative/verbal. So there HAS to be a quantitative component to understanding, and it has to be basically right. It can't be purely verbal or entirely based on analogies from everyday life.
But instead of using equations you provide this quantitative reinforcement with clear mental diagrams and grade-school arithmetic. It seems to work (with Escher's help :-)
Somehow your simple "Cosmic Lattice" toy-model of geometry reminds me of introducing someone to an ABACUS to help them get a concrete grasp of arithmetic.

At the moment there might not be anyone reading who WANTS to understand the basic concepts of expansion cosmology, but at least to me that seems to be OK because it is still a quiet place where you can lay out a draft and try different arrangements of material, and it's in electronic form so you can transfer to another venue whenever you want, and besides someone might drop in and ask questions, you never know!

I didn't realize until just now that you were a longtime SCF member, Burt. I didn't happen to follow the Twin Paradox thread at the time (October 2009). It was in Physics. You were quite right about the technical term "proper acceleration" as measured by accelerometer, and whipping around a black hole one twin can make a hairpin turn without experiencing acceleration. For relativists, proper=personal individual belonging to the observer or as experienced by the observer. Doesn't have to do with moral propriety or rectitude :-D
But that debate was all 3 years ago.

I wonder what will happen with this new stuff, here now in SCF AstroCosmo forum. I'd hate to see pedagogical stuff of this caliber go to waste. I expect it will not however--something will turn up. We'll see what happens.
Please just continue as you see fit--don't let my comment distract you from the main task of exposition.

[BurtJordaan]

Thanks Marshall, but do not worry, I am indeed using this board to try out new ideas for presentation. As you said, posting on a fairly quite place helps one to develop technique and I found that occasionally someone (including myself) stumbles upon a useful new insight.

An example: while experimenting with the accelerating expansion version of the lattice, using a spreadsheet, I plotted this graph for a constant Hubble parameter (i.e. pure de Sitter expansion), for our 'usual' distances and times against stretch S.

I was intrigued by the perfectly straight green D_now curve and had to revert to the equations to confirm that this was no fluke caused by the particular values chosen. It checked out and for constant H found D_now = (S-1)Y_now.

The chart is not complete with labels yet and note that the scales of some curves are different for convenience. I plan to write up the de Sitter case and then discuss the combination of these two by way of the lattice.

Note that 'LeanA25' does not work correctly for the pure de Sitter case (setting Y_inf = Y_
now and S_eq = 999999). It was not designed with de Sitter in mind, but I will look into why it does not handle it, because it should.

-B

[BurtJordaan]

In the previous lattice model, we have expanded the system by ‘growing’ each bar by a constant 1 meter per year. With the bars 1 km long, it means the growth was a fraction 1/1000 (0.1%) for the first year, but then it became 1/1001, or just 0.0999% for the next year and so on. This is somewhat like simple interest, where you do not earn ‘interest on interest’. Our present expansion seems to work more like compound interest, where the interest rate remains constant, but you earn interest on the growing total every year, or even every day. So let us see what happens if we grow each bar by a constant 0.1% of its ‘then’ length.

(c) Relativity-4-Engineers

If we start when the bars were 1 meter long and then grow them by 0.1% per year, how long will it take to reach the 1 km lengths of today? Since it is a compound interest problem, I asked our the lattice financial specialist^[1] how many years I need to grow $1 to $1000 at fixed 0.1% per year, compounded annually? She first frowned at the insignificant interest rate, but after me explaining the context, she pressed a few buttons on her tablet, smiled and said: “well, it had to be deposited 6911 years ago, probably originally recorded on a clay tablet”.^[2]

While I had her attention, I also asked her about the time required for our previous example (part 2, insight 7), where we observed a cube at stretch S=2, which is equivalent to asking: how long does it take for an investment to double at 0.01% compound interest? Her tablet gave 693 years, compared to the simple interest value of 500 that we obtained earlier. Before we do any further financial gymnastics, let us rather look at a graphical presentation of all the more interesting values.

Click image to view separately at better resolution

Note the value of the black (lookback) time curve for S=2 - it is about 700 years, just as the bean-counting tablet told us (693 yrs). At the same time, the green curve reads D_now = 1000 km and the orange curve D_then = 500 km (it has to be exactly half of D_now for S = 2). The curves T_Hub and D_hor sit on top of each other, at a constant 1000 years and 1000 km respectively.^[3] Recall that D_hor represents the cosmic event horizon distance, where a cube recedes at exactly the speed of lattice light. Note that the D_now ‘curve’ is actually a perfectly straight line, so it has a very simple algebraic equation: D_now = (S-1) x T_Hub.

Is this ‘compound interest’ model a good analogy of our real cosmos? Almost, but not quite - it is presently somewhat of a cross between this one and the ‘simple interest’ model. Anyway, our present universe seems to be well on its way to become exactly like the ‘compound interest’ model. In the next post, we will investigate how the two models can be combined.

Notes:

[1] I actually asked the question to this easy to use web-based financial calculator: http://www.ultimatecalculators.com/comp ... lator.html. Look inside the box for “Number of Compounding Periods”. It even gives you the financial equations!

[2] Even before formal writing was developed: http://en.wikipedia.org/wiki/T%C4%83rt%C4%83ria_tablets

[3] The fact that T_Hub = D_hor comes from our choice of the speed of light for the lattice, i.e. the convenient 1 km per year. We will see later that this is also valid when we revert to the speed of light as 1 light-year per year.

[BurtJordaan]

We have now looked at two possible schemes for the growth of our cosmic lattice: the simple interest and the compound interest scenarios. This chart for the latter has been posted last time.

Click on graph for a better resolution picture.

It seems that our cosmos has a little bit of both in it. With some juggling between the two cases, this toy-model can be a surprisingly close analogy for comprehending our place in the universe. Our universe possibly started out somewhat like the infinite lattice, except that there were no red cubes, just blue bars of indefinite length (the 'infinite vacuum' or 'void'), stretching at an extremely high 'compound interest' rate. In the greater scheme of things, this seems the more 'natural state' for our cosmos.

With no red cubes, this view is a little problematic, because how do we quantify the stretching of 'nothing'? Fortunately, quantum physics came to the rescue, because it allows elementary particles of energy (photons, gluons, quarks) to pop into existence out of the nothingness, but we will not go into that process here. It is somewhat like super-cooled water that may remain a liquid below freezing point, but then abruptly crystallizes or freezes.

Suppose the particles were all born at the same instant in cosmic time and were very evenly spread throughout the void. They then just had to move apart at the stretch rate of the local vacuum; after all, they came out of the 'stretching' vacuum. Now it is possible to quantify the 'compound interest' stretch rate and even the distance between particles (at least in principle). We can imagine particle-sized 'micro-cubes' separated by distances somewhere near the Planck scale,^[1] moving apart at the near the speed of light.

There is a catch, however. The mass-energy of the photons and the densely packed micro-cubes created a gravitational field that caused a drop in the ultra-high stretch rate. The moment matter and radiation appeared, the lattice abruptly changed from a compound interest growth rate to something like a simple interest growth rate. The percentage growth was still very large, but it was based on a small separations. We have seen before that simple interest, not allowing 'interest on interest', grows slower and slower in terms of percentage, but it obviously never quite stops growing.

The density and temperature of the lattice were extremely high and almost evenly distributed. However, quantum physics demands that there should have been some fluctuations in the density and this caused the denser areas of the lattice to have a slightly lower stretch rate than the less dense areas. In the cosmic background radiation, we observe these density differences as hotter and cooler spots. Over time, the mutual gravitational attraction of the more densely packed micro-cubes caused them to clump together to form the proper red cubes of our lattice; they are equivalent to the gravitationally bound clusters of galaxies that we observe today.

Eventually the distances between the cubes (equivalent to the voids between clusters) became so large that the mutual gravitational attraction could no longer overwhelm the natural tendency of the voids to grow according to the preferred compound interest rate. What we observe today is that radiation and matter has only some 30% influence on the growth rate and it is decreasing. Eventually, the voids will become so dominant that things will again resemble the 'infinite vacuum' or void that we started with; essentially, it would have gone full circle, perhaps ready to repeat itself.

Click on graph for a better resolution picture.

Shown above are the curves for the combined lattice model. They have exactly the correct shape for our real universe; we only need a change in the scale of the vertical axis to representative values. Note how the blue Hubble time and the red horizon distance curves have changed over from relatively low values in the past (S larger than one^[2]) to the constant value of 1000 in the future (when S is smaller than one).

S=1 represents 'now' and the value of the present T_Hub can be read of as roughly 850 years. The ratio between T_Hub now and the T_Hub in the far future (1000 years in this case) is very important in cosmology. In the next episode we will examine this ratio with proper cosmic scale and values plugged into the lattice model.

Notes:

[1] Planck length is the shortest time that has any meaning in quantum physics. If adjacent micro-cubes should move apart at the speed of light, their separation will increase by one Planck length in one Planck time. Fill the infinite lattice with such micro-cubes, one Planck length apart, with adjacent ones separating at the speed of light and you have something that possibly resembles the 'birth' of our universe.

[2] If we could take our lattice S to near infinity (the 'infinite past'), T_Hub would have settled to a constant one unit of Planck time and D_hor to a constant one Planck length. This condition could have lasted for a very long time. Some theorists think that this is how the real universe also behaved. In a way, our present phase may be just a transient between two different constant T_Hub values.

[Gregorygregg1]

This is the most understandable explanation of an expanding universe I have seen. The introduction of energy and mass into the void apparently alters the lattice according to relativity.
is it possible that we exist in a microclimate of dimension, and that the geometry of space changes increasingly at larger and smaller scales secondary to the quantity of mass and energy contained? In other words, that the shape of space is fractal?

[BurtJordaan]

The interesting thing about cosmic expansion is that the appearance and the "disappearance" of mass apparently do not alter the overall curvature of space - it seems to have remained flat throughout our observable past, despite the cosmological constant that dominates at present.

In a sense, the "microclimate" that you refer to is regulated by the cosmological constant, in order to keep the geometry flat (or at least very,very close to it).

[BurtJordaan]

This is a dummy post, just to get a better image quality into the next post. Somehow, I can get a reasonable resolution picture once it is uploaded as an attachment and then referenced in another post. A direct 'place inline' seems to produce an inferior resolution insert.

Please see next post, shortly...

Burt

[BurtJordaan]

S=1 represents 'now' and the value of the present T_Hub can be read of as roughly 850 years. The ratio between T_Hub now and the T_Hub in the far future (1000 years in this case) is very important in cosmology. In the next episode we will examine this ratio with proper cosmic scale and values plugged into the lattice model.

Before we go on and plug 'real values' into the lattice, here is one more important observation that we should make from the nice, round figures of this graph. The present T_Hub = 850 years is exactly the time it will take for all the bars of the lattice to double in length, should the present 'compound interest rate' be maintained. In technical terms, T_Hub is the slope of the green D_now curve at S=1. In fact, the blue T_Hub curve gives the "interest rate" for any time, not as a percentage, but rather as the time required for a doubling of all blue bars.

Does the real cosmos also work like this? You bet, just on a different scale of time and distance. When we change time intervals to billion years and the blue bars to lengths of one billion light-years each, we find that the present 'interest rate' is equivalent to doubling all blue bars every 13.9 billion years. This rate is still slowly decreasing, but should later settle on a steady doubling of bar lengths every 16.3 billion years. At the other end of the scale, by the time when the CMB photons were released (13.75 billion years ago), the doubling happened every 640,000 years. Not quite a 'stellar rate' by earthly standards, because even a pedestrian compound interest of 0.5% p.a. should double your investment every 139 years, provided that the bank did not go bust.

Here is the lattice curve with the real scale for the vertical axis.

Click on image for better resolution.

(will try to get a better resolution image inline...)

http://www.einsteins-theory-of-relativity-4engineers.com/images/Lattice%20realistic13-16.png

In the next episode, we will see what other interesting information this graph can offer.

[BurtJordaan]

This is a dummy post, just to get a better image quality into the next post. Somehow, I can get a reasonable resolution picture once it is uploaded as an attachment and then referenced in another post. A direct 'place inline' seems to produce an inferior resolution insert.

Please see next post, shortly...

I guess it may take the 2 hour grace period for editing before the image may become available in a quote...

Lattice realistic13-16.jpg

Burt

EDIT Jorrie, let me know if and when you want the dummy posts removed. But most likely you'll be able to take care of that. I don't know if the need for them eventually goes away. Maybe it is safer to leave them. Up to you. M.

Thanks to an appreciative reader for the alert. I didn't know about this graphics problem, and hope we can solve it.

[BurtJordaan]

I eventually got it to show inline properly. Sorry for the distractions, so for ease of reference, I repeat the important part of the other post here.

Does the real cosmos also work like this? You bet, just on a different scale of time and distance. When we change time intervals to billion years and the blue bars to lengths of one billion light-years each, we find that the present 'interest rate' is equivalent to doubling all blue bars every 13.9 billion years. This rate is still slowly decreasing, but should later settle on a steady doubling of bar lengths every 16.3 billion years.
Here are the lattice curves with the real scale of the vertical axis.

At the other end of the scale, by the time when the CMB photons were released (13.75 billion years ago), the doubling happened every 6400 years. Not quite a 'stellar rate' by earthly standards, because even a pedestrian compound interest of 0.5% p.a. should double your investment every 139 years, provided that the bank did not go bust.

[BurtJordaan]

After the battle to get the diagram inline with reasonable resolution, back to the physics.

We have seen that the blue T_Hubble curve represents the time required for all cosmic distances to double, should the compound rate be frozen at that stage. In practice, the rate has changed from extraordinary quick near the beginning of the expansion,^[1] to the sedate 7.2% per billion years of today. We get the 7.2% by taking 100 (%) and divide it by the 13.9 billion years for a size doubling that we have discussed last time.

At S=10, some 13 billion years ago (read from the black curve), the doubling happened in just less than one billion years (read from the blue curve). This gives an effective compound rate of some 100 / 1 = 100 % growth per billion years. On the left of the graph, the rate will eventually drop to 100 / 16.3 = 6.13% per billion years and we think this is where it will stay. The blue curve flattens out when S nears 0, which is equivalent to the infinite future.

The green D_now curve (how far an observed object is now) is related to the blue curve through green’s slope. Essentially, if we have any one of the two, we have the other one. The brown D_then curve is also available if we have D_now, because D_then is simply D_now divided by the stretch factor S. This is all very easy; provided that we have one of the three curves, we can get the other two. But how do we get hold of one of them?

This is where cosmological calculators enter the scene. Amongst others, we have http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A25.html, which is nicely optimized for simplicity and for the type of calculations that we were doing here. It is reasonably self-explanatory, so why not try it out? I got the following table out of it:

Click on table for a better resolution view

You will notice there is no T_look, because here T gives the cosmic time since the start of expansion. T_look is essentially just T_look = T_now – T, i.e. T_look = 13.755 – T. The parameter a is called the “expansion factor” and you can see that it is simply a = 1/S. It is left in there to give serious amateur cosmologists a ‘hook’ to the standard textbook stuff. The discussions on this forum may yet influence the calculator, because it is still a work in process.

I think it is enough for this episode. Next time we will zoom in on the brown (D_then) curve. It has some interesting characteristics of its own.

Burt

Note 1:
It is theorized that when matter and radiation appeared, near the end of the ‘inflation era’, the doubling happened in close to the Planck time, about 10^-43 seconds. Fortunately, by the time our lattice interest starts, things have dropped to more meaningful times. At about 380,000 years after expansion started, the CMB radiation was released to travel towards us. It then took 6400 years for a doubling in distances, which translates to around 0.01% growth per year. But to compare it to today’s values, we should express it in % per billion years, giving something like 100/0.0000064 ~ 15 million % per billion years. So do not think that cosmic expansion has always been quite as slow as today’s 7% per billion years.

[Watson]

This from another thread, the most accurate estimate of the expansion of the universe:

Space itself is pulling apart at the seams, expanding at a rate of 74.3 plus or minus 2.1 kilometers (46.2 plus or minus 1.3 miles) per second per megaparsec (a megaparsec is roughly 3 million light-years).

Can you put this in terms of your model?

(from this original post: viewtopic.php?f=72&t=23161 )

[BurtJordaan]

This from another thread, the most accurate estimate of the expansion of the universe:

Space itself is pulling apart at the seams, expanding at a rate of 74.3 plus or minus 2.1 kilometers (46.2 plus or minus 1.3 miles) per second per megaparsec (a megaparsec is roughly 3 million light-years).

Can you put this in terms of your model?

(from this original post: viewtopic.php?f=72&t=23161 )

If the 74.3 km/s/Mpc holds up to scrutiny, it would reduce the time required for a doubling in distance to 978/74.3 = 13.2 billion years, or about 7.6% per billion years "cosmic interest rate" (recall that we have used 7.2% per billion years before). Once it is analyzed with other observations, it may perhaps become the new "standard value" for Ho.

It's not quite "really, really fast", as SciAm sensationally writes. It all depends on perspective, of course, but by cosmic standards, it is slow. However, multiply this Ho by the largest distances that we can observe, and the present recession rate does become fast, really fast. Like in three times the speed of light...

Jorrie

[BurtJordaan]

Before we discuss the distance ‘then’, when light left the source, we have to take another look at event horizons. The dotted red D_hor curve is the distance of our cosmic event horizon at any specific stage in our past, present or future. More specifically, it is the changing distance where D_now is growing at one light year per year. In the accelerating expansion phase, once an observer has passed that horizon, all future contact is lost, in both directions.

On the other hand, the blue T_Hub curve sits at the changing distance where D_then was growing at one light year per year. However, the brown D_then curve was above the blue T_Hub curve for most of our past, meaning the distance to a source of light was growing faster than one light year per year. Yet, we observe sources all the way up to S=10 (and in principle, much farther); but how could that light ever have reached us?

Initially, those photons were being dragged away from us faster than they could make way in our direction. Gradually the expansion slowed down and eventually the photons found themselves at a distance where they were no longer being dragged away from us faster than what they could travel. This distance is exactly where the brown and blue curves cross and it explains the rather unusual form of the brown curve, because it has its maximum value at that point. From S=2.6 and on, the farther we look back in time, the closer to us things were when the photons left them.

Even stranger, consider two separate galaxies, both at stretch S=2.6 (D_then around 6 billion light years from the brown curve), but we observe them separated by (say) 10 arc seconds. Normally things will span a smaller angle if they are farther from us. Yet, if those same two galaxies were at stretch S=4, we would have observed them with an angular separation of 20 arc seconds. At that sort of range, things that are farther away look bigger! This has to do with the fact that the photons emitted towards us were first dragged away from us, before eventually coming towards us. In that time, the space between the two photon bundles stretched very fast, so that they were eventually reaching us at a larger angular separation.

We have now come almost ‘full circle’, because it is time to look at the infinite lattice again. The nice straight perspective that we see when looking down the ‘shaft’ (bottom left) is only valid for short distances on cosmological scales. If each of those blue bars were 1 billion light years long, it would not have been such a uniform picture. Remember that the light would have taken more than a billion years to cross each (expanding) blue bar. Because of the very rapid expansion in the past, the ‘shaft’ would eventually have flared out for the reasons that we have discussed in the previous paragraph.

In fact, it would have been only the first three or so cubes, looking down the shaft that would have appeared more or less as shown. Lower cubes would have appeared flared more and more, i.e. displaced away from the center of the shaft. One can also see this on the graph; the green D_now and the brown D_then curves more or less coincided for the first 3 billion light years or so, then they deviate (‘flare’) away from each other.

This concludes the story of the infinite cosmic lattice for now. The final episode will be a summary of all the insights gained from this analogy.

Jorrie

[Dave_Oblad]

Hello BurtJordaan,

First, I'd like to say this is very well done. Keep in mind, I'm only self educated, not an expert.

Now, I will offer a few ideas that may spark some insight for a more complete concept of what's involved.

1) I presume this Lattice is the fabric of Space-Time.

Issues: If this lattice is expanding then only a few possibilities present themselves:

A. More lattice rungs are being added at the edge of the Universe.
B. More lattice rungs are added periodically throughout the lattice.
C. The lattice rungs themselves are stretching without new rungs being added.

I dismiss (A) because that would increase the size of the Universe but the distance between the Galaxies would remain constant.

I dismiss (B) because that would screw up the symmetry, like playing chess on a board where on occasion each board square (lattice) decides to abruptly quarter itself.

That leaves (C) which I believe is your point anyway.

2) In order for Galaxies to be spreading apart this fabric must be having some effect on the matter involved. So the tricky part is showing how this lattice can get bigger without making the distance between particles also larger. If the distance between particles were growing then matter itself would have to grow along with the expansion giving a net gain of zero effect.

In other words, if you doubled the size of everything in the Universe, you wouldn't be able to detect any change, as all measuring rulers would also double in size.

So one must make a decision: This lattice has no effect on matter or it does. If it does, then at what Scale does it fail in growing the distance between particles. If it has no effect on matter, then the mean distance between Galaxies isn't really changing.

The solution is that the lattice does control the distance between particles. But in order to avoid a net gain of zero, we must accept that the lattice doesn't have a rigid universal symmetry. That in some places the rungs of the lattice are shorter than in other places.

So how can this be? Well, the lattice is actually 4 dimensional. But I know that's hard to show in an image. It has 4 axis directions of X,Y,Z,T.

Now the concept behind Relativity is that the size of a lattice 4D cube is always a constant volume. As Matter gets involved, the T axis stretches (time slows down) and the remaining 3 axis shrink, thus keeping a constant volume. This is the logic of why an object changes shape as it approaches the speed of light.

As Matter accumulates in an area, the lattice shrinks on X,Y,Z axis and stretches on the T axis. How do we know this is true? We know by experiment that The lattice can be flat, or curved, and twisted. Because Relativity predicts Gravity Waves can propagate by altering this lattice fabric, then we must also add expansion and compression to it's list of tricks.

So, if the early Universe was very flat and symmetrical then after particles began to appear, condensation took place due to electrical effects. As condensation reached a point where the lattice starts changing it's shape and shrinks around the particles allowing Gravity to become manifest.

So what is Gravity?

Here is an cross section (XY) of your lattice with a concentration of matter in the center:

Typical sample of Gravity affected lattice

Click image to enlarge.

Now imagine that the T axis is at 90' from this lattice (towards you). Next realize that matter/energy must always be in motion or it ceases to exist. It is basically coming towards you (the 90' thing) at the speed of light. During this travel it goes through many shifts in symmetry. At some point it completes a T axis shift and is now in a new location. I hate using the word Random, but let's assume the new position is a Random function. How far can it travel on the XY axis in the time allotted? I have drawn a circle to show the hypothetical maximum range it can travel on XY before it's new location is defined. It's like a cone with the point of origin being the sharp end of the cone and the circle drawn is the large end of the cone showing all the probable points it can reach in the T axis interval.

Notice anything unusual? Yes, there are far more places for it to relocate to in the direction of the greater lattice density. So Matter tends to migrate/gravitate towards anyplace the lattice is denser. So the variable density lattice explains Gravity.

Also note that if no new rungs are ever added to your lattice, that the increase in lattice density due to matter migration must come at a price of stretching the rungs length where the matter isn't. Namely between stars and especially between Galaxies.

This sounds like the effect of Dark Matter (denser lattice) and Dark Energy (lattice stretching). But this also means that if you could build a ladder spanning the distance between two remote Galaxies, and given that they have no relative velocity difference between them, that while it may appear the Galaxies are growing apart, the ladder is stretching along with them and not growing new rungs. So from this can we actually say that the Universe is stretching or expanding? Or are we simply seeing the effect of lattice condensation inside Galaxies at the expense of lattice stretching between Galaxies?

Since Dark Matter (dense lattice) creates a Gravitational Lens, then what kind of lens is being created by Dark Energy (stretched lattice). And given all that, how can we be sure of any form of distance measurement through all these deformations in our Universe? If our Galaxy is denser (smaller) than it was many billions of years ago and the light that left a distant but equal sized Galaxy has traveled all this way while we have all been shrinking during that time span, would the light appear red-shifted (longer wavelength), based on how far back in time it left it's home Galaxy?

Of course this is idle speculation, but I've heard others offer alternate explanations on why light red-shifts over great distances and time, without supporting the concept that the Universe is growing larger by the moment. Is the Hubble constant a measure of Universe expansion or local shrinkage? Fun thought that is.

One last thing.. as I have to head home now.. I'd lose the "Infinite" part of your lattice concept. Infinity can not exist by it's own definition. As a concept.. fine.. but in the real world.. it's no more than an unachievable goal.

Best wishes,
Dave :^)

[DragonFly]

Where are particles when they are 'between' the lattice?

[Marshall]

Where are particles when they are 'between' the lattice?

That's a philosophical question and one which Carlo Rovelli answered memorably in a 2009 seminar at Penn State.

but in this case I think Jorrie's lattice is simply an approximation or a toy model of reality. It's a good mental exercise which if you follow thru will prepare you to understand the mainstream standard model of expansion cosmology much better. So if it is just a toy model the question does not even arise. things can only be at lattice locations, so they hop. or something like that :biggrin:

We'll see how Jorrie answers, it's his model.

Rovelli had something more to say---it was regarding actual models of nature which are in some sense discrete. Several approaches to quantum gravity use discrete structures, networks, triangulations, building blocks of various types, to describe the quantum state of geometry. These are proposed as fundamental descriptions of reality, so then you can ask questions like the one you just did and it's more complicated to answer.

It was in this seminar series: http://relativity.phys.lsu.edu/ilqgs/
the May 5 2009 one. I think about 20 minutes into the hour. Someone else talks first for about 20 minutes then it's Rovelli's turn and he starts out by addressing this type of problem.

The nice thing about Jorrie's lattice is that he doesn't have to answer questions like this because the lattice is not proposed as the way nature actually works. It's like a playground jungle gym rather than a real jungle. You get the right kind of conceptual exercise by studying it. Then you can move on to more realistic setups if you want. Anyway that's my two cents. Just kibbitzing :^D

[BurtJordaan]

Interesting ideas, Dave. :-)

That leaves (C) which I believe is your point anyway.

Yup, that's it.

2) In order for Galaxies to be spreading apart this fabric must be having some effect on the matter involved. So the tricky part is showing how this lattice can get bigger without making the distance between particles also larger. If the distance between particles were growing then matter itself would have to grow along with the expansion giving a net gain of zero effect.

I think the cubes that does not expand make this rather obvious in the toy model. It's just the blue bars that stretches, but they are just space. Between cubes there are only photons, and they do "stretch", BTW. Their wavelengths stretch exactly with the length of the bars. Remember that the cubes are congregations of zillions of particles and they are being kept together by molecular forces and self-gravity.

The solution is that the lattice does control the distance between particles. But in order to avoid a net gain of zero, we must accept that the lattice doesn't have a rigid universal symmetry. That in some places the rungs of the lattice are shorter than in other places.

Nope, my lattice is perfectly uniform, all bars have the same length, so it is isotropic. It is only homogeneous on the large scale, somewhat like real life. And the lattice 'really' go on till 'infinity', hence it's name, coined by http://en.wikipedia.org/wiki/M._C._Escher, AFAIK.

I'm pressed for time right now, so I'll come back to your 4-D thoughts later.

Jorrie

[Dave_Oblad]

Hi Burtjordaan,

You said:

The red cubes represent the 'particles' of the perfectly flat, isotropic and homogeneous universe, while the blue bars are just distance markers (one could just as well have taken them away, but they help somewhat with the visualization).

I said: "C. The lattice rungs themselves are stretching without new rungs being added." and you agreed.

Now it would appear you are stuck between a rock and a hard place. You want to believe the particles exist and the rungs define the distance between them. You also support the idea that the rungs are informally expanding. But you don't accept that the distance between the particles can be expanding. You are in contradiction with yourself my friend. If you don't accept the idea that the particles are affected by the distance between the lattice rungs, then Matter/Energy isn't going to move further apart with the expansion of the Universe.

In other words: The expansion of the Universe would be invisible unless it affects the Matter/Energy contained in the Universe.

So where do you draw the line. Are those blue spots representative of Quantum Foam, Quarks, Atoms or Molecules? It really doesn't matter which you pick.. if the lattice affects matter (as you agreed) and the lattice is expanding, then so must the relative scale of Matter/Energy.

It's like that model of the expanding Universe. Just paint some Galaxies on a balloon, inflate it and watch as the Galaxies move apart. All's good until some child points out that the Galaxies are growing too. Thus.. if everything is expanding then nothing is expanding. So a quick correction is made to this model and the Galaxies are "Pasted" on the balloon. CHEAT!

Do you get it? You can't expand the Fabric of Space-Time without expanding the Scale of Matter/Energy relative to it.

If the lattice if densest at the center of a Galaxy then it would be very stretched near the perimeter. Imagine two objects of the same velocity placed in orbit around the Galaxy with one near the center and one near the perimeter. Because "Distance" is stretched near the outer edge, then one would observe remotely that the object near the outer edge must be moving much faster than the object nearer the center. In fact way too fast to stay in orbit around the Galaxy without a heck of a lot more gravitational field strength than can be justified by the visible matter. Now where have I heard that one before ;-)

The biggest problem I see being that if it is true that the Fabric of Space-Time has Variable Density, then a whole lot of Cosmology Science will be shown to be false and we will have to start all over again.. with a whole new paradigm regarding the true nature of the Cosmos. I'm betting I'll see that day.

Best wishes,
Dave :^)

[DragonFly]

What about the Planck scale? It seems to underlie all and is a firm size. One cannot even tell something form nothing at that scale, nor even much of anything at all, since the error of measurement is as large as what one would try to measure. Is the Planck scale - a more realistic model, if not reality itself - a lattice arrangement?

[Dave_Oblad]

Hey DF,

You talking to me? YOU TALKING TO ME??? YOU MUST BE TALKING TO ME...lol ;-)

I agree with the Lattice concept, down to the Planck Scale. That scale defines the relativistic functions of distance vs. time as far as the Speed of Light is concerned. And we already know Time is flexible/variable in a variable Gravitational Field. So the Universal Lattice can not be Universally Uniform.

Best to all,
Dave :^)

[DragonFly]

I was sort of just taking to the air (the thread at large). Just thought that the Planck scale would be a more real lattice. Unfortunately, that scale seems to dissolve everything that we can talk about at larger scales.

[BurtJordaan]

Now it would appear you are stuck between a rock and a hard place. You want to believe the particles exist and the rungs define the distance between them. You also support the idea that the rungs are informally expanding. But you don't accept that the distance between the particles can be expanding. You are in contradiction with yourself my friend. If you don't accept the idea that the particles are affected by the distance between the lattice rungs, then Matter/Energy isn't going to move further apart with the expansion of the Universe.

Dave, if you replace every red cube in my infinite lattice with one particle, perfectly homogeneous and isotropic (as I specified from the be beginning), then all particles will move apart just like the red cubes do. Each particle will feel the gravity from all other, but the expansion of the blue bars will take them apart, while their collective gravity will try to slow this "moving apart".

I took the lattice one step further when I discussed the fact that slight inhomogeneity between the particles could make them clump together into my red cubes. These things are now bound (molecular if small, plus gravitationally if large enough). Just like galaxies and clusters do not expand, the cubes don't either. I think you have confused the earlier 'single particles' with the later 'clumps of particles'.

... if the lattice affects matter (as you agreed) and the lattice is expanding, then so must the relative scale of Matter/Energy

.

The lattice does affect matter in the way I have described. The distances between cubes are increasing because they were born that way - recall that I said the natural tendency of the blue bars are to expand. The mutual gravity slowed the whole scheme down, but not before the bars were long enough so that the gravity became weak through the 1/r² law; weak enough for the natural tendency of the bars to overwhelm the effect of the mutual gravity. Not inside the cubes, because there the distance between molecules remain the same and they are not affected by expansion. This is what we see around us and this is exactly what I modeled - in 'toy model form', not to completely represent the cosmos.

Do you get it? You can't expand the Fabric of Space-Time without expanding the Scale of Matter/Energy relative to it.

I surely do not. I guess you could start a thread where you explain exactly what you mean by this view. If it is off the mainstream of science, you are still welcome, but then just do it in the 'Alternative Theories' section.

If the lattice if densest at the center of a Galaxy then it would be very stretched near the perimeter. ...

The center of galaxies are inside the red cubes and that domain is outside the scope of the lattice analogy. It is surely part of Astro/Cosmology, but I guess a separate thread would be required for that. I do not mind chatting away on that - it's the purpose of this board - but let's try not to defocus this particular thread, because it probably gives valuable insights to newcomers.

I will now go back to your interesting 4-D thoughts, very relevant to the lattice analogy, but will use a new reply in due time.

-J

[BurtJordaan]

What about the Planck scale? It seems to underlie all and is a firm size. One cannot even tell something form nothing at that scale, nor even much of anything at all, since the error of measurement is as large as what one would try to measure. Is the Planck scale - a more realistic model, if not reality itself - a lattice arrangement?

I have briefly touched on the Planck scale in my post of Oct 2nd: http://www.sciencechatforum.com/viewtopic.php?f=72&t=23060&p=219012#p218259.

Granted, it is just a lattice that starts at the Planck scale, not 'hovering' there... Problem is that without a theory of Quantum Gravity, we just do not know how to model things there.

-J

[BurtJordaan]

Hi Dave, coming back to 4D.

So how can this be? Well, the lattice is actually 4 dimensional. But I know that's hard to show in an image. It has 4 axis directions of X,Y,Z,T.

What you have described in the entire post is in line with standard Cosmology, based on General relativity. I just want to make a few comments to some of you thoughts.

Since Dark Matter (dense lattice) creates a Gravitational Lens, then what kind of lens is being created by Dark Energy (stretched lattice).

This is a very interesting question. While dark matter clump together (differently to normal matter, but they clump), dark energy is thought to be in the fabric of space, everywhere and hence don't clump. However, the stretching lattice does cause some magnification of distant objects. I have elaborated a little on that towards the end of this post: http://www.sciencechatforum.com/viewtopic.php?f=72&t=23060#p218902

If our Galaxy is denser (smaller) than it was many billions of years ago and the light that left a distant but equal sized Galaxy has traveled all this way while we have all been shrinking during that time span, would the light appear red-shifted (longer wavelength), based on how far back in time it left it's home Galaxy?

There is too much evidence that galaxies did not shrink after formation; they rather grew as more and more matter got trapped by their gravity. We observe some of this in the past (e.g. mergers), but by and large things were relatively stable for a long time (just inter-cluster distances expanding).

I'd lose the "Infinite" part of your lattice concept. Infinity can not exist by it's own definition. As a concept.. fine.. but in the real world.. it's no more than an unachievable goal.

True, but I'll stick to the 'infinite flat lattice' (not bend into a grand circle in 4 dimensions) for now. It's a lot simpler to work with. Otherwise, parallel lines will not remain parallel, even if you 'freeze-frame' the expansion and measure. If it's not infinite, let's call it near-infinite, large enough so that we will never find the edge.

BTW, the latest (Freedman 2012) data seem to favor a closed (hyper-spherical) cosmos with a circumference of at least 600 billion light years...

Regards,

Jorrie

[Dave_Oblad]

Hi Jorrie,

Just a quick observation. So we know Galaxies come in all sizes and many shapes. So you are saying that the Blue part of the lattice are Galaxies? Then what smaller scale does the expansion stop applying to stars or planets or moons, or atoms etc. Suppose we are talking about a real large Galaxy. One big enough to span between two of your Blue boxes? Then wouldn't expansion inflate the size of that Galaxy. And at what scale does the expansion stop defining the expansion?

As for the size of a Galaxy shrinking because the Fabric of Space-Time is condensing, how would you know.. since all the rulers that can be used are shrinking along with it? I think the problem is within our concept of distance. Yes, the Galaxy may grow larger as more matter is added. Internal rulers would indicate the diameter is increasing.. that's fine. But imagine the concept that by shrinkage, I am saying that distance is compressing, along with the content. That distance is non-euclidean.

Under these conditions a Galaxy may grow and shrink at the same time. However, the only way to locally measure this shrinkage is by using some form of exotic ruler, that doesn't change length regardless of it's speed or proximity to say.. a black hole. Such a ruler, in hand, would appear to grow as one approached a black hole. But the speed of light will appear constant. True, a bit hard to wrap one's mind around.

What defines the size of an Atom and why are they always compatible in size? Do you ever imagine what would happen if two large/gigantic hydrogen atoms were to try and bond with a small/tiny oxygen atom (water)? Of course not! Because something controls their relative scale when they are local to each other. That's the function of the lattice as I see it.

So if, as suggested, I were to find two remotely separated Galaxies that were stationary in respect to each other and spanned a hypothetical ladder between them, then by the expanding Universe concept, someday in the future the ladder will only reach half-way. And in my view, the ladder will stretch with the inflation of Space-Time.. without growing more rungs.. and always bridge the gap.

And by that token, a beam of light will always take the same amount of time to travel that gap, even though the remote Galaxy appears, due to an illusion of expanded space, to be moving further away.

Just to be clear.. I am not in disagreement with the measurements made by Science.. just the interpretation of them.

It wasn't long ago that we thought the Earth was the Center of the Universe. We had the precision in observations and the math to predict the exact location in the sky of the planetary positions for any reasonable distance into the future. We used the logic that because we had the math.. and the math made accurate predictions.. that the model had to be correct. And so we "knew" the Earth was the Center of the Universe, regardless of how strange planetary-retrograde motion appeared.

We never seem to learn to stop being so sure of ourselves. How many pioneers in Science have been banned from the Academic Halls and been labeled Mavericks or Heretics and suffered ruined careers, just to protect the current Status Quo?

Let's all hope we can stop doing that someday.

Anyway.. I feel I have demonstrated some minor logic flaws in your presentation. The solution is the concept of a variable density lattice.

Best regards as always,
Dave :^)

[cirenor]

Big gratitude for this thread, which I'm chomping through slowly but surely.

[BurtJordaan]

Hi Dave, some comments on your prior post:

Hi Jorrie,
Just a quick observation. So we know Galaxies come in all sizes and many shapes. So you are saying that the Blue part of the lattice are Galaxies? Then what smaller scale does the expansion stop applying to stars or planets or moons, or atoms etc...

The red cubes represent clusters of galaxies or superclusters (e.g. Virgo supercluster), which are gravitational bound (i.e. the galaxies are in some form of orbit around a common center of mass). There is an extremely tiny force caused by the expansion on cluster scale, but all it does is to make the (stable) orbit sizes marginally larger. Orbits do not expand with the cosmos.

As for the size of a Galaxy shrinking because the Fabric of Space-Time is condensing, how would you know.. since all the rulers that can be used are shrinking along with it?

A little hard to accept, for how would the laws of gravity work (also shrink?) It is also illogical to hold that things on atomic scales shrink. We observe no such effects on scales up to our own galaxy, where we see everything consistent with the standard view.

Under these conditions a Galaxy may grow and shrink at the same time. However, the only way to locally measure this shrinkage is by using some form of exotic ruler, that doesn't change length regardless of it's speed or proximity to say.. a black hole. Such a ruler, in hand, would appear to grow as one approached a black hole. But the speed of light will appear constant. True, a bit hard to wrap one's mind around.

Near black holes, space, time and rulers shrink, but only as viewed from afar. Nevertheless, the large scale does not work like black holes, of that we are certain. We observe large scale effects and local effects of black holes and everything seem consistent with general relativity. So why worry about some exotic scheme that might (perhaps) also explain it?

... Because something controls their relative scale when they are local to each other. That's the function of the lattice as I see it.

My lattice is not there to explain atomic scales; just the super large (cosmic) scales.

So if, as suggested, I were to find two remotely separated Galaxies that were stationary in respect to each other and spanned a hypothetical ladder between them, then by the expanding Universe concept, someday in the future the ladder will only reach half-way. And in my view, the ladder will stretch with the inflation of Space-Time.. without growing more rungs.. and always bridge the gap.

In my view, this ultra long ladder will eventually be broken apart by the expansion. You do not even need to 'tie' it to the two galaxies, you can place it in empty space and it will still be pulled apart by the accelerated expansion. This is outside the scope of this thread, but I may write one on it some time. There are many articles written about the "tethered galaxy" case. I will look up some references for you.

And by that token, a beam of light will always take the same amount of time to travel that gap, even though the remote Galaxy appears, due to an illusion of expanded space, to be moving further away.

Just to be clear.. I am not in disagreement with the measurements made by Science.. just the interpretation of them

I would take you up on your interpretation, but not in this thread, which is about basic, standard cosmology. Why not start an "Alternative Interpretation" thread, Dave? I would like to try to change your mind on that. :P

...

Anyway.. I feel I have demonstrated some minor logic flaws in your presentation. The solution is the concept of a variable density lattice.

But the whole lattice concept is one of variable density! The average density decreases all the time. I'm a little confused by your views, because I got the impression that you understand standard relativity and cosmology really well. Are you really trying to reinterpret all of it?

Kind Regards,

Jorrie