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This is a Blog on relativity and cosmology for engineers and the like. My website "Relativity-4-Engineers" has more in-depth stuff.

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A complete index to the Relativity and Cosmology Blog can be viewed here: http://cr4.globalspec.com/blog/browse/22/Relativity-and-Cosmology"

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Cosmic Balloon Application: the Cosmic Heart

Posted June 24, 2009 11:00 PM by Jorrie

Now that we have a 'design' and an algorithm for the cosmic balloon, it is easy to apply it some cosmological problems. The first application is about the discussion on the 'cosmic teardrop' or 'cosmic heart'[1] between GK and Physicist during the design phase of the cosmic balloon. Here are some numerically correct illustrations of the scenarios discussed, taken out of a spreadsheet.

The surface of the expanding balloon represents the entire cosmos at any particular time. Figure 1 (right) represents just a slice here, with the circumference of the circle representing 3D space. The interior and exterior of the balloon is a hyper-dimension into which the balloon expands or deflates.

When we look back in time (to distant sources), we essentially look 'inside' the balloon - not really inside it, because the balloon was just smaller at earlier times and we are still looking along the surface of the smaller and smaller balloon. The red and blue curves are photons coming from (say) our east and west sides, with the colors just to distinguish between them when they overlap in some scenarios.

This is easily simulated on the cosmic balloon. We just trace the paths of photons backwards along the surface of the shrinking balloon until we find the place where they originated (or until we find the BB!). For every time step, photons move a distance cΔt towards us along the surface of the balloon. Whether they approach or recede from us depends on the expansion rate and their distance from us.

For our present cosmos, even taking its minimum size, the 'heart' formed by the rays is not very remarkable - it is stretched very thin. The photons originated at the time of transparency of the cosmos (the CMB epoch), around 400,000 years after the BB, which is 'at the origin', for all practical purposes.

The balloon radius is 100 Gly, (shown on a linear scale) and the time of photon travel is 13.7 Gy. The 'two-gigayear-rings' show that the expansion was originally very fast, then slowed down somewhat and is lately picking up speed again. This is understood in terms of the effect of matter domination at earlier epochs and vacuum energy domination at later epochs. The two dotted radial lines represent where the regions of the photon emissions are today - a proper distance of 46 Gly from us. When those photons were emitted (or rather were released to move freely), the source regions were only 42 million light years way from us, just over 1000 times closer than today.

The only way to get a fuller 'cosmic heart' (as discussed in the previous thread) is to slow the expansion down to a crawl. This will happen with a flat Einstein-de Sitter universe, but it takes an extremely long time. Interestingly, there is a matter+vacuum solution to the Friedman equations that does exactly that within a 'reasonable' time (Figure 2, left). With matter energy making up 50% of the critical density and vacuum energy 200% of critical density (case 0.5,0,2.0), the expansion essentially 'stops' at 50 Gly radius - it actually approaches the 50 Gly radius asymptotically. For a period roughly the age of our present universe, the expansion would continue, but then the rate drops off towards zero. This would have made the CMB photons to travel almost the radius of the total cosmos, arriving after some 150 Gy. The half-circumference of the circle is ~165 Gly.

If we let the time continue for long enough (and if nothing disturbs the near-equilibrium, almost static situation), the 'heart' spreads around the balloon and eventually the origins overlap as shown in Figure 3 (right). For this scenario, it happens after roughly 200 Gy. This scenario means that we could have observed the same region by looking into opposite directions in the sky.

Apart from the first (present day) scenario, these may be hypothetical cases, but it illustrates the power of the cosmic balloon model rather well. It is quite simple to model and results like these flows very easily from it.

As an aside, the requirement for this situation is not so far from the observed case. There is no theoretical reason why the cosmological constant could not have been some three times higher and the matter density around double what we deduce today. However, there are observations that seem to rule out such a set of parameters.

Jorrie

[1] The 'cosmic teardrop' of observable space appears when the expanding universe is presented on a flat spacetime diagram, as shown in Figure 13.2 from Relativity-4-Engineers (shown in Figure 4, right). The parabola is equivalent to the surface of the hyper-sphere above.

The main reason for not seeing the 'comic heart' in publications is the fact that one is bound to the hypersphere and cannot show time properly on it. The 'teardrop' view does not suffer from that drawback. However, there are also certain drawbacks in the flat spacetime representation, like how a closed universe would be pictured.

My 'two-gigayear-ring' trick is a way around the time problem. The 'cosmic heart' originally earned its name from taking a flat LCDM cosmos, where the scale of R0 is arbitrary and setting R0 to a convenient value like the Hubble radius (13.7 Gly). It gives this pretty heart shape trace for the CMB photons (Figure 5, right). Much nicer than the 'streamlined bomb' shape when the proper (minimum) R0 is used, as in the main figure above.

The 'teardrop' is essentially just this 'heart', with the circular coordinates flattened out.

-J

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#1

Re: Cosmic Balloon Application: the Cosmic Heart

06/25/2009 6:48 PM

Excuse me teacher (I have my hand up and a blank look on my face), can I ask some stupid questions? What do the blue and red lines represent? Are they blue-shifted and red-shifted light? What do the angles between the 2GLY circles represent? I assume that I am at the point where the lines meet, and looking into space (which means looking back in time). Beyond that I am clueless. I have this to read, but I'm not very hopeful. The scale from -100 to +100 has the 100s on the surface of the sphere. They must be in my 3-D space, but where?

It would be nice if your figures were numbered, and you referred to them by number in the text. I must get past the first one before I can move on.

-S

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#2
In reply to #1

Re: Cosmic Balloon Application: the Cosmic Heart

06/25/2009 9:25 PM

Hi S, you asked: "What do the blue and red lines represent? "

Simply the right- and left-hand paths of photons from you to the CMB epoch. The only reason for the color difference is that they overlap in some scenarios and the colors make it clearer which is which. Sorry if it caused confusion!

"What do the angles between the 2GLY circles represent?"

I do not understand your question. What angles? It is time (2 Gy) circles, showing the radius of the sphere (read off the grid) at certain ages of this 'toy' cosmos.

"I assume that I am at the point where the lines meet, and looking into space (which means looking back in time)"

Correct. You are looking back at the CMB surface via the two hyper-spherical paths. In real life you do not 'see' the curved paths, so it looks like you are viewing the inside of a hollow sphere from its center. The scale from -100 to 100 Gly is for hyperspace. The surface of the balloon is your 3D present space. The surface of the 'heart' is observable space, taking light travel time into account.

Yea, figure numbers is a good idea, but the CR4 editor is not very helpful on that. One can embed the figure numbers inside the saved images (which is not a good idea). I'll try and put it in the text with the figures next to the text, if possible.

-J

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#4
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Re: Cosmic Balloon Application: the Cosmic Heart

06/26/2009 9:03 PM

Hi Jorrie, I must be coming across as an idiot. You have been doing this so long it must be obvious to you. I'll bet there are others in CR4 land who are not responding because they don't want to look like an idiot too. I am the 'scapegoat' for all of them.

I said "What do the blue and red lines represent? "

You said "Simply the right- and left-hand paths of photons from you to the CMB epoch."

I believe it was Phys who said one (blue) was with (me) looking 180° from the other (red). It doesn't matter whether I am facing North, West, South, or East or anywhere in between. Is that right?

I said "What do the angles between the 2GLY circles represent?"

You said "I do not understand your question. What angles?"

They are not really angles, they are curves. Let me rephrase the question. "Why do the blue and red lines curve out before 2GY and curve in after that?

In your balloon3 spreadsheet the 'cosmic heart' is labeled 'Observable Space'. Did you mean only the heart, or the entire circle? Now lets go to Figure 1 of this thread. If I was to put a compass on the balloon with the point where the blue and red lines intersect the surface, and draw a circle where the dotted lines touch the balloon, it draws my observable universe (3-D), and the entire surface of the balloon is the entire 3-D universe. Is that correct? But since the red and blue lines are only a point at the surface, there is nothing to see there. Do you see why I am confused?

OK, I see from the last paragraph that the surface of the heart is observable space, but it encompasses multiple time periods. It seems that I should draw a circle at a particular time, like in the above paragraph, to get the observable space for that time period. For today, the circle is infinitely small!

-S

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#5
In reply to #4

Re: Cosmic Balloon Application: the Cosmic Heart

06/26/2009 11:26 PM

Hi S. No, no, asking questions is just about the 'least idiotic thing' one can do!

I must apologize if my posts are less than crystal clear - Blog posts are not 'papers' or books - they are supposed to be brief, stimulate discussion and generate interest in a subject (if possible).

You understand the blue and red lines correctly now. I'm taking 'right- and left-hand' to be 180 degrees apart, irrespective of how you face. Since we are working with one-dimensional space here,* there are only these two photon paths possible - with us at the top of the circle, they are coming from either the left or from the right.

You asked: "Why do the blue and red lines curve out before 2GY and curve in after that?"

It is simply the expansion of the balloon that does that. Any photon coming towards us from afar will be taken more 'outward' than what it can make headway 'towards' us. It is only when it is closer that its speed wins over the inevitable outward movement in hyperspace. I don't like the 'crawling ant analogy', but just for moment, picture a photon as an ant trying to reach us from half-way around an expanding balloon...

"In your balloon3 spreadsheet the 'cosmic heart' is labeled 'Observable Space'. Did you mean only the heart, or the entire circle?"

Yes, the title refers to the curve(s). In the OP above I wrote: "Figure 1 (right) represents just a slice here, with the circumference of the circle representing 3D space."

You wrote: "Now lets go to Figure 1 of this thread. If I was to put a compass on the balloon with the point where the blue and red lines intersect the surface, and draw a circle where the dotted lines touch the balloon, it draws my observable universe (3-D), and the entire surface of the balloon is the entire 3-D universe. Is that correct?"

Yes, with the qualifier that the circle you have drawn represents the horizon of your observable universe today, but you cannot observe it as it is today. What we can observe today is defined by the blue and red curves. If there happened to be a galaxy** at 46 Gly distance today, we will observe it as it was ~13 Gy ago, near the center of the circle, when the observable universe was less than 1/1000th of its present size.

Do not shy away from shouting if it is still not clear.

-J

* One can think of the balloon's surface in 2D spatial terms, but as I have warned GK before, it becomes even more confusing - been there, done it. It is better to stick to a slice of the balloon, essentially a wire-frame ring. It is then clear that the dotted '2 Gy rings' are not somewhere on the surface of the larger balloon, but represents earlier, smaller balloons.

** Galaxies did not yet form at that time - we only observe the 'wrinkles', the density fluctuations, etc. as part of the CMB at that distance.

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#6
In reply to #5

Re: Cosmic Balloon Application: the Cosmic Heart

06/27/2009 1:42 AM

Jorrie, it's getting late, but I wanted to get a question or two off so that you can answer for tomorrow. I understand the smaller rings mean when the universe was smaller then, and that the wrinkles in the CMB are the 'seeds' of later galaxies. It's the charts that are giving me trouble. Making progress, but more to make. I like your description of the 'slice' of the balloon as a wire frame. The dimensions are a bit puzzling. In the opening post you said the circumference of the circle represents 3-D space, but the sphere represents 3-D, not the circle, right?. Is the circle 2-D? In this post you said "Since we are working with one-dimensional space here,*". Does the slice lower the dimension by how many (one or two)?

"Any photon coming towards us from afar will be taken more 'outward' than what it can make headway 'towards' us. It is only when it is closer that its speed wins over the inevitable outward movement in hyperspace."

I get that, but in figure 2, the blue line starts out nearly -180° 'away' from us, but figure 3 has the angle about -220. How can you more 'away' then 180? Again, what does that angle represent? If you could tell me what is plotted in balloon3 for the cosmic heart it might answer that (nothing is labeled for the X or Y axis for that 'graph'). I get the impression that more expansion causes the angle to be larger, yet in the opening post you said "The only way to get a fuller 'cosmic heart' (as discussed in the previous thread) is to slow the expansion down to a crawl." Did you mean relatively to the initial expansion?

-S

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#7
In reply to #6

Re: Cosmic Balloon Application: the Cosmic Heart

06/27/2009 2:35 AM

Hi again S, you wrote: "In the opening post you said the circumference of the circle represents 3-D space, but the sphere represents 3-D, not the circle, right?"

No, if we think 1-D wire ring model, the wire represents 3-D space, i.e., all the normal space there is. The radial dimension is fictitious hyper-space. I suppose the other surface dimension of the balloon is inherently there (so that we can have N-S-E-W, but not not up-down), but I choose to suppress it for clarity (as I wrote in the * end note above).

You wrote: "I get that, but in figure 2, the blue line starts out nearly -180° 'away' from us, but figure 3 has the angle about -220. How can you more 'away' then 180? Again, what does that angle represent?"

Ah, I now understand what you meant by the 'angle-question' before. I guess the simplest way to put is this: if the ring is not expanding, a photon will move around the ring at the speed of light. In time t is will cover a distance ct and a 'ring-angle' of ct/R. Over time, it can circumnavigate the ring ad-infinitum.

If the ring expands, the photon will spiral outward (as Fyz has also explained in the previous thread) and the ring angle it will cover depends on the expansion rate and profile. Fyz talked about a constant expansion rate, in which case the photon will only circumnavigate the ring if the expansion rate is slow enough.

Figures 2 and 3 of the OP represents another special case where the expansion rate is very fast in the beginning and then slows down to zero around R=50 Gly. Check this on the spreadsheet: set (0.5, 0, 2.0), Ro=100, Tmax=150 and then 200 - it should clear up any doubts.

-J

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#8
In reply to #7

Re: Cosmic Balloon Application: the Cosmic Heart

06/28/2009 9:34 PM

Hi Jorrie,

"if the ring is not expanding, a photon will move around the ring at the speed of light. In time t is will cover a distance ct and a 'ring-angle' of ct/R. Over time, it can circumnavigate the ring ad-infinitum."

Aha, now we're getting somewhere. I should have thought "surfaces", because that is where the 3-D space is. It's difficult when the hyper dimension is so 'forcefully' presented. Now the conversations between G.K. and Phz make sense. I can't believe it took so long to get it. So the blue and red lines represent the longest photon path to us (the oldest time) that we can still see now.

I have finished Lineweaver's inflation paper. I would not have 'got it' from his figures, though his text was easy enough to follow. He calls the effect of inflation (accelerating expansion) tunnel vision, but it's in every direction.

Concerning fig. 3 with the overlapping blue and red lines, you said that the same thing would be visible from opposite directions. I can imagine seeing the Big Dipper in the south as well as the north. But the same effect would occur at every direction I looked. Two Orion's, and every other constellation. Do you suppose that it would be as bright at night as in the day as Olber described? I assume it could be calculated. In any case I think navigation by the stars would be out of the question. I wonder if mankind would have made any cosmological progress in that situation. What do you think?

-S

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#9
In reply to #8

Re: Cosmic Balloon Application: the Cosmic Heart

06/28/2009 10:35 PM

Hi S.

"So the blue and red lines represent the longest photon path to us (the oldest time) that we can still see now."

Yep, that's it. We can also see many shorter paths in the same directions - essentially everything that's on the blue and red lines are visible to us, but at different times, depending on where they are. Adding one more dimension, so that the 'heart' has a surface, we can observe everything on the 2-D surface, but still not inside/outside the heart.

"I can imagine seeing the Big Dipper in the south as well as the north. But the same effect would occur at every direction I looked. Two Orion's, and every other constellation."

If the 'static universe' is not too big and old enough then yes, one would have seen multiple copies of just about everything (in principle), even Earth. The copies would have been at much reduced brightness.

"Do you suppose that it would be as bright at night as in the day as Olber described?"

'Olber's paradox' depends on age and size. For 'every line of sight to end on a star', you need an infinitely large and infinitely old universe. A small, static, closed universe of infinite age (with multiple paths to stars, as described) may perhaps approach the Olber's brightness. Star navigation would have been next to impossible because of the brightness masking individual stars. But then, mankind itself would not have been possible, I guess!

-J

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#3

Re: Cosmic Balloon Application: the Cosmic Heart

06/26/2009 6:06 AM

I have made some minor wording changes (as hinted by StandardsGuy).

I have also uploaded an improved version (more user-friendly) of the balloon model spreadsheet, with observable space (the 'cosmic heart') plot added as a bonus. You are welcome to download it and play around. Please read the opening post carefully to understand what the 'cosmic heart' represents.

It is still an experimental spreadsheet, so some issues may remain. It tries its best to automate the plot scales for various inputs (top of column C, yellow highlights), but some human scaling interaction may be required, done in column H. By looking at the expansion curve (left on spreadsheet), it is normally clear which of the column H parameters need to be tweaked for a better picture. The vertical pink line is an approximation of the age of the cosmos for the set of inputs entered.

The radius Ro (cell H1) is arbitrary for a flat universe (Ωo = 1), because the radius of curvature is then infinite and the model does not care what value you put in there. I have used the Hubble radius (1/Ho) as default, because it gives interesting shapes.

When Ωo >< 1, the model still does not care, but the results may be slightly wrong. As an example if Ωo > 1, curvature is negative, the cosmos is open and cannot be correctly represented by a hypersphere. Nevertheless, it is interesting to look at what it throws out!

Time max (cell H2) simply sets the time span of the run, which is arbitrary, but too short or too long a time will cause nonsense plots.

The 'mind of its own' of Excel also causes the circular coordinates to sometimes come out non-circular. This can be corrected by stretching the outer boundaries of the right hand chart.

It will not sell well, but it's a nice toy!

-J

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#10
In reply to #3

Re: Cosmic Balloon Application: the Cosmic Heart

06/29/2009 7:23 PM

Jorrie,

Have you ever assessed Dr. Russ Humphreys white hole cosmology theory? Any merit to it?

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#11
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Re: Cosmic Balloon Application: the Cosmic Heart

06/30/2009 12:05 AM

Hi Guest.

Yes, I have and found his model seriously wanting! I think Dr. Russ has admitted that later on.

He has taken a different direction lately. I remember reading a later paper from him, but cannot remember much of it, except that it was technically reasonably well though out. It seems that his quest is to reconcile the billions of light years of observable cosmic distances with a young Earth and a young cosmos. I think it is a futile exercise.

-J

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#12
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Re: Cosmic Balloon Application: the Cosmic Heart

06/30/2009 4:54 PM

thank you

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#13

Re: Cosmic Balloon Application: the Cosmic Heart

06/30/2009 7:19 PM

Jorrie, in the previous thread, you said that Ω was very close to 1 before inflation (as I have also read elsewhere). If =1, then that would make the universe infinite I think. But if it evolves away from 1, the universe would then become finite. That is illogical. So can we conclude that Ω was not exactly 1? We talked before about an infinite universe expanding, which to me is also illogical. I think to some cosmologists, math is their god, and they 'obey' their god without regard to logic or common sense. I liked Lineweaver's comments about initial conditions (inflation doesn't solve problems, it just changes them to other problems). For example, if Ω started out at .9999999999999, that would be an initial condition that is just as hard to justify (if not more so) than .9995 now. I don't have any problems with God setting initial conditions on the 'seed' he planted.

Are you considering any other applications in this thread?

-S

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#14
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Re: Cosmic Balloon Application: the Cosmic Heart

06/30/2009 11:39 PM

Hi S, you wrote on Ω=1: "If =1, then that would make the universe infinite I think. But if it evolves away from 1, the universe would then become finite. That is illogical."

If Ω=1 identically, it does not evolve away from 1. Then the cosmos start out infinitely large but still expands, which is not a contradiction. Here 'expands' does not necessarily mean growing larger (infinity + 1 = infinity). But there is nothing stopping everything to move way from everything else...

I prefer the initial Ω=1.00000....000000001 (some 65 zeros before the one, at least). This makes the universe finite, just about as large as can be and it is 'flat' for all practical purposes. Maybe God also preferred this...

-J

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#15
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Re: Cosmic Balloon Application: the Cosmic Heart

07/01/2009 9:23 PM

"infinity + 1 = infinity"

No, no, no, the math god is angry! Nothing can be added to infinity. It's all there can be. This is like saying "I don't believe in infinity". You must have faith! I think I know what you are going to say. You will give a different definition of infinity and bring up some scientists who proved something that I can't refute. If only Cantor was here to defend me! BTW did you know that those 3 scientists who tried to prove that some infinities were bigger than some other infinities all died in the nut house? I don't suppose this would help:

Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." (found here)

That is the problem with mathematics.

"Then the cosmos start out infinitely large but still expands, which is not a contradiction."

I beg to differ!

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#16
In reply to #15

Re: Cosmic Balloon Application: the Cosmic Heart

07/01/2009 9:52 PM

Hi S.

""infinity + 1 = infinity"

No, no, no, the math god is angry! Nothing can be added to infinity. It's all there can be."

I do not want to add something to the end of the "infinitely long line" - I simply want to add something in-line, right in front of me. Politely ask the line elements to shift one up.

Yea, I know, it will take an infinitely long time for the line elements to shift up, so I can never add something in. If I try to find the end of the line, I'll look forever and won't find it.

Ah, I know. The line length must be infinity minus one; then I can add the one...

-J

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