The True Mathematical Structure of Space-Time

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Subtitle: The Three Theses for the True Mathematical Structure of Space-Time.

Subject: The Space-Time Metrics of Post-modern Physics.

Introduction

What are the true nature of space and time? Are there aspects of these entities that are really beyond what clocks and meter sticks tell us? These are foundational questions that lie at the heart of physics, and they determine if we will ever progress in the fundamental understanding of the universe.

Now, this article is about the true mathematical structure of space-time in the universe, and it is one of my articles that will further address the fundamental questions presented above. And as you may know, in physics, “space-time” is a composite geometrical entity comprising of space and time in a mathematical union.

Let me more explicit, in physics, whenever we discuss “space-time”, we are talking about space and time as two unified entities and as they emerge from the mathematical framework of relativity. And in this article, we will be discussing space-time as it emerges from absolute relativity which is the post-modern theory of relativity. 

The true mathematical structure of space-time is a very important topic in physics. Physics since the advent of relativity has been particularly concerned about the true structure of space-time.

A Brief History

After Albert Einstein published his theory of special relativity, it wasn’t long before another physicist called Hermann Minkowski developed the space-time description of relativity. This was a purely geometrical description of relativity and the metric he proposed for the relativistic description of space-time is usually referred to as the Minkowski metric. I will represent it later in this article.

Hermann Minkowski

Now, the Minkowski metric is an Euclidean metric. It shows us how light underpins the relativity of uniform motion in a flat, Euclidean space-time. And according to this metric, there are three dimensions of space and one dimension of time which are fused together to form the 4-D space-time.

However, in 1915, Einstein presented the theory of general relativity which was to be the extension of special relativity. In this theory of general relativity, the Minkowski metric becomes an approximation of the Reimannian metric that describes the motion of accelerated frames in a curved, non-Euclidean space-time.

So, in general relativity we have the Reimannian metric which also represents how light also underpins accelerated motion in a curved space-time. In general relativity, space-time is no longer flat like it is in special relativity. This is the basic mathematical difference between special relativity and general relativity.

Now, why I say that light also underpins accelerated frames in general relativity is because of the insistence even in general relativity of light being the speed limit of the universe, and even gravity which the theory attempts to described is taken to move at the speed of light. So, the speed of light is the only limit of motion in both special relativity for uniform frames and general relativity for accelerated frames.

Also, in general relativity, space has three dimensions while time has just one dimension. So, the idea of the 4-dimensional space-time continuum is therefore common to both special relativity and general relativity.

I want you to take note of these things I am bringing to your remembrance in this section because they will be relevant as we proceed in this article, for they will show us the relevance of the new concepts post-modern physics introduces into our understanding of space-time as a composite entity of both space and time.

So, above we have two successions, the first is special relativity and the second is general relativity. However, there is a third, which is very, very important for what we are going to discuss in this article. The third event in the progression of relativity came about when Einstein wanted to unify light and gravity, or when he wanted to incorporate electromagnetism into his theory of gravity.

He couldn’t accomplish this feat, but a tangible or would I say a popular attempt in this direction came from Kaluza and Klein who found out according to their theory that they could incorporate electromagnetism into Einstein’s theory if they added another dimension of space. This fourth dimension of space was to make the idea of a 4-dimensional space-time continuum to become the idea of a 5-dimensional space-time continuum.

Kaluza and Klein

Theodor Kaluza (left)  and Oskar Klein (right)… source

Einstein found this idea very impressive and he saw it as the first light in the direction to the unified field theory. But there was a problem. The fourth dimension of space Kaluza and Klein proposed could not be experimentally proven. They had assumed that the fourth dimension of space is rolled up and for this reason it is beyond our reach to actually discover or prove.

Also, the Kaluza-Klein model or theory did not make any new predictions concerning light or gravity. This among other reasons led to its failure as a unified field theory despite the interest the theory spurred in the research on the fifth dimension.

So, are there really extra dimensions of space or of time waiting to be discovered but have not been properly represented? What is really the problem? These questions among others are what take us to the true post-modern description of the mathematical structure of space-time.

Background

In this article and to resolve the above questions, we will describe the mathematical structure of space-time for uniform frames and then we move to describe that for accelerated frames.

The post-modern descriptions of the mathematical structure of space-time to follow will be purely relativistic and they are extracted from Absolute Relativity which is the new post-modern theory of everything.

However, for uniform frames, which we will begin with, we shall start with the Minkowski metric for the relativity of uniform frames.

The Space-time Structure for Uniform Frames

According to Minkowski metric, the space-time for uniform frames is represented as,

\Delta s^2=c^2 \Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2 \;\;. \;\;\;. \;\;\;. \;\;\;. \;\;(1)

In the above Minkowski metric for uniform frames, the product of light c and time is used to derive a kind of spatial dimension which completes the metric and shows how space and time are related and fused to form the space-time continuum. It is now a common tradition in modern physics.

However, what is very important in this article is how we interpret the above according to special relativity and the difference between this interpretation and what we now have in absolute relativity.

In the above metric of space-time in special relativity, space and time are interpreted as purely physical entities, and light c is simply a non-mechanical wave which is known as the speed limit in the universe. This is simply how the above metric is interpreted.

But in absolute relativity we have a new interpretation of the metric for the relativity of uniform frames, and the metric for uniform frames is presented thus,

d s^2=c^2 d t^2-d x^2-d y^2-d z^2 \;\;. \;\;\;. \;\;\;. \;\;\;. \;\;(2)

The above metric in post-modern physics is referred to as the uniform metric for uniform frames, and though it is mathematically similar to the Minkowski metric, it however differs from it in interpretation. This is why it is referred to as the uniform metric in order to importantly distinguish it from the Minkowski metric.

So, what is the interpretation of the uniform metric? In the above uniform metric, the space-time interval is related to uniform space and uniform time and not just space and time like for the Minkowski metric. Uniform space with its three dimensions are represented as dx, dy, and dz, while uniform time is represented as dt. And light c is a non-mechanical wave which is the limit of inertia for only uniform frames and does not extend to accelerated frames.

I just had to present to you the post-modern interpretation of the uniform metric for uniform frames before I now ask you to look again at the difference between the modern interpretation of space-time for uniform frames and the post-modern interpretation of space-time for uniform frames that is directly above.

Satellite

Pixabay

It is important to see how light c in the mathematical structure of the space-time for uniform frames is a limit of inertia only for uniform frames and not for accelerated frames as modern relativity proposes, for underlyingly and according to the Minkowski metric light is a speed limit not just for uniform frames but also for accelerated frames.

Also, I want you to see that the change of the essence of light c from just a speed limit in the Minkowski metric to a limit of inertia in the uniform metric comes with a change in the essence of space and time. What you are seeing above is the absolute nature of space and time as they apply to uniform frames.

The above shows you that the relativity of uniform frames takes place in a form of space and time called uniform space and time. (And in post-modern physics, uniform space and time are indicated by the letter “d” as applied above. And the whole idea of space and time differences become implicit in the equation.) The relativity of accelerated frames will be shown to take place in another form of space and time. 

This interpretations and discoveries cannot be found in modern relativity where the same form of physical space and time applies to both uniform and accelerated frames. I want you to begin to see how the aspect of form is introducing us to the qualitative nature of space-time. The Minkowski metric is solely quantitative while the uniform metric is both quantitative and qualitative.

The Minkowski metric is solely quantitative while the uniform metric is both quantitative and qualitative.Click To Tweet

So, the equation (2) shows us uniform mathematical representation of the uniform space-time interval and not just the space-time interval. We must make this translation from the purely quantitative Minkowski metric of modern physics to the balanced quantitative and qualitative uniform metric of post-modern physics, for it is by this new and balanced description that we can move to realize the true space-time structure for accelerated frames. 

The Structure of Space-Time for Accelerated Frames

When we move to accelerated frames, we make three important realizations, which are:

  1. The relativity of accelerated frames is underpinned by another non-mechanical wave in the universe and not light.
  2. The relativity of accelerated frames occurs in another form of space.
  3. There is an extension of the form of time in the relativity of accelerated frames. 

The above three realizations are absolutely fundamental to the new space-time structure for accelerated frames. And a good look at the third realization above shows us how to extend the aspects of space and time. What we should extend to arrive at a unified field theory of the universe is the form of time and not the dimension of space as Kaluza-Klein had tried. I will discuss this further as we proceed.

In post-modern physics, the mathematical structure of space-time for accelerated frames can be generally represented as,

\Delta s^2=u^2 \.dt^2 \Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2 \;\;. \;\;\;. \;\;\;. \;\;\;. \;\;(3)

I want you to really look at the above metric which is referred to as the accelerated metric for accelerated frames. From the first realization, the new non-mechanical wave which underpins the relativity of accelerated frames is referred to as the gravi-electromagnetic wave and it is represented in post-modern physics by the u.

(Also, the above accelerated metric is as it applies in the electrical universe. I chose it because it is the one most apt to give us the general description of the space-time structure for  accelerated frames.)

https://www.echaandscience.com/three-absolute-quantities-absolute-motion/

From the second realization, the relativity of accelerated frames occurs in another form of space called accelerated space and which is represented in the accelerated metric and for the three dimensions of accelerated motion as Δx, Δy and Δz. (The delta symbol “Δ” is used to indicate accelerated space and time and so any difference in accelerated space and time become implicit in the metric.) 

And from the third realization,  we have the two forms of time, uniform time dt and accelerated time Δt attached to the gravi-electromagnetic wave. And if you notice in the accelerated metric the two forms of time are not separated by a comma. This is to indicate that we are dealing with two forms of time and not two dimensions of time. There is no such thing as two dimensions of time in the universe.

Also, according to post-modern physics, the accelerated metric can be contracted thus,

\Delta s^2=u^2 \.d \tau ^2-\Delta x^2-\Delta y^2-\Delta z^2 \;\;. \;\;\;. \;\;\;. \;\;\;. \;\;(3a)

In the above contracted accelerated metric, dτ2=dt2Δt2. In post-modern physics, tau time dτ is simply the product of uniform time dt and accelerated time Δt. I really want you to see these things and the new physics emerging in this post-modern era. 

These three realizations discussed above complete your understanding of the mathematical structure of space-time or the accelerated metric for accelerated frames.

The third realization is what Kaluza-Klein should have achieved. They should have extended the form of time and not the three dimensions of space. And as seen for the two forms of space above for the uniform metric and accelerated metric, they both have three dimensions. The dimensions of (absolute) space are three and not any further, and this importantly satisfies our intuitive understanding of the universe. 

In post-modern physics, light which underpins the relativity of uniform frames moves in uniform space and it is attached only to uniform time dt, while the gravi-electromagnetic wave which underpins the relativity of accelerated frames moves in accelerated space and it is attached to the two forms of time, uniform time dt and accelerated time Δt. 

So, according to absolute relativity, we have the uniform metric for uniform frames and the accelerated metric for accelerated frames. Understanding these two metrics is all there is about the true mathematical structure of space-time in the universe and for all frames.

We have the uniform metric for uniform frames and the accelerated metric for accelerated frames.Click To Tweet

The space-time metric for accelerated frames show that the gravi-electromagnetic wave u is the limit of motion for all accelerated frames and not light c which is the limit of motion for all uniform frames.

The Two Non-mechanical Waves and the Euclidean Space-Time Metrics

Now, let’s discuss something important, which is how the two non-mechanical waves are partly responsible for why both uniform frames and accelerated frames are described by Euclidean space-time metrics and not Reimannian metrics. We will discuss next how absolute space and time are responsible for the remaining part.  

So, using the terms of absolute relativity, how is the linearity of gravi-electromagnetic wave in the accelerated space-time continuum established just as the linearity of light in the uniform space-time continuum is established?

This is a very subtle question that shows you something about how the two non-mechanical waves maintain geometrical linearity in the space-time continuums for uniform and accelerated frames.

Now, listen to me very carefully, light in the uniform metric for uniform frames is a linear non-composite wave. This means that we have just light in the space-time metric for uniform frames. However, in the accelerated metric for accelerated frames, the gravi-electromagnetic wave is a linear, composite wave unlike light. This means that the gravi-electromagnetic wave is a composite wave consisting of light and gravity.

For gravi-electromagnetic wave u, light and gravity are related by the Pythagoras theorem or what I call the orthogonality principle because of its deep extensiveness in post-modern physics. So, the square of u in the accelerated metric for accelerated frames is as linear as the longest side of a right-angle triangle of which the other two sides are occupied by light and gravity respectively.

(I want you to read the article below on non-mechanical transportation to further grasp this.) 

I want you to understand this for it will reveal to you how there are two non-mechanical waves in the universe govern the two kinds of motion in the universe. So, in the uniform metric for uniform frames, light is a non-composite straight line, while in the accelerated metric for accelerated frames the gravi-electromagnetic wave is a composite straight line consisting of light and gravity which are related by the orthogonality principle.

In the uniform metric for uniform frames, light is a non-composite straight line, while in the accelerated metric for accelerated frames, the gravi-electromagnetic wave is a composite straight line consisting of light and gravity.Click To Tweet

This is an introductory understanding of how the two non-mechanical waves are partly responsible for why both uniform and accelerated frames are described by a Euclidean space-time metric and not the Reimannian curved space-time metric.

Also, the understanding of the linearity of the two non-mechanical waves in the uniform and accelerated metrics will be important when I begin to show you the space-time diagrams for uniform and accelerated frames according to the subtle laws of absolute relativity. 

Now, let’s discuss how absolute space and time are also partly responsible for the why both uniform and accelerated frames are described by only Euclidean geometry. This will reveal the entire ramifications of the great truth that the universe is geometrically unified. 

The Great Conceptualization and the Euclidean Space-Time Metrics

There is a new understanding of the universe that is emerging from this article so far and which you are conceiving, and which is that the existence of two non-mechanical waves in the universe and the existence of absolute space and time lead to the geometrical harmony of all frames and of the extensive universe.

This is the greatest consequence of the true mathematical structure of space-time. We had before now thought that we needed two geometrical principles to understand uniform and accelerated frames, but this is not true. When we come to understand the true mathematical structure of space-time we will realize that the Euclidean geometry applies to all frames, both uniform and accelerated frames.

So, again, the geometrical harmony of the universe is the greatest consequence of the true mathematical structure of space-time. And the two non-mechanical waves discussed above are partly responsible for this. Now, the new discoveries about absolute space and time are also partly responsible for the geometrical harmony of the universe. Let’s now discuss how.

The geometrical harmony of the universe is the greatest consequence of the true mathematical structure of space-time.Click To Tweet

In the space-time metrics for uniform and accelerated frames according to modern relativity, we find that space and time have only one form. This is the basic result of our understanding of relative, physical space and time. And this our understanding of only one form of space and time is what is called the conceptualization in post-modern physics.

So, according to the conceptualization, space and time are relative and they have only one form each. This recognition of only one form of space and time resulted in the geometrical disharmony in modern physics. This is why uniform frames followed Euclidean laws while accelerated frames followed Reimannian laws.

However, in post-modern physics, we encounter the great conceptualization, which informs us that there are two forms of space and time. It is these two forms of space and time that resulted in the geometrical harmony in post-modern physics. Also, you must bear in mind that while the one form of space and time are attributes of relative, physical space and time; the two forms of space by time are attributes of absolute, metaphysical space and time.

So, without the discovery of the true nature of absolute space and time and the two non-mechanical waves in the universe, we would not have discovered the true mathematical structure of space-time and the geometrical harmony of the universe.

We can even go further to see how post-modern physics corrects the oversight in modern physics. Modern physics by holding on to one form of space and time, and also one non-mechanical wave in the universe could not expose the true mathematical structure of space-time, and thus discover the geometrical harmony of the universe.

But post-modern physics by espousing the two forms of space and time and the two non-mechanical waves in the universe discovers the true mathematical structure of space-time and the geometrical harmony of the universe. These are important distinctions between modern physics and post-modern physics.

This is why above and in the three theses of non-mechanical transportation you learnt that light which is the first non-mechanical wave moves in uniform space which is the first form of space, while the gravi-electromagnetic wave which is the second non-mechanical wave moves in accelerated space which is the second form of space.

So, without the two forms of absolute space and time referred to as the great conceptualization, it would have been impossible to describe both uniform and accelerated frames using a Euclidean space-time metrics as above.

Now, in the space-time metrics, it is still important to know the nature of space and time as separate entities being inserted into the metrics. In the Minkowski metric shown to you for uniform frames and even in general relativity, space is as it can be measured using meter sticks and time is as it can be measured using clocks.

These are what relative, physical space and time are according to modern physics. So, we have relative, physical space and time inserted into in the mathematical structure of space-time according to modern relativity.

But in the equations (2) and (3) for the uniform and accelerated metrics, the space and time inserted are different. What we have in the uniform and accelerated metrics are absolute, metaphysical space and time, and they are not according to what meter sticks and clocks tell us. In the post-modern mathematical structure of space-time, space in its two forms is that which flows with regards to inertia, while time in its two forms is that which flows regardless of inertia.

These are the true natures of absolute space and time missing in classical and modern physics, and they are what are inserted into the uniform metric and the accelerated metric. You must understand this for it further shows you the background upon which the true mathematical structure of space-time is based. The true mathematical structure of space-time cannot be realized from physical space and time for they are not the true space and time of the universe.

This brings us to a crucial accomplishment of post-modern physics. Now, just as modern relativity accomplished the unification of relative, physical space and time to produce the physical space-time continuum, so has post-modern relativity conceptualised as absolute relativity accomplished the unification of absolute, metaphysical space and time to produce the metaphysical space-time continuum.

So, you must see the equations (2) and (3) as the mathematical representations of the absolute space-time continuum for uniform frames and accelerated frames respectively. This is important.

hostgator

The mathematical structure of space-time for uniform frames and accelerated frames in post-modern physics are absolute and metaphysical. “Absolute” in the sense that they are derived from absolute space and time, and “metaphysical” in the sense that they are therefore quantitative and also qualitative, unlike physical space and time that are solely quantitative.

And even ontologically absolute, metaphysical space and time take deeper meanings and they reach as the true representations of reality more than relative, physical space and time. I will discuss these vital aspects later in this blog.

Now, I can’t end this section without informing you that all these discoveries about the true mathematical structure of space-time lead to the realization of the experiential nature of the universe. This is different from what the space-time of modern physics inform us. The space-time of modern physics derived from special relativity and even general relativity only reveal to us a mechanistic, non-experiential universe which does not represent the true nature of the universe.

But the space-time metrics of post-modern physics expose to us the organic, experiential universe. It is unlike what we have known of the universe! So, the translation from the space-time metrics of modern physics to the space-time metrics of post-modern physics is not just a mathematical translation, it is an epistemological translation that alters how we understand the universe. It is a translation that goes to the epistemological foundation of science.

Everything I reveal to you in this blog about the universe are all expositions of the experiential nature of the universe. So, even when you become doubtful and pessimistic, I want you to know that what is being revealed to you is a physics that is not mechanistic and non-experiential that you have always known, but what is being revealed to you is a physics that is organic and experiential.

The true mathematical structure of space-time is organic and experiential. All these align with the new foundation of space-time studies. The space-time of post-modern physics are based on the great conceptualization which informs us of the two forms of absolute space and time.

So, when I say that the true mathematical structure of space-time is based on the great conceptualization, I simply mean that the true mathematical structure of space-time is based on the two forms of space and time.

The Three Theses for the True Mathematical Structure of Space-Time 

In fact, let’s present the three theses that encompasses the true mathematical structure of space-time and also in a good way bring together all we have discussed so far in this article. Below are the three theses upon which the true mathematical structure of space-time is based:

  1. Thesis I: The true mathematical structure of space-time is based on the great conceptualization (or the two forms of space and time).

  2. Thesis II: The true mathematical structure of space-time incorporates the two non-mechanical waves in the universe.

  3. Thesis III: The true mathematical structure of space-time is entirely Euclidean. 

It is evident that the third thesis is the remarkable result of the first two theses. And coupled with what I have informed you about the experiential universe, it implies that in post-modern physics, we are dealing with Euclidean geometry at an experiential level. We are dealing with how Euclidean geometry determines our internal experiences of the universe and not our external, physical observation of the universe. 

This is important, for classical and modern physics failed to give the true description of the experiential nature of the universe. I have discussed the experiential universe and its connection to gravi-electromagnetic orthogonality or wave in this article. Let’s now proceed to summarize this very great scientific article. 

Summary

This scientific article exposes to you the true mathematical structure of space-time for both uniform frames and accelerated frames. And it reveals that for both frames, their space-time metrics are similar and are perfectly Euclidean. This is the geometrical harmony of all things.

Also, this article reveals that the true mathematical structure of space-time is Euclidean for all references frames because we have two non-mechanical waves and two forms of space and time in the universe. These are important discoveries that are not found in modern physics, as discussed above.

Also, as you may have noticed, in this article, I stuck with the representation of the invariant interval in post-modern physics, and I did not go further. I want to let you know that I will go further in my future articles. I just want us to build this new edifice of post-modern physics line by line and concept upon concept till it has reached the high heavens.

So, the true mathematical structure of space-time is Euclidean and for all reference frames.

Until next time.

For post-modern physics!

– M. V. Echa



M. V. Echa

M. V. Echa

My message is the universe, my truth is the universe, and this blog contains all you need to know about the universe, from the true nature of reality to the long-sought unity of the cosmos — which is the big picture!