## Expansion of the Universe in 4 Dimensions

Discussion of theories involving time as a dimension, time travel, relativity, branes, and so on, usually applying to the "real" universe which we live in.

### Expansion of the Universe in 4 Dimensions

Hi. I am new to forum. So, I am not sure if any similar subject has been discussed here before. I can remove the thread if it is a duplicate. My native language is not English, so also apologies for any typos, in advance.

I have been thinking about the expansion of the universe a lot lately and I believe that I have developed a different look on the subject. It requires some higher dimensional imagination capability; therefore, I think it wasn’t quite understood where I have tried to explain it before. Since this forum is focused on higher dimensions, I thought some of you might find it interesting and relate with it easily. It will get quite long in order to have a clear explanation, though.

The idea is simple; what if our universe is expanding in 4 spatial dimensions instead of 3? To visualize what it would be like, I have gone 1 dimension below, as most people do. So, I have imagined our universe like a balloon in a 3D environment and replaced our 3D space with the balloon's 2D surface, therefore we also became 2D creatures living on this surface.

When this 3D balloon is expanding, as 2D creatures living on its surface, we only notice that the surface is being stretched and all the points on this surface will be moving away from each other, similar to what our astronomers define what happens when the universe expands today.

But while the surface is stretching for the 2D creatures on it, the balloon’s radius is also increasing. This increase is happening on 3rd dimension direction and the creatures on its surface are not able to observe this motion since they are limited with their 2D reality. If we apply this to our 1 higher dimensional real universe, it means that; when we measure the expansion of the universe, we only measure the stretching of its 3D surface on a 4D hyper sphere but cannot notice any increase on its diameter which takes place in 4th dimension direction. (This seems easy to understand, but it is usually the point where people miss to grasp and get lost. So, if you want to continue, it is important to relate with this part well.)

But of course, there is a link between the surface and the radius. If you increase one of them, the increase of the other should be possible to be calculated. So, I draw a section of this balloon as below: The black arc in this sketch represents the surface section of our 3D balloon. It extends in –x and +x directions completing a circle. It has a radius R and there are 2 point objects; "A" and "B" standing on it. Since the radius is quite big, its center point (let’s call O) is not visible here but I guess you’ve got the picture.

This was showing what the static version of the balloon looks like. If this balloon would be expanding as explained above, after an amount of time we should have a scenario like the one below : While the balloon is expanding, A and B would be following the paths shown with the red lines. Such a movement would increase the L distance to L’, balloon’s radius will increase from R to R’ and t - t’ amount of time will be passed.
We want to calculate the relation between R and L when one of them increases. So, in order to have a clear picture, let’s get rid of the unnecessary parts of this sketch: “O” represents the center of our balloon. The distance between A and B increases when the balloon expands. We know the expansion rate of our universe today which is about 70 km/sec per each megaparsecs (3.26 million light years). This will correspond to increase of the distance between A and B in our model. If we could know how much the radius of the balloon has increased during this process, we could calculate the radius itself using the similar triangles method very easily.

For some reason, I wanted to set the speed which the radius increases to the speed of light, so I decided to try this value. If we insert all our values into our sketch, we get this: This is showing what happens after 1 second of time while our balloon is expanding. Since its radius is increasing with speed of light, BB’ must be equal to 300.000 km. So, let’s calculate what is our radius represented by |OB| above. OAB and OA’B’ are similar triangles in the sketch. So we can write:

|A’B’| / |AB|=|OB’| / |OB|
(|AB| + 70) / |AB| = (|OB| + 300.000) / |OB|
1 + 70 / |AB| = 1 + 300.000 / R
R= 300.000 x |AB| / 70
R = 300.000 x (3.260.000 x 9,47376 x 10^12) / 70
70R= ~ 9,26534 x 10^24
R= ~ 1,323 x 10^23km
R= ~ 13,97 x 10^9 light year.

So our balloon has a 13,96 billion light year radius. If this radius is expanding with speed of light every moment, and if it has been the same since the beginning of the universe, it would take 13,97 years for it to expand to this radius value.

13,97 billion years…

I don’t know this was convincing for you or not, but reaching to a value which is really close to age of our universe after such a simple calculation using only the speed of light and expansion rate of the universe was too much of a coincidence for me. So, I have decided to pursue this idea and see where it leads. But before that, let’s try to find out if this model is consistent with our modern Physics theories.

First thing you can argue about the model can be; nothing with mass can travel with speed of light according to today’s Physics theories. We have used the analogy of us being 2 dimensional creatures on the surface of the balloon before, but this was actually only for easier understanding. The basic version of the model only includes space, speed of light and expansion of the universe. We didn’t introduce any concepts such as; mass, gravity, energy, force etc. yet. If you accept the fact that the only thing expands in the above scenario is space, then there is no contradiction. Because in modern Physics, there is no speed limit to how fast the space itself can move.

Another argument can be that; we don’t have any observational evidence about such physical motion around us. If the objects were moving with that speed, they would be infinitely length contracted and we would notice that. The answer to that is simple; everything is moving with the same speed in the same direction, therefore it is not possible to notice any relativistic effects. It is not so difficult to grasp that I guess.

You can also argue that there is no evidence in our current Physics theories that shows we are moving in such speed. I guess that is not true. I can show you something in front of us all the time but we were not seeing it this way:
Many of you probably know that the coordinates of any event in space-time is expressed as x, y, z, t in modern Physics . Space-time is a 4 dimensional environment and t represents the 4th spatial coordinate. But since x, y, z has a unit of distance, the “t” coordinate which has a unit of time, has to be converted into a spatial coordinate, as well. The conversion rate for this purpose is; -yes you guessed it correct- is the speed of light “c”. Since the speed of light has a unit of distance/time, when you multiply it with time (c*t), the remaining value has a unit of distance, as it is supposed to be. Therefore, the coordinate of an event is usually expressed as x, y, z, c*t. This also implies that; even if an object is stationary in x, y, z coordinates in space-time, after “t” amount of time it will cover a distance in 4th dimension direction which is equal to c*t.

Let’s keep that in mind for a second and go back to our model. What happens to an object in our above scenario when an object stands still on balloons surface? It will stand still on the surface but at the same time it will be dragged outwards while the balloons radius is increasing, right? And if we want to calculate the amount of this drag, we should multiply the time passed with the speed which our balloon’s radius expands. We have set this speed to the speed of light when we have calculated the age of the universe, so this distance should be equal to c*t, as well.

So, this shows that the way our current Physics models use to calculate the distance between 2 events in space-time is exactly the same way which I have used the above scenario. I think this is a quite interesting coincidence.

Even though it surprisingly complies with the current Physics theories, I still have many unexplained questions about the theory. One of them is; the sign of time. Time has an opposite sign compared to all other spatial coordinates in Physics. So, if you call spatial coordinates with a + sign, the time has a – sign or vice versa. In above model, the displacement of any object shall be calculated with; S^2=x^2 + y^2 + z^2 + (c*t)^2 whereas in Physics, it is calculated with; S^2= x^2 + y^2 + z^2 - (c*t)^2. It is very difficult for me to visually explain how this opposite sign works physically and it is probably the answer to many of my troubles but anyhow, I am still trying to figure out. Time will tell the answer.

Before finish, I would like to mention a little bit about how this model can be useful. When you try to explain things in a new model, one of the arguments of the Physicists usually is; “What does your model explain better than the current model does?” After trying to understand what happens when the objects move on surface of the balloon in such a universe model, I have reached to an explanation about motion based time dilation using simple Pythagorean Theorem calculations. I will not go into the details of it here not to make this already long post any longer. You can read more about this in my blog post here: https://erens.quora.com/1-Do-We-Travel-in-Time-Direction-With-Speed-Of-Light. In addition to that, I also tried to show how you can reach the basics of gravitational time dilation in this post: https://erens.quora.com/Gravitational-Time-Dilation-Due-to-Expansion-of-the-Universe-in-4th-Dimension.

Sorry this was quite a long post, but I have tried to be as clear as possible. It will be nice to discuss with anyone who gets interested.

P.S. I think the essence of this post fits into the Spatial Dimensions section better. But I saw that the description of Non Spatial Dimensions section includes the real universe subjects, so I have posted it here. Admin can move the thread where he/she feels it fits into, if that matters.
eren_c
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### Re: Expansion of the Universe in 4 Dimensions

Hi Eren

Welcome to the forum. I had a quick look at your essays.

This kind of question troubles me, because i have a reasonably good grasp of the geometry.

1. Were space finite, like the surface of your balloon, you might see the same thing in different directions. This applies not only to the sphere, but where the space is any of the other weird shapes proposed. We have not observed any part of this.

2. Space, as far as can be determined, does not go far from the Euclidean space, that is, it's essentially flat.

3. Space is rated as expanding at a linear rate. This would happen in the balloon model, if the expansion is rated as per radius rather than per volume. This is the basis of Hubble's constant.

4. There are actually two theories of relativity, 'special relativity', which is the space-time thing you are talking of, and 'general relativity', which says gravity exists because of curvature.

5. What does 'red shift' mean? A photon has the energy of hc/l (where l = wavelength), and a red shift makes it go to hc/(l+L), would make it less energetic. Where does this energy go to.

The use of hubble's constant says that distances are increasing with time. This means that the volume is increasing with the cube of time, but because it's so long ago. While stars and galaxies do have proper motion, a good deal of what we see is due to the expansion of space.
The dream you dream alone is only a dream
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\(Latex\) at https://greasyfork.org/en/users/188714-wendy-krieger wendy
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### Re: Expansion of the Universe in 4 Dimensions

Hi Wendy,

Thanks a lot for your comments. I partly agree with what you wrote but I also need clarification on some of the topics.

1. Were space finite, like the surface of your balloon, you might see the same thing in different directions. This applies not only to the sphere, but where the space is any of the other weird shapes proposed. We have not observed any part of this.

I couldn’t see how this can be possible unless you are standing on the surface of a spherical object with huge gravity which bends light so much that light starts to orbit the object and comes at you from different directions showing the same item on the other side of the sphere as at multiple places at once. Is there any geometrical shape which can cause such effect without having huge gravity? I just ask to learn.

2. Space, as far as can be determined, does not go far from the Euclidean space, that is, it's essentially flat.

I’ve read before that the space is considered flat in Physics. I have my own reasons to believe that it has a curvature even though it is tiny. I have no way to prove that of course, so I have no point to argue.

3. Space is rated as expanding at a linear rate. This would happen in the balloon model, if the expansion is rated as per radius rather than per volume. This is the basis of Hubble's constant.

As far as I know, the space is rated as expanding at an accelerating rate. https://en.wikipedia.org/wiki/Accelerating_expansion_of_the_universe Did you mean something else with linear rate?

In 3D balloon model, if the radius increases with a constant speed, surface increase should be related with square of the radius (4πr²), but this happens in 2 directions. If we look at what happens in 1 direction only, it should increase constantly. Maybe this was what you meant. In case of a 4D balloon, the surface volume should increase with cube of r (2π²r³). But this is because the increase happens in all 3 directions and again if we look at 1 direction, it should still happen with a constant rate. This doesn’t match with our current observations, either. (I am not sure how does π having a square term effects 1 directional rate, though. Couldn't decide.)

If you have a look at the Hubble’s Law page in Wikipedia (https://en.wikipedia.org/wiki/Hubble%27s_law) you can see that the expansion rate has been measured differently more than 15 times since it is discovered. Furthermore, just last year, a group of scientists have claimed that they have found evidence that the expansion has a constant not an accelerating rate. As far as I know, this claim has been pulled back later on and we are left with an accelerating expansion again. So, this tells me that the value and even the nature of expansion is not fully understood yet. One day, they can claim again that it has a constant nature, or they may say it even has jerk instead of acceleration. I guess we have to wait more to find out.

4. There are actually two theories of relativity, 'special relativity', which is the space-time thing you are talking of, and 'general relativity', which says gravity exists because of curvature.

I agree. What I‘ve tried to explain above enters into STR field since it doesn’t include any concepts other than space, speed and time, as I wrote before. I have tried to enter into gravitational time dilation subject in the post that I’ve linked above which gave me a lot of hints about how gravity can arise in such a model, but at some point, it required the knowledge on the fundamental structure of space which I am not sure, if we will ever completely figure out or not. So, I am stuck at it for now.

5. What does 'red shift' mean? A photon has the energy of hc/l (where l = wavelength), and a red shift makes it go to hc/(l+L), would make it less energetic. Where does this energy go to.

I don't have any good answer to that. I have a different view on the light and speed of light being related with higher spatial dimensions but they are controversial, so I prefer not to argue over them.

The use of hubble's constant says that distances are increasing with time. This means that the volume is increasing with the cube of time, but because it's so long ago. While stars and galaxies do have proper motion, a good deal of what we see is due to the expansion of space.

I couldn't understand exactly what you mean with so long ago part. But I agree that good deal of what we see is due to the expansion of space. This is another nice part of this model for me. Having such a motion in 4th dimension direction presents a lot of default energy to our universe model. It automatically replies the questions about where did all this energy come from. Because it is already in it, since the beginning.
eren_c
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### Re: Expansion of the Universe in 4 Dimensions

Hi Eren,

I have been trying to tackle cosmology from a number of unusual sources. But it's useful to start at the other end: the photon, and red/blue shift.

We suppose two places, earth E, and a far away place F. A photon from F is emitted, and arrives at E red-shifted. Red-shifted means it has a longer wavelength L, and thus the energy is E=hc/L. The explanation of this is due to the doppler effect, and that galaxies like F are moving faster as things get further.

The doppler effect is real between a moving thing and its local background. The largest account of a blue-shift is 1150 km/s, which is to assume that 300 comes from us moving around our galaxy, about 300 from F moving around its galaxy, and 600 coming from the two galaxies moving towards each other. This is an edge-on pair of galaxies heading towards each other. The largest shift of true red shift is about this order too. Most of the time we would expect much lesser true shifts.

Part of the shift is caused by F moving in its frame of reference, that the photon at F is stretched by its source moving away from it. Likewise, we get a further increase if E is moving towards the photon, but these together, do not account for the full red-shift, and we know that we are actually seeing the photon from F at E. This means something else is causing the loss of energy in the line FE.

1. The universe is expanding, and as a result, everything, including the photon, is stretching. This would cause the observed red-shift, but leaves no clue to where the energy goes.

1a. We now need to consider, that if space is a balloon, what kind of symmetry it holds, and what makes it expand.

1b. Space is made of small plates, and these are being created.

2. Space is not expanding, and the loss of energy is due to some sort of friction at a large scale.

=====================================================================================================

1. One must assume, that if space is expanding in some higher dimension, then the universe as we know it, is the surface. Light would not need to travel in a straight line relative to the whole balloon. Instead, it would be like an ant on the surface, follow the shortest path on the surface. That is, light itself is part of the fabric of the surface.

Even if the sphere expanding at the speed of light, we could see other points. If you draw a circle around an event on the ray through OA, it will eventually cross the ray at OB.

This is a fairly common presentation, and one could assume that the change of wavelength in FE is because the tangents at E and F are at different angles, so the photon leaving F would be orientated at the angle of F, even when we see it at E. We see it in the same size, against a contracted length-background, which means that it has stretched, or red-shifted.

But i can not to the life of me, construct a model of space with generally E3 space, that expands in any 4D space, at a constant rate that comes to this.

2. Part of the GRT model, is that space is curved. Very few people actually give an explanation of what this curvature looks like, but the true nature is that if you get a cloth and a ring, and poke the cloth into the ring, so that a larger circle on the cloth matches the ring, then you pretty much have a model of how space is curved. What happens, around each mass M, the circumference of a circle is not just 2pi(R ) but 2pi(R+GM/c²). This second term is the amount of cloth you poked into the ring.

If you now suppose that space is in tension, and draw a unit square around part of a circle around M, then you get a greater length on the inner part than on the outer side of the square, and the difference gives GM/R^2. What happens is that the cloth on a measure x of any circle is 2pi x(r+ m)/2pi r, gives 1 + xm/(2pi r), where m=GM/c² . So the inner bit is pulling in by 1 +xm/(2pi 59.5), and the outer bit is pulling out by 1+xm/2pi(60.5), The net force is xm(60.5 - 59.5)/2pi (60.5 * 59.5) = xm/2pi(3599.75). This is the inverse square law.

A straight line divides the local circumference in millimetres, not degrees. Since in the example above, the square has the inner side slightly longer than the outer side (a trapesium, of shape (60.5, 60, 59.5, 60) the photon travelling through the square will exit on the other side, so that the heights outwards add to 59.5, rather than 60. This means that it would be deflected towards the heavy mass. If R is the closet distance it comes, the deflection is 4R/m, or 4GM/c²r, measured as radians. This is gravitational lensing.

If we suppose that instead of a solid fabric, space is formed of small platelets, then the density of the platelets define the curvature, there are more platelets where the curvature is more negative. A thing then more or less stumbles randomly into these platelets, and by the reasoning above, gravity is caused by matter moving randomly to platelets.

Geometry itself does not explain gravity, so you have to create a proportional tension.

-----------------------------------------------------------------------------------------------------------------------------------------------------------------------

2. The universe is not expanding, but the red shift is the cause by friction.

The trick here is that the photon, travelling from F to E, is indeed affected by doppler effect of the motions of F and E, but it slows down as a result of travelling from F to E. This means that when we see a red shift, we are actually seeing a loss of energy, and this loss of energy is given by the photon slowing down.

The measure cH has the dimensions of acceleration, this is about 10^-10 of earth's gravity (which is light-speed / 354 days, or c/1 yr). We now suppose that the photon is in flight for t years, which gives an apparent red-shift velocity of cHt. This is tc/ hubble-time. We could then suppose that the energy loss of a photon is thus E' = E cHt gives the energy as E = hc² Ht / L, and that the wavelength is reduced by say Ht (time of flight / hubble-time)

The calculations are free and fancy here, but pointing to the notion that redshift is not caused by extra space being created, but that space has a slight drag on the photon.

3. Where does all this energy go?

If the slight drag is caused by the gravitational field, then what we are seeing is not so much that the photon is loosing it to free space, but to the gravitational field itself.

This is important, because, gravity, as an attractive field, is a source of energy. To conserve energy, we sink it from the drag photons.
The dream you dream alone is only a dream
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\(Latex\) at https://greasyfork.org/en/users/188714-wendy-krieger wendy
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### Re: Expansion of the Universe in 4 Dimensions

Hi Wendy,

Thanks for your reply. I know and agree on the things that you said about the doppler effect. But I can't decide if the energy is really lost when there is doppler effect or is it happening from our point of view only. In the case of the simplest example of doppler effect, when a car passes us, what happens to the energy of its sound waves before and after it passes us? Does it loose energy just because it has passed us? Or if we start to travel after that car, does its sound wave magically gain its energy back? Can our own actions effect the energy level of a wave? I might be mistaken but it sounds similar with the case of universe's expansion.

1... Even if the sphere expanding at the speed of light, we could see other points. If you draw a circle around an event on the ray through OA, it will eventually cross the ray at OB.

This is a fairly common presentation, and one could assume that the change of wavelength in FE is because the tangents at E and F are at different angles, so the photon leaving F would be orientated at the angle of F, even when we see it at E. We see it in the same size, against a contracted length-background, which means that it has stretched, or red-shifted.

But i can not to the life of me, construct a model of space with generally E3 space, that expands in any 4D space, at a constant rate that comes to this.

It is not clear to me what you meant by this. The things you wrote before your last sentence tells me that the change of wavelength complies with this model so the model can be useful to demonstrate that but the last sentence says anyway you don't believe that such a model can lead to this. Maybe I couldn't understand the part before your last sentence.

I guess what you have described is a hyperbolic geometry in the next topic. I am having hard time to visualize anything with hyperbolic geometry and couldn't get into the maths you wrote, so I must skip that.

2. The universe is not expanding, but the red shift is the cause by friction.

The trick here is that the photon, travelling from F to E, is indeed affected by doppler effect of the motions of F and E, but it slows down as a result of travelling from F to E. This means that when we see a red shift, we are actually seeing a loss of energy, and this loss of energy is given by the photon slowing down.

As far as I know the speed of light in vacuum is invariant and its speed is not affected by doppler effect, only its frequency. For the loss of energy, I think as I wrote above.
eren_c
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### Re: Expansion of the Universe in 4 Dimensions

I don't know whether it truely relates here, but I had recently a similar problem. It does neither relate to Doppler nor to macrocosmic distances. Rather it simply happens at a very locale scale when light is refracted at some change of medium. The theorem of Snellius there says   sin(alpha1)/sin(alpha2) = c1/c2 = n2/n1,   where alphak are the angles wrt. the surface normal, ck are the velocities (of light) within the respective medium, and   nk = cvacuum/ck   are the respective refraction indices.

My problem then was, as   c = lambda * f   (wavelength times frequency), which of those would have to change according to the change of c, and which would keep the same. And, depending on the definition of color by wavelength or by frequency, would the color change while refraction or not?

--- rk
Klitzing
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### Re: Expansion of the Universe in 4 Dimensions

Snell's law.

The photon equation is E = hf. (energy = planck's constant * frequency). The frequency never changes. What changes is lambda, so we get c/n = lambda/n × f.

When the law is illistrated, one shows parallel wavefronts at an angle to the surface, which move into a separate set of parallel wavefronts at a different angle. Since the wavefront replace each other at the rate of 1/f, we see that lambda_1 = x sin(alpha_1) and lambda_2 = x sin(alpha_2) where x is the spacing on the surface.

What we get next is lambda_1 f = c_1 = c/n_1, and lambda_2 f = c_2 = c/n_2, or lambda_1 / lambda_2 = n_2 / n_1. or n_1 lambda_1 = n_2 lambda_2.

Note that f does not change, and E = hf is still observed. Likewise deBroglie's equation is E = hc_1/lamda_1, = h(c/n_1) / lambda_1 = hc/(n_1 lamdbda_1). But from the previous equation, the factor in the denominator does not change for passing through media of different refractive media, and therefore Snell's law is not a source of energy loss.
The dream you dream alone is only a dream
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### Re: Expansion of the Universe in 4 Dimensions

The doppler effect is real. In particular, it is due to the relative motion of the source and medium, and to the listener and the medium.

The proper medium for sound is the air. If you are travelling parallel to a fast train, you would not hear an effect. Instead, you would see the train as it is at your level, but the sound of the engine would be as if it were L/324 seconds ago, where L is the distance between you and the train, and sound travelling at 324 metres/second.

But the doppler effect here is a bulk effect. It is the risings of sound-crests are increased or decreased by the relative speeds of you and the engine to the air. The engine compresses them in the direction of travel, so that you get (c-v), and you when you cross the crests, at a rate of (c+u).

Since we suppose there is a proper medium, then we should measure proper speed against it. What happens is that were space expanding, then the apparent speed of F is based on the proper speeds of E and F, and that new space is being created between E and F. While we expect the proper speeds of E and F to produce a doppler effect in the way that it does for sound, there is nothing in the expansion that ought expand things. Even if it did, the missing energy still has to be explained.

We can see that proper motion is apparent because there is a blue shift. The blue shift is smaller, it only goes as much as 1150 km/s, which is consumate with the speeds and rotations of galaxies locally. But if this is proper motion of the stars being dragged across space, it does not explain the EF bit.

If you take some wave, and put pins in at the peaks, stretching the underlying space would also stretch the wave. This might cause the doppler effect here too, but here there is a loss of energy in flight, rather than the source and target being dragged across space.
The dream you dream alone is only a dream
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### Re: Expansion of the Universe in 4 Dimensions

I have got four ojections.

1.) The radius of the universe - if it has spherical topology - is much larger than the size of the visible universe. We know this from measurements of spacetime curveture on large scales. So in your model the balloon would have to expand much faster than the speed of light.

2.) The reason the size of the observable universe is close to the age of the universe is simply that this is the time, that light had, in order to reach us.

3.) The expansion of the universe is not constant. It was likely exponential in the first instances, decreased from there and is now once again increasing. It is not driven from the outside, but by the energy densities of the universe.

4.) For a simple spherical geometry it's compelling to think in terms of a spacetime embedded in higher dimensions, but order to describe every aspect of spacetime curveture you need not just 4, but 46 dimensions. Clearly this makes it a very unlikely and unhandy model.
What is deep in our world is superficial in higher dimensions.
Teragon
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### Re: Expansion of the Universe in 4 Dimensions

Hi Teragon,

The starting point of this idea was the motion of space with speed of light in 4th dimension, I had no balloon idea at the beginning. I was thinking about a linear motion since the c*t term in relativity was pointing that out to me. I have started to use the balloon analogy for the ease of visualization and saw that the curvature of a spherical shape can be useful to explain things like gravitational time dilation. I have explained how this works out in the blog that I have linked in the original post.

But later on, when I thought about the propagation of light in such a universe model, I found many problems with it due to this spherical shape. Roughly, it is about the time has to stop for the light (relative to us) according to modern Physics. That means, in order for time to stop for the light, it shall be trapped in a circular path in my model while our objects are travelling outwards of the sphere. That would mean either we can never observe the future events or they should be already there since the beginning of the universe, so we just happen to hit the light in our future during our outwards travel. And that leads to many problems about determinism and causality. It is difficult to explain all the details in a short post and it won't be clear with such a short explanation so I won't go any further. In short, now I believe that it is not a spherical geometry but something else with a curvature. The things that I have noticed, points me out an inside-out spherical geometry. I heard that hyperbolic geometry can be described like an inside out-sphere and I also heard that Minkowski space has such a geometry. At the moment, I am not able to visualize that geometry, so I am just stuck at it.

Just as a thought experience, one possibility comes to my mind is that all of these things are happening inside the event horizon of a black hole. That would be a good explanation for the motion with speed of light since this is the expected speed of the objects inside the event horizon, as far as I know. And due to the singularity at the the center of that black hole, the space shall be bent more and more while you are travelling towards the center. That sounds like, the more you travel to the center of the black hole, instead of the space around you to be getting tightened as you would expect from a regular geometry, the space can remain the same (if not expanding). That would prevent the objects to be getting closer to each other while travelling towards the center of the black hole. The distance between them may even increase, I am not sure. That could also be a good explanation for that inside-out spherical geometry. Of course, all of this is a speculation since we cannot know anything about the events beyond the event horizon. Just wanted to share.
eren_c
Mononian

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