The Nature of Space by M. V. Vasilyeva

Mr Kley,

Thank you for your interest in my essay. It was very late for me when I saw your question. In my reply I used wrong terms in the sentence which should have read: both 'attractive' and 'repulsive' gravity emerge from the same property of the hypersurface of the 4D structure of space that seeks to minimize its hyperarea.

You asked, "Can you offer up any specific experiment or present unexplained phenomena that may be illuminated by your vision of 4D Space/Time?"

Having thought your question over, I decided to address the wave-particle duality, with which physics replaced the paradox of space. This duality is best revealed in the double slit experiments. I would like to use this opportunity to demonstrate that, from the 4D perspective, there is no mystery why both matter and light appear to move in waves. In my essay I only briefly mention this in the Flatland analogy.

Before we begin, it is important to appreciate that a hypersphere has 3-dimensional surface, each point of which is equidistant from its center, in the same way as each point of the 2D surface of a sphere is equidistant from the center of the sphere. This topological fact is what makes the 3D space we perceive invariant in all 3 directions and precludes the possibility of selecting a preferred reference frame. The dynamic nature of the 4D structure also makes it unsuitable for the role of the absolute reference frame (just like sea captains can't use ocean as a reference frame and must rely on the external clues such as stars or GPS).

The other aspect of the 4D geometry worth remembering is that form a 4D perspective, each point of the volume of a cube is visible in one sweep just like we grasp each point of the 2D plane in one glance. In a sense, from a 4D perspective, a 3D volume of a cube appears flat, similar to a 2D plane seen from 3D.

With this in mind, let us see how a double slit experiments works out in 4D, on the simplified analogy of the Flatland Plane (the Flatland analogy is indispensable, because 4D is virtually impossible for most people to visualize).

First, I would like you to please take a look at the following image from google images, since I can't post images here:

http://ej.iop.org/images/0295-5075/94/2/20004/Full/epl13428fig1.jpg

The top image, (a) is a side-view snapshot of a hexagonal lattice aggregate of bouncing droplets on a vibrated liquid bath that interact via the surface waves they emit and form various types of stable crystalline clusters [1]. Suggested by Andrew Norton, it is indeed an excellent model of the Flatland nanoscale with droplets representing nuclei confined to the outer, empty side of the Plane (in the "headroom"). Please take a good look at the image again and imagine 3 such aggregations, at some distance apart from each other. These 3 aggregations of droplets/atoms represent the solid structure of the plate with 2 slits in between.

Now please appreciate the fact that light waves, an electron, an atom or a molecule, all move along the surface of the Plane (or hyperplane in 4D) that contains the EM field. The difference between a light wave and, say, an atom, is that the light wave is the transverse disturbance in the surface itself (just like a transverse wave in the surface of water) and as such is entirely confined to the EM field it contains, while "matter" (stuff with intrinsic mass) glides just above this surface in the 3rd dimension (3rd dimension in the Flatland analogy and in the 4th dimension in our world). Nuclei are integrated into the surface by their electron clouds interacting with the EM field in it. The electron cloud makes the indentation a nucleus makes in the surface locally even with the rest of the surface and at the same time acts as a sort of a roller or better yet, a surf-board on which the nucleus surfs the light waves themselves. (In this context, the increase in the inertial mass at high speeds is mostly due to the electrons interacting with the EM field contained in the surface.)

Thus there is no difference in the path a particle of light or a material object takes. A light particle is a convenient abstraction that stands for the momentum of a light wave in space (or, a hyperplane). A material object, such as an atom, can also be represented by a point that too follows the same path. It is the structure of space itself that dictates all movement by expelling the deformation introduced into it locally into the direction that gives (even though, of course, there is a difference in the transverse disturbance that we perceive as light and a longitudinal disturbance we perceive as mass).

In this model it is essential to realize that everything we perceive directly or with the help of our technology are the projections onto the 3D surface of the hyperplane. Matter moves along the same surface as light waves themselves. Thus the interference pattern seen in the double slit experiments is the reflection of the fluid nature of the structure of space itself.

I also would like to emphasize that the strict separation between matter and space of the model I propose is applicable only at low energies of our own experience. At very high energies, the number of space dimensions grows in proportion to the energy density. At high energies, in higher number of dimensions, the geometries of space and matter intermix and are best described by the multidimensional models of string theories.

Again, I thank you for your interest in my essay and for giving me opportunity to address this important question, which I could not do within the constrains of the requirements.

References

[1] A. Eddi, A. Boudaoud and Y. Couder, (2011, doi:10.1209/0295-5075/94/20004), Oscillating instability in bouncing droplet crystals. http://iopscience.iop.org/0295-5075/94/2/20004

Hello James!

Thank you for your interest in my essay. I like the vividness with which you describe how you perceive the world. I will certainly read your essay and will comment in your thread.

Mr Kley, I replied to your question in a comment below.

Thank you for your interest to my essay.

Thank you Ke Xiao for your feedback. Even though it was somewhat difficult for me to read your highly technical essay, I did grasp its main point, namely that the wave-particle duality paradox lies in the structure of spacetime itself. Here we are in agreement; and while you amply speak to the professionals, I offer a geometrical representation of the same on the simplified analogy, making it accessible to the lay public.

Xie xie!

Dear Amazigh Mabrouk Hannou,

I tired reading your essay but had a very hard time understanding your "accent in print". This is because, even though I know several languages, I have not studied yours. I think it is important to have our papers checked by a native English speaker before submission. This becomes especially important when introducing new ideas.

Regarding your question of gravity, my approach is purely geometrical and is inline with Einstein's general relativity, which treats gravity as a curvature in 4D spacetime. My take on it is that this curvature is due to surface tension of a 4D ocean of perfect fluid that seeks to minimize its hyperarea (similar to water tension in 3D).

In a topological sense, in 4D, there may be another reason for 3/4 - 1/4 distribution of energies. It may be due to the fact that a convex 4-space can be broken down into 4 contiguous, adjacent 3-spaces. This means that a convex 4D object, such as a hypersphere discussed in my essay, can be broken down into 1/4 for its 3D surface and 3/4 for its bulk. (The same does not work for a sphere, but works, with different numbers, for all n-spaces where n>3.) The same overall distribution of hypersurface to hypervolume, 1/3 to 3/4 respectively, applies to other convex 4D objects, even though actual proportions may vary a bit when they deviate from a hypersphere, which has the minimal surface area.

Take care!

Dear Dr. Perez,

very sorry for the delay. First I overlooked your post, and now I got lots of work... I very much value your feedback and want to continue our discussion. I will answer your questions as time permits and will post them in the end of the thread, so they won't get lost.

First, regarding the structure of space and what it is "made of", there is a concept in topology where a structure not only fully occupies a space but also defines it. That's how I mean it. And yes, like everything else, the structure of space is a particular expression of energy, just like charge is an expression of energy and so is mass, etc.

You say: "It seems that you haven't realized that by conceiving space with an internal structure as a fluid or strings you are contradicting the background independence of the general theory of relativity."

Not at all. In fact, the model I propose is just an extension of the geometry of GR. It suggests that the 3D space GR deals with is the surface of a 4D structure. It seems that our difficulty lies in accepting the reality of the 4th spatial dimension and Minkowski spacetime was a step in the right direction.

You say: "That space has this kind of internal structure means that space is not geometry as GR states."

-?? Geometry implies structure. Geometry is all about a structure. There is no geometry without a structure.

You say: "You: "The high energies at first instances after the BB correspond to higher-dimensionality of space." From where did you get that? The BB theory is based on GR and it assumes that since the beginning of time the universe is 3+1 dimensions, no matter how hot or cold the universe was."

Thank you for bringing this up. I did overlook this point, as we often do when an issue appears self-evident to us. And so I have a question to you in turn: what does BB have to say about how exactly the whole of the universe fit into a point and then a very small volume which then expanded? Or, where did that given 3D space come from in the first place, i.e. how does it appear out of nothing? And what constitutes the expansion of space, i.e. what is expanding and exactly how? And, not to forget, why 3D? Why not 4 or 5? Does BB have anything to say about all this? Not to my knowledge. The theory is completely silent on all these questions.

In contrast, my conceptual model gives answers to all these questions. Namely:

The universe fit into a point because it was compressed into an infinite number of dimensions. As it was cooling off, the number of dimensions diminished while the length of the remaining dimensions increased. This is what constitutes the expansion.

To illustrate this, I give you a simple example: you can take 8 cubes (edge length = 1 unit) and stack them in 3D so they make up a larger cube with the edge length of 2 units. You can arrange the same 8 cubes in 4D to make up a tesseract (a 4D cube) with the edge length of 1 unit. So, if the length of the edge stands for the lengh of a dimension, you just squeezed 3D space with dimension length of 2 units into a 4D space with the dimension length of 1 unit. The opposite process is when you take a tesseract and rearrange its volume in 3D. The size of the dimensions in 3D is twice as large. This explanation is as simple as it gets.

In the similar fashion, when the whole volume of the universe is compressed into an infinite number of dimensions, the length or size of those dimensions approaches a point. And as it cools off, the number of dimensions diminishes, i.e. the volume hidden in higher dimensions trickles down into the lower dimensions increasing their length. This constitutes the expansion.

I have to run now, but in the next post I will show why 3D. And I will certainly address all of your other questions. Thank you!

Mr. Fernando,

That sounds very interesting. I will read your essay as soon as I get a free moment and will comment in your thread.

Take care!

Mr. Hoang Cao Hai,

Thank you for your interest in my essay. Unlike your very advanced and far reaching work, I do not have ready answers about the concept as fundamental as space. How about you? What is space according to you? And what is your opinion about the reality of a 4th spatial dimension?

Dear J.R.,

Thank you for your interest in my essay. You say "...plus time as a possible 4th dimension", but my model stands on the reality of the 4th spatial dimension at low energies of own own experience. -?

Regarding your question of compressibility, the answer is yes and no. I conceive of a basic unit of spacetime as a 3D volume, which is incompressible as such. However, you can pack these "bricks" by arranging them in higher dimensions and it is in this topological sense that space is compressed.

If you read my post addressed to Dr. Perez just above yours, I give a simple example of how a 3D volume can be packed into 4D, resulting in the decrease in the length of the dimensions. The same process can be repeated for 5D, 6D, etc, all the way to infinity. As the number of dimensions grows, their size (or length) decreases, approaching 0. The same process going in the opposite direction explains how space expands (i.e. as the number of dimensions decreases, the length of the remaining dimensions grows).

Dr. Perez,

here I continue our discussion and will show now how the model I propose explains "why 3D?"

As I stated in the reply to J.R. above, the basic unit of spacetime I propose is 3D. From topological point of view, 3D is the minimal volume that is preserved in all n-spaces where n > 2. I have shown in the example above how 3d volumes can be packed into 4D with the result that the volumes themselves are preserved, while the length of the dimensions decreases. That's how, in principle, you can pack the whole of the universe into what can amount to nearly a point, as the number of dimensions grows to infinity.

Please keep in mind that each n-space, in topological sense, has its own characteristics and properties. Mathematicians often overlook this and thus loose the appreciation of the role of topology in physics. Laymen too, usually make a mistake when starting learning 4D visualization, which they do by heavily relying on analogies, applying the relationships in 2D->3D to 3D->4D. Most people do not realize the crucial difference between 2D->3D and 3D->4D, and that is: you can pack 3D volumes into 4D, just as I showed on the example above, where all 8 cubes are aligned face to face and edge to edge, thus filling in the whole of the 4D volume, while their own 3D volume is preserved fully intact. The same is not true about 2D->3D. You can take an infinite number of 2D planes and no matter which way you stack them, they will still amount to exactly 0 3D volume. What 2D planes can do is to enclose a 3D volume and with the size of their area give an indication of its shape.

And so 3D is the basic unit of volume, in which all other n-volumes (n>2) can be expressed. It is in this sense that 3D can be viewed as a given, always present as a subspace of any other n-space.

Now, we have a dynamic, vibrating structure that is initially packed in infinite number of dimensions (corresponding to its high energy density) and so the question becomes, as it cools off and expands, at what number of dimensions it will settle. I claim it is 4D, because of the unique properties of this space.

4D contains the largest number of regular polytopes (a.k.a. platonic solids), which is 6, compare with 5 in 3D and 3 in all other n-spaces (n>2). This is important, because it speaks of the number of symmetries n-space permits. In addition, in space of an even number of dimensions rotation takes place around a point, a plane, or some other axis-space of an even number of dimensions, while in space of an odd number of dimensions the axis of a rotation is always of an odd number of dimensions. Thus 4D, unlike any other space, offers the maximum number of symmetries, which is important for a structure that seeks to harmonize the vibrations of its components.

True, this needs to be backed by a proper theorem. Perhaps it, or something similar, already exists. If not, it should be proven. Namely: a dynamic vibrating structure of N-dimensions will find its lowest energy state in 4D.

Assuming this is so, what we get is a hypersphere, which geometers call a 4-sphere and topologists, a 3-sphere. This differences in definitions may be confusion for a layman, which is why I use 'hypersphere'. Also, calling the object a 3-sphere, topologists are mainly concerned with its 3D surface, while I insist on considering the object as a whole.

Now, the surface of a hypersphere is 3-dimensional, each point of which is equidistant from its center, making this 3-space continuous and invariant in all 3 directions.

At low energies of our own experience, matter (stuff with intrinsic mass) is confined to the surface of this hypersphere, with nuclei sort of gliding just above it in the 4th empty dimension, supported by their electron clouds. The EM field is confined _entirely_ to this 3D surface, making it in effect a 3D display that "shows" what is attached to it (other details can be found in my essay).

To summarize, the word we perceive is 3D, because it is the surface of a 4D object (a hypersphere). It is the shape in which an N-dimensional dynamic vibrating structure found its lowest energy state. If 5D were a space corresponding to the lowest energy state, then we would be crawling on a 4D surface (in the 5th dimension). But we "crawl" on the 3D surface, in the 4th spatial dimension, aware only of this surface and seeing only what is attached to it.

In the next post I will address the questions you asked above and also demonstrate the advantage of this setup in various problems in physics.