Looking Again at the Balloon Analogy

[liometopum]

Has anyone here ever asked these questions about the balloon analogy?

1. If the universe actually was the 3D surface volume of an expanding 4D hypersphere, how big would the radius be? (let the radius of the observable universe be 46.25 billion light years)

2. How fast is it expanding? (let the age be 13.8 billion years , via Planck mission)

3. What is the fractional change in the circumference of this hypothetical 4D hypersphere? That is, divide the change in the entire circumference by the circumference.

The answers are eye-catching.

 

[Drakkith]

If the answers are eye-catching, does that mean you've already got them? If so, why don't you post them?

 

[marcus]

Has anyone here ever asked these questions about the balloon analogy?

1. If the universe actually was the 3D surface volume of an expanding 4D hypersphere, how big would the radius be? (let the radius of the observable universe be 46.25 billion light years)

2. How fast is it expanding? (let the age be 13.8 billion years , via Planck mission)

3. What is the fractional change in the circumference of this hypothetical 4D hypersphere? That is, divide the change in the entire circumference by the circumference.

The answers are eye-catching.

We already discussed some of these questions in one or more other threads. Sorry I don't have time to go back and find links for you.

BTW People unfamiliar with GR think that distances cannot increase faster than c, but as you probably know there are distances to stuff which we observe which are increasing several times c, it is not forbidden (distance increase is not like relative motion with SR speed limit).

Also BTW one does not assume that the 3D hypersphere is immersed in 4D space. One can calculate a LOWER BOUND on the RADIUS OF CURVATURE (as was done in one of the WMAP reports), with 95% confidence. I think it was WMAP5.

One should not assume that RoC is a real distance in a real 4D space. It is what would be the radius of the 4D ball if the 3D hypersphere was actually the surface of a real 4D ball. But we do not assume the 3D hypersphere has any "inside" or "outside", all existence is on it,AFAWK.

One calculates the RoC lower bound from the estimate of Ω_k. If the 95% upper bound on |Ω_k| is, say, 0.01, then the lower bound on the RoC equals the Hubble radius R divided by sqrt|Ω_k| which is 0.1 max.

So if R = 14 GLY, the RoC must be AT LEAST 140 GLY. So the circumference must be at least 2π times that which is 44/7 times 140, which is like 880 billion LY.

So one would expect that the circumference (at least 880 Gly and likely much larger) is increasing at many times the speed of light. Easy calculation 880/14 = 63
So the circumference would be increase at least at the speed 63 c. This is just a lower bound on the speed assuming that estimate of Ω_k.

There may be no circumference, the U may be spatially infinite, but if it is finite and a hypersphere as you suggested then the circumference of that hypersphere is likely to be increasing much faster than 63c.

That answers at least a part of your questions. We've gone over this before, maybe one of the others has a link to earlier discussion. Hopefully if I've made any errors someone will correct me on this.

 

[marcus]

If the answers are eye-catching, does that mean you've already got them? If so, why don't you post them?

Heh heh I agree with what I think Drak is saying. Personally I can't think of anything especially surprising or "eye catching" coming up here, but if you think you have some surprising answers PLEASE POST THEM.

 

[liometopum]

the answers

I have the details, with equations, attached.

1. The hyperverse radius is about 27.59 billion light years.
2. the speed of radial (4th dimension) expansion is 2c, twice the speed of light.
3. The fractional increase of the hyperverse circumference matches the Hubble constant.

 

[liometopum]

To Marcus...from your first post..."3D hypersphere is immersed in 4D space"

I think you missed the whole idea. I am talking about a hollow, expanding, 4D sphere. It surface would be three dimensional, our universe. If we take the volume of the observable universe (with us in the center of a 3D sphere), and then ask "if our 3D spherical universe is the actually the surface volume of a 4D sphere, how big is this 4D sphere?"

The answer is that the radius of this hypothetical, expanding, hollow, 4D sphere would be about 27.59 billion light years.

Given an age of about 13.8 billion years, this hypothetical "hyperverse" would be expanding at, what appears to be exactly, twice the speed light.

Now, let me address the immediate thought... relativity applies to motion within the universe. This is not motion within the universe.

The 2c radial expansion rate suggests that the speed of light is a function, one-half, of the rate of radial expansion. It is the radial expansion of space that determines the speed of light.

And this should suggest to you that the radial expansion of space is the foundation of time.

But let us continue.

What is the fractional increase in the circumference of this hypothetical hyperverse? By 'fractional', I mean the total increase in the circumference, divided by the total circumference. This gives us the increase per point along the circumference.

The answer is the Hubble constant! And that is a powerful result.

Remember that the Hubble constant is given in terms of distance per time per distance. The distances cancel out, and at its core, the Hubble constant is a dimensionless number per unit time. So there is no inherent conflict in speaking of the Hubble constant being the rate of circumferential expansion.

 

[Chalnoth]

I have the details, with equations, attached.

1. The hyperverse radius is about 27.59 billion light years.
2. the speed of radial (4th dimension) expansion is 2c, twice the speed of light.
3. The fractional increase of the hyperverse circumference matches the Hubble constant.

The initial conditions are wrong. Even if the model of our universe as the surface of a 4D hypersphere were correct, the radius of said sphere at the time inflation ended is unknown. There may also be a mistake in how you estimated the radius changes over time (In reality, $R_c \propto a^2$), but I didn't look in detail at what you did.

Given the hypersphere model, the way you'd get at the radius would be to measure the spatial curvature. The radius of curvature at the present time can be defined as:

$$R_c = {c \over H_0\sqrt{\Omega_k}}$$

Here $\Omega_k$ is the fractional difference between the true energy density of our universe and the critical density (which is a function of the expansion rate). This parameter is currently measured to differ from zero by no more than about 0.005. This corresponds to a curvature radius of about 200 billion light years, at a minimum.

 

[Bandersnatch]

I have the details, with equations, attached.

If I correctly see what you did there, you took the observable universe and you mapped it onto a 3-sphere hyperarea for no apparent reason. Unless you think the observable universe is the whole universe(it isn't).
It's like taking your home town, and mapping it to the surface of a sphere so that it's fully covered, and then trying to come up with conclusions about geometry of the Earth. You'll get nonsensical, or "eye-catching" results as well.
The correct approach is to measure the curvature first, then use it to find the radius - which is what Marcus and Chalnoth have talked about(by the way it's >200 Gly now? I always seem to be 30% behind the current best measurements).

 

[liometopum]

A gift?

I also think that the whole universe is much larger than the observable. And, yes, Bandersnatch, I am taking the 3D volume of the observable universe and wrapping it around in one higher dimension.

The point of the post is this: if the observable universe was a hyperverse, we get striking results: a 2c hyper-radial expansion rate, and a circumferential expansion matching the Hubble constant.

What I want to share is the observation that a balloon analogy, made to match what we can see, what we an measure, the observable universe, gives amazing, ’eye-catching’ values. These are highly unlikely to be random results. A circumferential expansion rate matching the Hubble constant, by itself, should draw the attention of some of you. And the 2c hyper-radial expansion gives us a clue as to why the speed of light is the maximum velocity in the universe; the hyper-radial expansion rate appears to set the speed limit in the universe.

Is this a coincidence or a clue?

I vote clue. I have been on this for 4 years and have all sorts of conclusions, but that is beyond this post, and the forum rules.

You can just blow the observation off, or you can ponder the results. Maybe the universe is telling us something here. You can accept it as a gift from the universe, or throw it away.

 

[Drakkith]

There's no conclusion here because you are just arbitrarily mapping the observable universe to the surface of a hypersphere for no real reason. If we assume that the universe is larger than the observable universe (we have no reason to believe otherwise, and I've read about evidence suggesting that the universe is at least several tims larger than the observable universe) then none of this is correct. Furthermore, the speed of light has nothing to do with the expansion rate of the universe. It is a local speed limit, not a cosmological speed limit, and the expansion rate of two opposite sides of a hypersphere should have nothing to do with the speed limit along its surface.

 

[liometopum]

radial expansion is local and global

Drakkith, I think you are off on all the points.

1. Comment: “There's no conclusion here because you are just arbitrarily mapping the observable universe to the surface of a hypersphere for no real reason.  “
——-It is not arbitrary. I post here under the cloud of censorship, as I have had posts closed previously, for even hinting at independent work, so I am afraid to say anything in this regards. With that aside, I can safely state that it is well known that the universe, as the surface of a hollow, expanding, 4D hypersphere, gives a positively curved, and finite universe, without an edge or center, within the universe (the edge would be everywhere, and the center nowhere within the universe). It is ‘easy’ to explain expansion with a balloon model, except for the extra dimension messing people up. The bread model has edge issues. Many cosmologists speak as if there are only three dimensions, and that leads to issues of edge and center, and violation of the cosmological principal. The universe as the 3D surface volume of a 4D sphere solves those things. It is quite reasonable to use this model.

2. Comment: “If we assume that the universe is larger than the observable universe (we have no reason to believe otherwise, and I've read about evidence suggesting that the universe is at least several times larger than the observable universe) then none of this is correct.”
——Response, part 1: The ‘no reason to believe it is larger’….. The universe is measured at about 13.8 billion years old. How long has the light been traveling to us? Not that whole time. The universe grew during that 300,000 or so years before it could move freely, and it grew a lot. We only see to the edge of the ‘fog’, so to speak. The fog bank is large.
——Response, part 2: If the universe is larger than what we see, the fact remains that a hyperverse the size of the observable universe is expanding hyper-radially at 2c, and circumferentially at the Hubble constant. I am still working on what that means, but the observation stands. You can, and are, trashing it, and that is fine with me, but the observation needs to be placed out in the open. It means something. Maybe our measurements are some how tied to the size of the observable universe. But this is no coincidence; it is telling us something if we listen.

3. Comment: “Furthermore, the speed of light has nothing to do with the expansion rate of the universe. It is a local speed limit, not a cosmological speed limit, and the expansion rate of two opposite sides of a hypersphere should have nothing to do with the speed limit along its surface.”
——-Drakkith, I think that you too don’t have the correct image in your mind of what I trying, sadly, and not too well, to describe the hyperverse as being. With a hollow, expanding hyperverse, every point in space is on the edge of space, in direct contact with the, presumably, nothingness into which it is expanding. The 2c radial expansion is occurring at every point in space. It is a local phenomenon, and global as well, as every point everywhere is moving hyper-radially at 2c.
——In a universe that is the surface of a hyperverse, everything is moving hyper-radially at 2c, but not in any of our three dimensions. The direction is into the fourth dimension.
——Your statement about the separation of the two opposite sides… nooooo. That would be 4c. The 2c is from the center to the edge, and the edge is everywhere. The 2c expansion is local, and everywhere.

And thank you for taking the time to talk about this! I enjoy it. I won’t likely be able to respond further until this evening.

 

[bahamagreen]

I can't find the site to reference, but maybe someone up on topology can comment if they recognize this...

What I saw was a derivation that rotation of a four (might have been five) dimensional space results in a hyperbolic universal expansion of the three (might have been four) dimensional subspace (probably didn't get all that right).

The point was that our observed three dimensional expansion through time was what one would expect from this higher dimension rotation. And this suggested that at the higher dimension there need be only rotation with no expansion; the "transformation" of higher rotation to lower dimension expansion means we in the lower dimension subspace we only measure and observe universal expansion without rotation. I'm sure I didn't get all that right, too.

Anyone recognize something along these lines?

 

[Bandersnatch]

Disregarding all other issues, observation should always be the ultimate test of a proposition. A closed universe with the radius of curvature you specified would be detectable with the instruments we have. Both WMAP and PLANCK point to a remarkably flat universe, with the lower bound on the curvature radius in the vicinity of 200 Gly.
This ought to be enough for you to ditch the idea, no matter how you like it.

 

[Chalnoth]

You're misapplying the balloon analogy. Nowhere in the balloon analogy is it implied that the observable universe covers the entire balloon.

Yes, it is the case that the radius of curvature is increasing at a rate faster than c (by my calculations, if the radius of curvature is currently 200 billion light years, it is increasing at a rate of about 14c at present). This may seem nonsensical, but that's just because people tend to think the speed of light limitation applies to every speed we might ever want to write down. This isn't the case: in General Relativity, the speed of light limitation is the statement that nothing can outrun a light ray moving through vacuum. And this hypersphere certainly isn't outrunning any light rays due to the increase in its radius when the light rays are confined to be within the surface of the sphere.

Another way to put it is that there is no unique way to write down the distances or speeds of far-away objects. There are ways to write down these distances and speeds, but there isn't anyone preferred method. And if there is no unique definition of a speed, then how can it possibly obey any hard limits such as the speed of light?

In General Relativity, relative speeds are well-defined at a single point. This is why the rule is that nothing can outrun a light ray. But you can easily define speeds for far-away objects that never exceed the speed of light, sometimes exceed the speed of light, or always exceed the speed of light. It all depends upon what definition of speed you use.

 

[Bandersnatch]

Let me point out a math error as well.
You took a 3d euclidean space and mapped it onto a non-euclidean 3-sphere as if it were an euclidean space too.
It might be easier to visualise by taking the dimensions one notch down.

Say you've got a flat surface, on which you draw a circle or radius $R_0$. The area enclosed by the circle is the observable universe, and equals $\pi R_0^2$.

You want to map it onto a regular, 3-dimensional 2-sphere and find out what is its radius $R_H$.

To do that, you say the area of the 2-d observable universe and the surface area of the sphere must be equal. $\pi R_0^2=4\pi R_H^2$ from which follows that $R_H=1/2 R_0$.

However, now let's try and find what is the distance between the same observer and the edge of the universe before and after mapping.
Before, it's just $D=R_0$. After, it's half the circumference $D=\pi R_H$.
Which leads to $R_0=\pi R_H$. Substituting earlier calculations, we get $R_0=1/2\pi R_0$.

 

[Drakkith]

——-It is not arbitrary. I post here under the cloud of censorship, as I have had posts closed previously, for even hinting at independent work, so I am afraid to say anything in this regards. With that aside, I can safely state that it is well known that the universe, as the surface of a hollow, expanding, 4D hypersphere, gives a positively curved, and finite universe, without an edge or center, within the universe (the edge would be everywhere, and the center nowhere within the universe). It is ‘easy’ to explain expansion with a balloon model, except for the extra dimension messing people up. The bread model has edge issues. Many cosmologists speak as if there are only three dimensions, and that leads to issues of edge and center, and violation of the cosmological principal. The universe as the 3D surface volume of a 4D sphere solves those things. It is quite reasonable to use this model.

My issue isn't that you are mapping the universe onto the surface of a hypersphere, but that you are mapping only the observable universe onto this surface.

——Response, part 1: The ‘no reason to believe it is larger’….. The universe is measured at about 13.8 billion years old. How long has the light been traveling to us? Not that whole time. The universe grew during that 300,000 or so years before it could move freely, and it grew a lot. We only see to the edge of the ‘fog’, so to speak. The fog bank is large.

I said that there's no reason to believe that the universe isn't larger than the observable universe, not that there's no reason to believe it is larger. In other words, the universe as a whole should be larger than just the part we can see.

——Response, part 2: If the universe is larger than what we see, the fact remains that a hyperverse the size of the observable universe is expanding hyper-radially at 2c, and circumferentially at the Hubble constant. I am still working on what that means, but the observation stands. You can, and are, trashing it, and that is fine with me, but the observation needs to be placed out in the open. It means something. Maybe our measurements are some how tied to the size of the observable universe. But this is no coincidence; it is telling us something if we listen.

I don't see how you're coming to this conclusion. You can't map only the observable universe onto a hypersphere and leave out the rest. Since we don't know the size of the entire universe, the only way to get the size of the hypersphere is to look at the curvature.

Drakkith, I think that you too don’t have the correct image in your mind of what I trying, sadly, and not too well, to describe the hyperverse as being. With a hollow, expanding hyperverse, every point in space is on the edge of space, in direct contact with the, presumably, nothingness into which it is expanding. The 2c radial expansion is occurring at every point in space. It is a local phenomenon, and global as well, as every point everywhere is moving hyper-radially at 2c.
——In a universe that is the surface of a hyperverse, everything is moving hyper-radially at 2c, but not in any of our three dimensions. The direction is into the fourth dimension.
——Your statement about the separation of the two opposite sides… nooooo. That would be 4c. The 2c is from the center to the edge, and the edge is everywhere. The 2c expansion is local, and everywhere.

I get what you're saying, and it still doesn't make any sense. This 2c expansion is from the center to the edge, not between any two points on the surface, meaning that the recession velocity between any two points on the surface depends on the distance between them. The fact that there is a maximum velocity along the surface has nothing to do with any radial velocity.

 

[liometopum]

Disregarding all other issues, observation should always be the ultimate test of a proposition. A closed universe with the radius of curvature you specified would be detectable with the instruments we have. Both WMAP and PLANCK point to a remarkably flat universe, with the lower bound on the curvature radius in the vicinity of 200 Gly.
This ought to be enough for you to ditch the idea, no matter how you like it.

I stated earlier, that yes, the whole is larger than the observable. I agree with the idea. It is not a question of a radius of curvature to me, it is the observation itself, that a hypersphere, whose surface volume matches that of the observable universe, has a circumferential expansion rate that matches the Hubble constant. The math matches what we measure. Is that not an observation?

It is understanding why it is, that is the challenge.

That the "observable hyperverse" has an expansion rate that matches the Hubble constant is highly unlikely to be a coincidence. There is a reason for it, whatever it might be.

 

[liometopum]

You're misapplying the balloon analogy. Nowhere in the balloon analogy is it implied that the observable universe covers the entire balloon.

Hi Chalnoth. Misapplying? Is there a rule somewhere about how to use the analogy? Just make the balloon's 3D surface equal to the volume of the observable universe. What is wrong with that?

It seems nobody has ever bothered to do what we are looking at here, and thus nobody has ever faced the problem, or seen the opportunity.

 

[liometopum]

Yes, it is the case that the radius of curvature is increasing at a rate faster than c (by my calculations, if the radius of curvature is currently 200 billion light years, it is increasing at a rate of about 14c at present). This may seem nonsensical, but that's just because people tend to think the speed of light limitation applies to every speed we might ever want to write down. This isn't the case: in General Relativity, the speed of light limitation is the statement that nothing can outrun a light ray moving through vacuum.

I agree with you, and see no issues here.

And this hypersphere certainly isn't outrunning any light rays due to the increase in its radius when the light rays are confined to be within the surface of the sphere.

If I understand you correctly, you appear to understand something that most people cannot get... the relativity applies to motion within the universe, not the motion of the universe. I really like how you stated it!

Another way to put it is that there is no unique way to write down the distances or speeds of far-away objects. There are ways to write down these distances and speeds, but there isn't anyone preferred method. And if there is no unique definition of a speed, then how can it possibly obey any hard limits such as the speed of light?

Your comment gave me a thought. Maybe a private message is in order? Thanks for the input.

 

[Chalnoth]

I stated earlier, that yes, the whole is larger than the observable. I agree with the idea. It is not a question of a radius of curvature to me, it is the observation itself, that a hypersphere, whose surface volume matches that of the observable universe, has a circumferential expansion rate that matches the Hubble constant. The math matches what we measure. Is that not an observation?

No.

The radius of curvature increases as:

$${dR_c \over dt} = R_c H$$

Here $R_c$ is the radius of curvature and $H$ is the Hubble parameter. This is the case in any homogeneous, isotropic, expanding universe, no matter its features.

 

[Chalnoth]

Hi Chalnoth. Misapplying? Is there a rule somewhere about how to use the analogy? Just make the balloon's 3D surface equal to the volume of the observable universe. What is wrong with that?

The radius of curvature of that model is far, far smaller than the radius of curvature we observe. It doesn't match reality.

 

[Chalnoth]

If I understand you correctly, you appear to understand something that most people cannot get... the relativity applies to motion within the universe, not the motion of the universe. I really like how you stated it!

It's more than that. Even within our observable universe, by the usual definitions most of the galaxies we observe are now and always have been moving away from us at faster than the speed of light.

Your comment gave me a thought. Maybe a private message is in order? Thanks for the input.

I generally don't like to have physics discussions in private.

 

[liometopum]

Yes, it is the case that the radius of curvature is increasing at a rate faster than c (by my calculations, if the radius of curvature is currently 200 billion light years, it is increasing at a rate of about 14c at present). This may seem nonsensical, but that's just because people tend to think the speed of light limitation applies to every speed we might ever want to write down. This isn't the case: in General Relativity, the speed of light limitation is the statement that nothing can outrun a light ray moving through vacuum. And this hypersphere certainly isn't outrunning any light rays due to the increase in its radius when the light rays are confined to be within the surface of the sphere.

Yes, I agree. And I like how you view that a 2c hyper-radial expansion is not a violation of relativity.

 

[liometopum]

Let me point out a math error as well.
You took a 3d euclidean space and mapped it onto a non-euclidean 3-sphere as if it were an euclidean space too.
It might be easier to visualise by taking the dimensions one notch down.

Say you've got a flat surface, on which you draw a circle or radius $R_0$. The area enclosed by the circle is the observable universe, and equals $\pi R_0^2$.

You want to map it onto a regular, 3-dimensional 2-sphere and find out what is its radius $R_H$.

To do that, you say the area of the 2-d observable universe and the surface area of the sphere must be equal. $\pi R_0^2=4\pi R_H^2$ from which follows that $R_H=1/2 R_0$.

However, now let's try and find what is the distance between the same observer and the edge of the universe before and after mapping.
Before, it's just $D=R_0$. After, it's half the circumference $D=\pi R_H$.
Which leads to $R_0=\pi R_H$. Substituting earlier calculations, we get $R_0=1/2\pi R_0$.

I think it is you who has made the math error.
In your example, you take a 2D circle, and the observer is in the center. Then you wrap the 2D circle around a sphere, of suitable size, and claim that the distance is one-half the circumference. It is not; that is wrong.

Look at it this way. In your mind, take a sheet of paper that is cut in a circle. Place it top center on a proper-sized ball. Then fold the paper around to the other side. Do you see it? Are you going to place the entire perimeter of the circle on that one point, at the bottom? No way.

 

[liometopum]

the observable hyperverse acts like the whole should act

I don't see how you're coming to this conclusion. You can't map only the observable universe onto a hypersphere and leave out the rest. Since we don't know the size of the entire universe, the only way to get the size of the hypersphere is to look at the curvature.

Yes, we can map the observable onto a hypersphere. The results from this seemingly inappropriate model, are astounding. Can you explain it? I can't, yet at least, but that is what we get. I am trying to get the word out that something is up here, and it is likely very important in any attempt to understand what is going on in the universe.

I get what you're saying, and it still doesn't make any sense. This 2c expansion is from the center to the edge, not between any two points on the surface, meaning that the recession velocity between any two points on the surface depends on the distance between them.

The recession velocity of the surface is the Hubble constant, that is the fractional increase in the circumference of the observable hyperverse.

The fact that there is a maximum velocity along the surface has nothing to do with any radial velocity.

Yes it does. The two are intimately connected. We cannot have a hyper-radial expansion of twice the speed of light without some connection to the speed of light.

Drakkith, the observable hyperverse acts like the whole should act.

 

[liometopum]

The equation for the Hubble constant

No.

The radius of curvature increases as:

$${dR_c \over dt} = R_c H$$

Here $R_c$ is the radius of curvature and $H$ is the Hubble parameter. This is the case in any homogeneous, isotropic, expanding universe, no matter its features.

Let me give you the equation for the Hubble constant, based on the observable universe being an observable hyperverse:
$$H = {2c\over R_H}$$
That is a simple and powerful equation for the Hubble constant.

 

[liometopum]

It's more than that. Even within our observable universe, by the usual definitions most of the galaxies we observe are now and always have been moving away from us at faster than the speed of light.


I generally don't like to have physics discussions in private.

I will not discuss it then.

I wanted to say that you are probably the first person to understand, or at least state it, that a 2c hyper-radial expansion is not a violation of relativity.

I have been shocked by the resistance I have received about it. It reminded me of what Feynman said in one of his books (I have the audiobooks) of when he went to Brazil to teach. He'd lecture and the students would duly take notes. Then he'd ask them if they understood what he said, and all they could do was repeat what he had told them. I think there are lot of "The Physicists from Brazil" around.

So, thank you!

 

[Drakkith]

I have been shocked by the resistance I have received about it.

I don't see why. All you've done is claim you've found some meaningful cosmological clue that hundreds or thousands of professional scientists have failed to find despite years of constant searching. In addition, it doesn't even appear that your understanding of the math is adequate if some of the above posts are correct, and you're making claims without supporting them.

 

[Jorrie]

Let me give you the equation for the Hubble constant, based on the observable universe being an observable hyperverse:
$$H = {2c\over R_H}$$
That is a simple and powerful equation for the Hubble constant.

Yes, but as more than one respondent have indicated, demonstrably based on an unphysical definition of $R_H$. The observable universe can be seen as a surface patch (radius some 46 Glyr) on a larger hypersphere with observed radius anywhere from 200 Glyr (the lower bound) to an imaginary number (negative curvature geometry), including "infinity" for a flat geometry.

It is not clear to me what your variant of the balloon analogy is to achieve.

 

[Bandersnatch]

I think it is you who has made the math error.
In your example, you take a 2D circle, and the observer is in the center. Then you wrap the 2D circle around a sphere, of suitable size, and claim that the distance is one-half the circumference. It is not;

Sure it is. Shortest distance from pole to pole. Whether you have a flat or wrapped space, the distance to the farthest points you can see ought to be the same. Otherwise you're dealing with two different universes, only one of which is real.

And yes, when mapping the circle to the sphere you should stitch the whole circumference to a single point. Look at any simple map of the Earth, with the polar regions stretched along the top and bottom, where in reality these are only points(Mercator projection). It's even more pronounced in polar projection, where the whole circumference represents a single point.

That's why this simple approach doesn't work - you lose "folds" of the circle as you try to wrap it, so that in effect it must have less area.
This is exactly the same issue, albeit in reverse, as with trying to draw a flat map of the spherical Earth - the areas don't match, because the geometry is different. Any flat map of the Earth is a distortion of some aspect of the original spherical surface.

Now look at your numbers. You've got a radius of the hypersphere that produces circumference that doesn't match the radius of the observable universe. Looks like the circumference of the "surface" of the hypersphere(2∏R_H) is larger than the circumference of the "surface" of the hypersphere you started your calculations with(2*46Gly).


Anyway, the whole thing being in conflict with observation, means this whole discussion is unnecessary. That you so easily dismiss the ultimate objection says a lot about what we can expect.
All it's going to be about is you defending your exercise in numerology, and us telling you you've been chasing a red herring.

 

[liometopum]

Sure it is. .

and

Substituting earlier calculations, we get R0=1/2πR0.

Hi Bandersnatch:
In the case of a circle with area pi*r^2, and a radius of one unit, the distance from center to edge is one unit. A sphere with surface area (4*pi*r^2) has a "hyper-radius" of 1/2. That is, the distance from the center to surface is 1/2. We agree on that.

In the remodeling of the surface area, the observer that was in the center of the circle stays on the surface of the sphere. His distance has not changed.

 

[liometopum]

Any flat map of the Earth is a distortion of some aspect of the original spherical surface.

And maybe the universe is a distortion of 3D space? This was never my thought, but it applies.

Now look at your numbers. You've got a radius of the hypersphere that produces circumference that doesn't match the radius of the observable universe.

I am not sure what that is saying. But with that aside, they are different objects, yes.

Anyway, the whole thing being in conflict with observation, means this whole discussion is unnecessary. That you so easily dismiss the ultimate objection says a lot about what we can expect.

What observational conflict? There is no conflict. We take what we can see, and then calculate a volume. Then, we ask "how big would a 4D sphere be whose surface area matches what we see". That is it. There is no measurement conflict.

All it's going to be about is you defending your exercise in numerology, and us telling you you've been chasing a red herring.

What I see, and what I would like you to see, is that this pseudo-hyperverse gives us the exact numbers we would expect to find for the universe, if the universe were indeed the 3D surface volume of a 4D hypersphere. Exactly the numbers. You can blow it off, or take it as an insight. There are people who will find this interesting, and hopefully, valuable in there own pursuits. Let them decide if it numerology or not. But if you cannot explain it, then maybe you should be the one to let it go.

And let me say too, that I actually appreciate your input and challenges!

 

[liometopum]

Yes, but as more than one respondent have indicated, demonstrably based on an unphysical definition of $R_H$. The observable universe can be seen as a surface patch (radius some 46 Glyr) on a larger hypersphere with observed radius anywhere from 200 Glyr (the lower bound) to an imaginary number (negative curvature geometry), including "infinity" for a flat geometry.

It is not clear to me what your variant of the balloon analogy is to achieve.

Welcome Jorrie:

Yes, I agree the 'observable hyperverse" is a patch. No doubt, and I knew it from the start. But the results are noteworthy and should be on the table for others to see.

What does it achieve? I will give you something, but I have to go now. I will be in touch shortly.

 

[Chalnoth]

But the results are noteworthy and should be on the table for others to see.

I don't see how. It's a bunch of equations that don't (and can't) say anything about reality. How can that be "noteworthy"?

 

[Drakkith]

I think this thread has run its course. We get claims similar to yours all the time and they never lead anywhere. I recommend leaving the advancement of science to the trained scientists and not chasing a red herring just because something "looks interesting". In any case, PF rules prohibit speculation such as this. Thread locked.