Mr Kley, I replied to your question in a comment below.
Thank you for your interest to my essay.
Mr Kley, I replied to your question in a comment below.
Thank you for your interest to my essay.
Dear M. V. Vasilyeva,
It is basically alright. The de Broglie wavefunction is described in the four dimensional spacetime. The slit takes one dimension and the light-beam takes another dimension, therefore, the equations of the double slit interference can be described in 4-2=2 dimensional space-time. Of cause you can still argue about the width of slit and the 3D surface electron cloud, and so on. My essay "Rethink the double slit experiment" have the detailed calculations compared with many experiments. There are real math-physical calculations, not a graphic illustration. The most important part is the connection between the two slit by the cross-linked angle, which is derived from the particle scattering to the wave function. In that way, the particle-wave duality paradox is linked to the space-time. I have my email in the essay, please send an email if you like to talk more.
Yours
KX
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 M. V. Vasilyeva,
In Coherently-cyclic cluster-matter paradigm of universe, the dimension of time emerges from 1D eigen-rotational string that demonstrates 3D tetrahedral-brane transformation from 2D membrane surfaces by the eigen-rotational phases of that string.
With best wishes,
Jayakar
Dear M. V. Vasilyeva,
" To remove water from a cup, it is necessary to fill it instead of water by the air. Further, in my
interpretation to remove air from a cup, we should fill the cup instead of air by the ether. Thus,
the cup can not be empty, only one medium can be replaced by another. Ether, according to
Aristotle, "is more subtle substance" than the air."
I also think the gap or space is not empty but full of energy; Nothingness does not exist.
I read your essay with carefully. It is interesting approach and I would like to have your view point about a gravity question : what do you think about gravity and space or dark energy, which relationship do they maintain between them.
Do you think that the expansion of the space is a force opposite to the Gravity ?
http://www.fqxi.org/community/forum/topic/1552
Thank you and lall the best.
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 Ms Vasilyeva,
I write in French and I translate into English.
To be honest, I must admit that it is advisable for me what you have just made as remark.
The reason is that I discovered this contest only later.
Then, I had only two choices, to quickly write the article in my language and then translate it in a record time, or not participate at all, and wait until next year.
I chose the second option. I said to myself that if someone is interested in my ideas, we are human and I can always catch me up later.
What is important, is the idea and not the way that it is formulated.
And I sincerely believe that the idea which I have formulated will mark the history by its relevance and will make smile also by its shape, these are the vagaries of life.
If you are interested by any side of this model please let me know it, I would answer, with pleasure, your expectations.
I hope my translation is enough good for more understanding.
I wish you all the best.
P.S. your 1/4 and 3/4 interest me.
Ms. Vasilyeva,
You say:
"But in this model there is no gravity per se. What we call gravity emerges entirely from the interaction of mass (which is displacement of volume) with the surface tension of the structure wanting to minimize its surface area."
What are your thoughts on dark energy as opposed to gravity?
Jim
Mr Hoover,
I am not certain what is meant by dark energy currently (it keeps on changing). Trying to see from where you are coming to it, I read your essay and saw this picture, which I also found on google images:
http://www.theepochtimes.com/n2/images/stories/large/2011/05/23/549761main_pia14094-43_946-710.jpg
This is an artist rendition of a 4D model of space, with one dimension removed, the green grid representing gravity (= curvature of the 3D surface on which "matter floats") and purple grid representing dark energy that pushes on matter, pressing it into the surface, displacing a volume and causing it to curve. That's how I understand it. The articles that came with the same image says, "This contradicts an alternate theory, where gravity, not dark energy, is the force pushing space apart. According to this alternate theory... gravity becomes repulsive instead of attractive when acting at great distances." I agree that gravity becomes repulsive at great distances (where voids form in my 4D model of space). This is stems from pure 4D geometry.
I personally do not agree that the universe is expanding, especially at ever-increasing rate. First, why don't we see it right where we are? The whole idea is based on the redshift and in my essay I offer another explanation to it.
But I understand what you're interested in and that is the enigma of UFOs and how they move as if inertia is none of their concern. I happened to see 2 small saucers at a very close range. I saw them about 25-30 meters away and then I ran towards them and got even closer. I was an adolescent then, and my impressions were vivid and observations sharp. Because of this I do not disregard other people's accounts.
I absolutely agree with you that there is a way to manipulate the curvature of the 3D surface (on which we live, according to my 4D model of space) which we call gravity. For this we need to have a better understanding of space, which we don't. In fact, the situation is just deplorable, as this contest demonstrates. The establishment is so entrenched in a particular way of looking at things that they are incapable of even considering anything else. And now with this idea of ever-expanding universe, which is happening everywhere except just where we are... People got Nobel prizes, defended their PhDs and built their academic careers on it, which means that they will defend it tooth and claw, all the way till the end. It will be another generation, after this one goes away, before an alternative view on things could even have a chance.
Dear Vasilyeva
Just to remind you that I have replied to you in my previous post above (Sep. 11, 2012 @ 17:14 GMT). I thought you may have overlooked it.
Israel
Yes, Mr. Perez, my apologies! I did overlook your post. I am reading it now and will certainly reply.
Dear Vasiliyeva,
I read your essay with a great interest. I agree that role of geometry in physics has not been fully appreciated. However, my view differs from yours in that in place of empty space, what is there is the field (in the sense of Newton's vacuum and Leibniz' plenum). The field plays a role in every interaction without exception.
I too have taken a Geometric approach. However, unlike yours, it concerns very simple Geometric relationships leading to trignometric expressions between related phenomena. It makes relativistic phenomena quite understandable visually. I am sure you would like my essay
The gist of my essay: http://fqxi.org/community/forum/topic/1549
1. It identifies the PRIMORDIAL Foundational Problems in Newtonian Mechanics (NM) that runs through ALL BRANCHES OF PHYSICS. (Please see the short attachment "Primordial Foundational Problems").
2. It eliminates the problematic concept of POINT-MASS (common to NM, QM, SRT) to allow internal structure for a particle. This in turn enables to resolve the other interconnected primordial problems.
3. The result: By taking these two steps, ALL THE EQUATIONS OF SRT are DYNAMICALLY derived by identifying the trignometric relations within the energy-momentum equation, and by restoring Galileo's principle of relativity. (I request you to have a glance at the attachment - "Geometrodynamics of Energy" to verify this claim). - See also comment by L.B Crowell below.
4. This achievement will establish that I have not just treated these problems at the level a speculative discussion as in other essays, but that the problems discussed are real problems, by virtue of their solution leading to the unification of NM and SRT (by finding an equation of motion which is equally valid for slow and very fast motions).
Here is the impartial comment made by Ben Dribus (essayist in no 2 position): "One thing I will say is that it appears as if you made an honest effort to answer the question posed by the essay contest rather than just writing down your favorite ideas about physics. You will notice that I made a similar effort..... I am not sure why it was rated so low, but my impression is that many authors automatically rate other essays low to boost their own standing".
Here's the comment made by LB Crowell (essayist at no. 20 position): "The calculations I just looked at and they seem alright. ...... Your procedure appears to be some euclideanization of relativity. At the end you arrive at equations which are the same as special relativity".
In order to enable follow up of your comments easier for me, I request you to reply to this in my thread (under my essay) : http://fqxi.org/community/forum/topic/1549
Best regards,
VirajAttachment #1: 3_Primordial_Foundational_Problems.docAttachment #2: 3_GEOMETRODYNAMICS_OF_ENERGY.doc
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 M.V.
Well written essay
Aside from the seemingly obvious properties of space existing in three (XYZ) dimensions plus time as a possible fourth dimension, (I seek intrinsic mechanical properties) do you perceive space as being a compressible medium? J.R.
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.
Dear Vasilyeva (this is part 1)
Thanks for your reply. I will try to express myself so we understand each other as well as possible.
Before 1905 people believe that space was synonymous of utter emptiness, nothingness. This was known as the Newtonian space which was mathematically represented by a 3D Euclidean space. As we all know, Euclidean space is structureless, it is nothing but the mental abstraction of physical objects (shapes, lengths, points, planes, etc.). It is thought of as a background composed of no physical entities and no internal structure; simply because nothingness cannot have structure, no energy or no substance. In the XIX (also in the XVII century) this emptiness was assumed to be filled with the aether. Some sort of material fluid having well defined mechanical properties. Then EM fields were thought of to be states of the aether (but physicists never conceived that the aether could play the role of space itself). At that time, there were several models for the aether, some people proposed that the aether was a gas like air, but they soon realized that in order to support light waves (shear or traversal waves) the aether should be a liquid or solid. In 1887-1890 Heinrich Hertz modified Maxwell equations to account for this feature. In this way he succeeded in creating a consistent model of the aether. Unfortunately, his formulation was ignored by the mainstream of the time.
After 1905, most physicists started to abandon the idea of the aether. By doing this they not also rejected the material character of the medium but also erased from their minds the idea that the material aether could be space itself. Instead, physicists accepted Newtonian space and filled it with EM and gravitational fields. In 1908 Minkoswki proposed his space-time. This space-time was also made up of nothingness, it was only a geometrical representation (a manifold) similar to Euclidean space so that astronomical objects interacted at a distance (instantaneously) without the need of any mediator. And again this space was unaffected by matter and filled with EM fields. Einstein soon realized this deficiency (following Mach philosophy) and then developed the GR. In his theory, space is no longer a static infinite vessel but a dynamical one that changes form depending on the matter-energy content [Here the matter-energy content means any other field different than gravitational]. These fields are introduced in the gravitational equations (Einstein' equations) through the energy-momentum tensor. Solving the equations give us the form of the metric tensor (in general non-Euclidean geometry) that defines the shape and properties of space-time. This tensor plays the role of space and paves the way for the displacement of matter and fields. Under this theoretical framework, space-time is conceived as a different physical entity if compared to fields or matter. So Einstein was categorical: If we have no matter and EM fields, we are left only with the gravitational potentials (metric tensor); and if we have no potentials we are left again with nothingness, total emptiness which for him was inconceivable as well. Then, according to the GR, the metric tensor has only a relational character but not substantial. In 1917 he introduced the cosmological constant in his equations to counter balance the force of gravity. He wanted to have a static universe for, without the constant, his universe will collapse. Thus, this constant represents a perfect fluid or the energy filling the relativistic space (that is, a space without EM fields or matter) and it is found that the energy density caused by the cosmological constant is about 10^-29 J/cm^3. In his lecture delivered at Leyden in 1920 he reintroduced the notion of aether, meaning not a material fluid in the old sense but a gravitational aether in the sense of the metric tensor. The metric tensor is then playing the role of a dynamical empty vessel for the motion of matter and fields.
Things look different from the point of view of QM. From here, one realizes that there is a state of lowest energy called the ground state and one can show that the vacuum state has a minimum energy different from zero, actually, its density is approximately 10^91 J/cm^3, a difference of 120 orders of magnitude with respect to the density predicted under the cosmological constant. In other words, from the perspective of QM the vacuum is full of a huge amount of positive energy. The time-energy uncertainty principle allows particles (virtual particles) to be created out of the vacuum for a very short period of time. In a certain sense, physicists say that these particles are created out of nothing. As I said in my previous post there is a game of words (a linguistic problem). When physicists talk about creating matter out of nothing they really mean that virtual particles are created out of the quantum vacuum which is some sort of energy reservoir called the zero-point field (take a look at the essay of Luis de la Pena and Cetto to understand this concept). So, we have a big problem here. Relativity says that the energy of its empty space should be 10^-29 whereas QM says that the energy of the vacuum should be 10^93. To hide the problem, physicists arbitrarily argue that the energy of the universe, by an unknown mechanism, cancels out to zero. They say that at the BB the same amount of positive energy (matter and fields) and negative energy (relativity space) were created, both adding to zero. To me this is non-sense since I support the idea that the energy of the universe is not zero and that there is no negative energy (take a look at this video for illustrative ideas according to the mainstream: http://www.youtube.com/watch?v=WQhd05ZVYWg&list=PLB58F0D021A12F173&index=29&feature=plpp_video).
Israel
Part 2.
Now, let's talk about cosmology and the Big Bang (BB). At the beginning of the XX century most astronomers thought that space was euclidean (or Minkowskian) and that the universe was infinite in extension. This conception however lead us the so-called Olbers' paradox. How to get out of this puzzle? Well, as time went by astronomers started to estimate the velocity (through the red shift of the corresponding spectrum of light emission) of distant galaxies. They found that the more distant the galaxies were the more red shifted they appeared. Astronomers also reported that most of the observed galaxies were showing a red shift. This trend was interpreted (based on the available theoretical framework) as if the galaxies were moving away from each other. In short, they had two big problems in need of an explanation: the red shift observed in galaxies and Olber' paradox.
In 1915-16 Einstein developed his theory and later some solutions to Einstein's equations were discovered. Some of these solutions suggested that in an expanding universe light should lose energy that should be observed as a red shift. One of these solutions matched with the observations and they arrived at the consensus that space was expanding. Then Lemaitre proposed that if we play a cosmological clock backwards the universe should compress in a single point of infinite density, that is, the BB singularity. A singularity is a point where we have an infinity (and when we have an infinity we do not know what happens there). Some other people added that if the universe started with the BB then there should be a relic cosmic background radiation (CBR).
Let's stop here since there are two versions about the universe. There are many people who claim that there is no BB at all and that the universe is very old. They contend that the red shift, Olber's paradox and the CBR can be explained without invoking space expansion and therefore there is no need of BB. In this version the universe is static and possibly infinite. Who shall we believe? Unfortunately, these people have been ignored by the mainstream (later we will discuss their proposal). For the time being we shall consider that the BB took place. If so, what is the real situation about the BB? It is common to hear in popular media that the universe started out of nothing. And when a physicist is asked: how something can be created out of nothing? They usually replied with the virtual particle scenario, in which something is creating out of "empty" space (i.e., quantum vacuum). Above I have explained what they mean by that. However, in the case of the BB, it is supposed that space-time was also created along with the universe, so this situation is different from the previous one, because in this case we have no reservoir from where the energy can be extracted and therefore their argument becomes ambiguous. But since they assume that the total energy of the universe is zero, the BB is not violating the energy conservation (this point is debatable) and the creation of the universe can come from zero energy or nothing.
Beyond this intuitive picture the real situation is that, at the singularity, nobody knows what happened or what the laws of physics are. At this point simply, GR breaks down completely (also at the black hole singularity). This means that nobody knows if the universe really bang. The theory has nothing to say and therefore there are no answers to the questions: What was before the BB? 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? (see this video of the mainstream: http://www.youtube.com/watch?v=4Y3rNm3QSVU&list=PLB58F0D021A12F173&index=28&feature=plpp_video)
You also ask: 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?
What expands is the space itself (the space of relativity). This is one of the solutions of Einstein's equations, the so-called Friedmann-Robertson-Lemaitre-Walker metric in 3+1 dimensions. But again this "space" is represented by the metric tensor which differs from the quantum vacuum. We are talking about two different notions of "empty" space. In the former case, when we have no fields and matter we only have the energy of gravitational potentials (metric tensor), but in the case of QM we have the energy of the zero-point field. QM presupposes that there is some background whereas relativity denies it.
As an alternative you are proposing that space has an internal structure made up of a fluid or strings. You also say that "there is a concept in topology (I would like to know what this concept is) 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." Your notion is still quite ambiguous. One can have the energy-momentum tensor equal to zero (vacuum equations, see Milne and de Sitter universes). This means that even if there were no matter-energy and fields in the universe we could still have the metric tensor (gravitational field). In this sense I say that the space of relativity is mere geometry, it is not a substantial entity as in the case of the quantum vacuum.
Israel