The Brane Theory as It should be

The brane theory is the derivative of the string theory, a mathematical model which was built to simulate the empirical particle interactions. The theory holds on the premise that the most basic indivisible objects underlying all matter are tiny vibrating segments or loops of the one-dimensional string-like entity.

In its later development, string theorists discovered that the strings do not freely move throughout the whole spacetime continuum but are constrained along the surface of membrane-like objects, coined as "branes," whose dimensions may extend over some, but not all of their embedding space's dimensions.

The branes are not just loci of matters and forces interactions, but also real objects which can be slack, wiggling and moving, or stretched tight. Branes are distinguished not only by the number of their dimensions in which they extend but also by their charges, their shape and tension¹.

Despite the development intensity of the theory, the brane theorists are not yet to know whether their branes exist in the real world. They are like a blind man wondering about the big picture of the elephant after having felt the ear and the trunk of such animal.

1.   The spontaneously symmetry breaking of the spacetime

The brane theory proves that the spacetime world model as currently conceptualized is wrong. It is the existence of a 3-brane embedded in such spacetime that makes the latter's dimensions differentiated. Otherwise, the spacetime's dimensions are equal, in the sense that the spacetime is perfectly symmetric, homogeneous and isotropic ^a).

Paradoxically, as Lisa Randall remarked, many physicists think otherwise²:" They did not want to include branes in a physical realization of string theory because brane violated their intuition that all dimensions are created equal. Brane distinguishes certain dimensions – those along the brane are different from those that extend off it – whereas the known laws of physics treat all directions the same. Why should string theory be different?”

In fact, following the Minkowski’s discovery [1908] on the inextricability of the space and time, physicists have confusedly taken for granted the inequality of the dimensions of the spacetime, the union of the space and time³. They thought that the light cone system set up in every point within such spacetime could establish order and preserve the causality.

The special relativity, as currently conceptualized, is inconclusive. In order to explain the inequality of such continuum's dimensions, the physicists should include in their world creation scenario the spacetime's spontaneous broken symmetry. As it happens, the 3-brane comes into being in between the two spacetime's distinct halves, in which the dimensions along the brane become spatial and those off it temporal dimensions.

It is just like the separation of oil and water which we can observe in our daily [3-dimensional ambient] life ^b). The dimensions along the interface, because of the effect of the interfacial tension, would be different from those off it.

The gravity constant (G) that we are familiar with is nothing but the interfacial tension of the 3-brane embedded in 4-spacetime. Consequently, the gravity field like the other classical fields is trapped along the brane, not propagate off it as what the brane theory currently assumes ^c). The fields that may propagate off the brane are quantum fields including those of Higgs.

2.  Why a brane traps matter and classical fields?

Randall took for the analogy of trapping matter on a brane among other things water droplets on a shower curtain which travel only along the curtain's surface.

We wish here to provide a better and physically more appropriate analogy for describing the braneworld model. Matters are analogous to tiny flashes appearing and disappearing on the surface of a giant TV-screen.  The impact of streams of electrons fired onto and hit the screen resulting in the generation of these tiny flashes

The seemingly trapped matters on the brane are tiny sparks appear and disappear on the TV screen-like brane. These tiny sparks, which are quantum in size^d), are generated as the effect of the quantum fields (Higgs fields) hitting in the normal direction through the brane.

The quantum fields are themselves generated as the result of the constant interplay between the positive and negative energies located at the opposite sides of the brane (Figure-1A). As the quantum fields piercing through the brane, the classical fields including those of light are generated, under the right-hand rule, propagating along the surface of the brane (Figure-1B).

3. Why there exist various dimensional branes?

So far we have shown that the special relativity theory ultimately leads us to the union of the spacetime and energy.  The spacetime has faded away into a mere shadow to become just the geometry of the energy, the only independent reality in nature.

Some physical theories identify the existence of higher dimensional manifolds (spacetimes). It leads us to the conclusion that energies of higher dimensions corresponding to such spacetimes should exist as well.

The energy in itself inherently consists of a pair of positive and negative components. As the opposing energy components tend to segregate, all of those spacetimes are highly unstable. The segregation of the positive and negative energies causes the spacetime to split in two, creating a brane in between.

The splits of those spacetimes are taking place starting from the highest dimensional spacetime down to the lowest one, which is nothing but our universe, the 3-brane (Figures-2 and 3).

We may imagine an interface of oil and water as a 2-dimensional slice of a 3-dimensional liquid, similarly, a 3-brane as 3-slice of 4-dimensional spacetime, 4-brane as a 4- slice of 5-spacetime and so on. We should underline that the dimensions along the brane are always spatial and off the brane temporal, and never mix them up ^e).

4. Why the branes exist in a pair?

The brane theory includes the existence of two parallel branes bind higher-dimensional worlds such as Horava-Witten and Randall-Sundrum brane-worlds. In such world models, the standard model particles are constrained on one brane, and non-standard particles are sequestered on the other branes.

Why does such a pair of parallel branes exist?

A brane, like a piece of paper, has a very thin thickness and two opposite sides. The two parallel branes that the physicists refer to are not independent of each other but just two different sides of a single brane.

As one side of the brane faces positive energy, and the other side faces the negative energy, the two brane's sides have accordingly opposite charges (Figure-4). For example, under a 4-ambient space, the positive side of the 3-brane contains matters while the negative side sequesters antimatters. This kind of brane world-model can be extended merely to the higher dimensional world. Within the 4-spacetime framework, the distance between these two sides of brane might be equal to Planck distance of 10^-33 cm (10^-44 second), and it would be much more significant in a higher dimensional spacetime.

5. Why 10- or higher-dimensional branes exist in nature?

The brane theory consists of two theories of10- and 11-manifold. To explain why such duality may arise, we have to refer to the mathematical concept of hypersurface whose geometry is equivalent to that of the brane. We define a hypersurface as a multi-dimensional surface having one or more dimensions lower than those of the embedding space.

A rule dictates that an n-dimensional curved hypersurface does have a solution if and only if it is embedded in an ambient space having at least ½ n(n+1)-dimensions⁴. Accordingly, a 4-brane (spacetime) requires an ambient space containing at least ten dimensions.  It seems that the 11-ambient space provides enough room for such a world model has a solution than the 10-ambient space does.

Notes:

a.   In the ancient cosmology, the condition where the spacetime's dimensions are still equivalent, in the sense that the time is not yet divided (there is no present, past and future), is termed as chaos and the spacetime 'eon.'

b.    Surprisingly, this kind of analogy had been proposed a long time ago, since the dawn of the history, by Enmeduranki, the king of Sippar, Babylon, who lived and reigned before the Flood⁵

c.  As the graviton is a close-loop string having no ends, the whole parts of its length are pinned down on the brane, on the contrary of what happened in the brane theory.

d.   We guess it is more appropriate to use the term of quark as the acronym of quantum spark, the underlying nature of the fundamental particles, not limited to those which compose the nucleons.

e.  It may lead to the establishment of multidimensional times theory which is naturally more appropriate than one-dimensional time.

References:

1.    Randall, L.: "Warped Passages," Harper, New York, 2006, p. 305-306.

2.    Idem, p. 307

3.  Einstein, Albert: The Meaning of Relativity, Princeton University Press, New Jersey, Fifth Edition, 1954, p. 31

4.   Sokolnikoff, L.S: "Tensor Analysis," Wiley Toppan, Second Edition, New York, 1964, p.205

5.  Wright, J.E.: The Early History of Heaven, Oxford University Press, Oxford, 2000, p. 43.

The Spacetime Misconception and the Crisis in Physics

The mainstream physicists are still unable to recognize the true nature of space and time, albeit their recognition of the union of the two. The physicists have taken for granted the union which forms a four-dimensional continuum as such as representing the actual universe.

Physically, such a continuum should be homogeneous and isotropic in the sense that all of its dimensions are equivalent. However, physicists seem to lose their physical sense as they assume that the four-dimensional continuum has different dimensions. We can see this confusion from Einstein's statement on the inextricability of the spacetime:" the non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space coordinates with the time coordinate. On the contrary, we must remember that the time co-ordinate is defined physically wholly differently from the space co-ordinates"¹.

The dimensions of such spacetime continuum should be equivalent and their intrinsic nature is [undivided] time-like in a sense that there is no present, past, nor future ^a). It is precisely the condition which prevails in the world model  (spacetime) that mainstream physics have adopted. The notation of [unsplit] spacetime is better to be replaced by the 'eon' which is more appropriate to describe such undivided time which is quasi-eternity (Figure-1).

Einstein himself was worried about the absence of the concept of Now in modern physics as he said to his friend philosopher Rudolf Carnap. The latter wrote²:"… Einstein explained that the experience of the Now means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics [… ], so he concluded that there is something essential about the Now which is just outside the realm of science".

It is the background why physics is now in crisis.

Creation by Separation

Physicists are forced to set up a fundamental structure consisting of light cone at every point within the spacetime for the purpose to establish order within otherwise a chaotic world model.  The physicists have to establish such odd construction to preserve the causality because they miss identifying a critical step within the chain of the creation process, i.e., the act of separation, a common phenomenon in physics,  which is often called "symmetry breaking."

At school, we have learned this separation phenomenon, for example, in the electrical process where equal amounts of positive and negative electricity form if we rub a glass rod with a piece of silk. The glass rod becomes charged with positive electricity, and we find a precisely equal negative charge on the silk. This empirical fact shows that friction does not generate but only separates the two kinds of electrification d).

We may think this positive and negative electricity as two fluids that are present in all bodies in equal quantities. In non-electrical neutral bodies, they are everywhere present to the same amount so that their outward effects are counterbalanced. In electrified bodies, they separate. One part of the positive electricity has flowed from one body to another, just as much negative has flowed in the reverse direction ³.

Analogously, the same phenomenon happened in the cosmic creation. The four-dimensional spacetime, which physicists have assumed to be intact,  has spontaneously broken its symmetry as a result of the split of related energy into its positive and negative components (Figure-2). As such the spacetime was split in two, creating a three-dimensional [hyper] interface in between the two halves, transforming the dimensions along the interface into spatial ones.

It is just like the separation of oil and water where we can observe an interface taking place between the two.

Geometrically, we can imagine that the nature of dimensions along the interface is different from those within the bulk of oil and water because of the tension that arises at the interface.

Analogous to this three-dimensional oil-water system, we may posit that the nature of the gravity constant in our four-dimensional world is nothing but the interfacial tension of the 3-[hyper]interface.

Transversality of Light and Hypersurface

The concept of hyper-interface or more generally the hypersurface can be borne out based on the phenomenon of transverse waves. The weird phenomenon that hardly anybody thinks about is the transversality of light waves in which particles vibrate at right angles to the direction of propagation of the wave.  The transverse waves are taking place either on a surface of a liquid (water wave) or as the vibration of a stretch string, and not in the interior of a substance (body). However, as light waves propagate in the [interior of] space, there should be an explanation of this paradox.

Numerous experiments have proved the transversality of light waves. It should lead us to the conclusion that the medium wherein the light propagates should be surface-like. How come that it could be? We live in the interior of a body, not on the surface of something.

The answer lies in the concept of hypersurface that the mathematicians have introduced as a point of departure in the generalization of the concept of space, long before physicists surmise the multi-dimensionality of the spacetime. We may conceptualize the space as a 3-manifold  as a 3-hypersurface embedded in a 4-enveloping space. We can easily extend this concept to any higher multidimensional space (Figure-3).

Now, we have a proper place for light as a transverse wave to propagate on the hypersurface. It is a three-dimensional [hyper] surface in which photons vibrate at right angles (along with the time dimension) to the direction of propagation of the wave across the hypersurface. From this relativity point of view, we see the space as a 3-hypersurface vibrating to and fro in the time direction.

As the propagation of light waves indicates, we are dealing in this case not with waves in the interior of a substance but with phenomena on a surface (hypersurface or hyper-interface) or motions of whole configurations (like a vibration of strings). We have already a string theory which seems going nowhere and now tends to converge into a "brane" theory. We wish to suggest to shift the theory into a more proper hyper-interface theory.

It is in this context that we should develop the current brane theory.  The brane is more like a hyper-interface rather than like a piece of paper floating in the air or in the bulk of something that conceptualized in the current brane theory. Besides, the gravity force should act only along the surface of the brane and not out of it crossing through the higher-dimensional bulk.

Notes:

a.    The ancient creation myths referred to such condition as chaos.

References:

1.   Einstein, Albert: The Meaning of Relativity, Princeton University Press, New Jersey, Fifth Edition, 1954.

2.    Barbour, Julian: The End of Time, Phoenix, London, 2001.

3.    Born, M: "Einstein's Theory of Relativity," Dover Publications, Inc., New York, 1962.

The Cosmic Inflation Never Happened

The Big Bang theory holds the premise that the universe originated from a singularity which came into being out of nothing through a single massive explosion. The concept of the minuscule size of such primeval singularity born from the thought projection of the current universe's expansion backward far in time.

The theory has at least two dubious primary grounds. The first one is the speculative concept of nothingness. The Big Bang theory presumes, violating the first law of thermodynamics, that energy (and matter) was created out of nothing. This idea came from the mindset that the creation of the universe (4-spacetime) was the beginning of everything. Notwithstanding, the theory takes for granted that the quantum fluctuation which stimulated the primeval explosion held in the nothingness before such creation.

The second speculative ground is about the size of the universe which can be shrunk indefinitely backward in time from the current size into a singularity. Close to the moment of creation, the size exponentially shrunk about 10⁶⁰ smaller just within 10⁴ seconds, from 10^-33 to 10^-37 second posterior to the explosion, the rate of which was exceedingly faster than the speed of light¹.

As we have elucidated so far, there can be no such thing as nothingness. The energy, as the only reality in nature, can neither be created out of nothing nor destroyed into nothing. The universe was born as the result of the interplay between the opposite (positive and negative) energies that created the universe and everything within, not out of nothing.

We can mathematically describe energy in its pure condition as waves' spectrum of different frequencies and amplitudes expressed in terms of Fourier series or its complex form, the Laurent series:

                         f(z) = F⁺(z) + c₀ + F^–(z)

It is a wave function expressed as the sum of its positive frequency (F⁺(z)) and negative frequency (F^–(z)).

Globally, we can depict this wave function in terms of Riemann sphere, the positive frequency F⁺(z) extends holomorphically into the southern hemisphere, and the negative frequency F^–(z) extends holomorphically into the northern hemisphere, where the equator represents the real coordinate and the longitudinal circles its imaginary time coordinate.

The domain of the positive and negative frequencies, however, does not fully extend to the poles, as the Riemann sphere has an annulus of convergence which excludes the domain around the zero points (singularity) as well as the infinity (Figure-1). 

 This pure mathematical analysis indicates that the split of the energy cannot create a stable interface (hypersurface) from the beginning when the energy started to split up to a certain period where it reaches the minimum size (represented by the Riemann sphere's inner ring of convergence).  The interface created in this period would instantly dissolve into energy.

It is only after reaching this limit that the hypersurface comes into being where it stabilized until it reaches its maximum size (represented by the Riemann sphere's outer ring of convergence).

The doomsday comes when the hypersurface reaches its maximum size.  At this particular time, the hypersurface becomes extremely unstable that makes it break down into pieces dissolving back into pure energy.  We illustrate these phenomena in Figures-2 and 3A.

This cosmology scenario avoids the need of either the concept of singularity to represent the beginning of the creation or the big crunch at the end of the universe's life as well as the everlasting expansion where the universe has no dead end.

This cosmology theory also excludes the need of the concept of cosmic hyper-inflation in the early period of the creation (Figure-3B), as the baby universe was born in an exceedingly larger size than that of the singularity which the Big Bang theory presumed.

The interplay between the positive and negative energies generates quantum fields across through the interface (hypersurface) ^a), located in between the two, perpetually creates quantum sparks (fundamental particles), the building block of the universe. As the split (hypersurface) area is enlarging with time, new matters are created in the expanding horizon keeping the average matter density per area almost constant.

This cosmology concept resembles the one of the continuous creation, steady-state expanding universe put forward by Fred Hoyle².

Notes:

a) We use the split area, interface, hypersurface, and space interchangeably.

References:

1.  Guth, A.: "The Inflationary Universe," Basic Books, New York, 1997

2.  Gregory, Jane: "Fred Hoyle's universe," Oxford University Press, New York, 2005

Multidimensional Time and Hypercomplex Numbers

Long before physicists embarked on the study of higher-dimensional spacetimes, 19^th-century mathematicians had firmly established the geometry concept of multidimensional metric manifolds. Many of these concepts were straightforward generalizations of ideas on the properties of surfaces embedded in the three-dimensional Euclidean manifold.

To simplify things, the mathematicians have introduced a multidimensional surface-like concept called hypersurface for modeling multidimensional space embedded in a higher multidimensional ambient manifold. A flat m-hypersurface can be appropriately embedded in an (m+1) space, but matters become more complicated when one comes to consider curved hypersurfaces. A curved m-dimensional hypersurface requires an ambient space whose dimensions are at least equal to or greater than ½m (m+1) 1.

Accordingly, a 4-dimensional curved spacetime requires at least a 10-dimensional ambient space. The spacetime's point position and, hence, the curvature of the spacetime is completely defined through a collection of numbers associated with the coordinate system set up in such 10-ambient space which we are more familiar with as the metric tensor's independent components of such 4-spacetime.

So what is so startling about it is when we explore the micro realm we would be confronting with the same 10-dimensional ambient space. Alas, in the later development, such as in that of the superstring theory, physicists made a blunder as they wrongly assumed the curly nature of the extra dimensions of such ambient space,  which made them going nowhere

The same fate happened to Big-Bang theory as physicists firmly exclude the existence of the universe's surrounding spaces. In doing so, physicists throw away the more significant part of the system, and this might be the reason why the theory incorporates only five percent of the total mass and energy that it actually should be.

Now the only option to cope with this impasse is jumping off the ship and abandon not only about the curly nature of the extra dimensions but also the one-dimensionality of time.

As the last article has deliberated,  those multiple temporal dimensions are the results of a series of successive symmetry breakings occur which had created different worlds, each of which had its respective temporal dimension (Figure-1).

Quaternion and Octonion

Now, how do we describe the structure and the geometry of such multiple temporal dimensions? To do this, we need to build a coordinate patch within such ambient space framework. To start with, let us deal with our 3-dimensional physical space embedded, as it should be, in a 6-ambient space. In such a case, we assign a coordinate patch consisting of three real space coordinates x₁, x₂, and x₃ and three imaginary time coordinates whose basis i, j and k.

If we denote x= x(x₁, x₂, x₃), then we can define any world point in such 3-physical space as:

q = x + ui + vj + wk,

expression is found to be nothing but the quaternion; a generalized complex number discovered a long time ago by Hamilton who established the geometry and the algebraic structure of this quaternion in 1843.

If we express the time variables u, v and w proportionally to the speed of light c_i of the respective temporal dimensions t_i then we can write:

q = x+ ic₁t₁ + jc₂t₂ + kc₃t₃,

This quaternion describes a general vector within a 6-dimensional space expressed as a function of space and time coordinates. Quaternions, therefore, describe a 6-dimensional vector space over the real numbers, depicting the dynamical geometry of 3-space embedded in 6-ambient space.

Similarly, we can define the 4-spacetime whose ambient space is ten dimensional through a coordinate patch consisting of three real space coordinates and seven imaginary time coordinates.

Again if we assign a space coordinates as x= x(x₁, x₂, x₃) and i, j, k, l,m, n, and o denote independent imaginary numbers as the coordinate basis representing seven different time coordinates, then we can define any point located at the 3-space in such coordinate patch as:

q = x + ai + bj + ck + dl + em + fn + go

, where x, a, b, c, d, e, f  and g are real numbers. Graves and Cayley had already discovered this expression, known as double quaternion or octonion, long time ago in 1845, although they did not know about the physical implication of it.

If we express the time variables a,b,c ... g proportionally to the speed of light c_i of the respective temporal dimensions t_i then we can write:

q=x+ ic₁t₁ + jc₂t₂ + kc₃t₃ +lc₄t₄ + mc₅t₅ + nc₆t₆ + oc₇t₇

Octonions form a 10-dimensional vector space over the real numbers, depicting a 3-physical space embedded in 10-dimensional ambient space.

In a later development, the original notions of quaternion and octonion are further modified and generalized through what so-called Clifford and Grassmann algebras applied to any higher dimensions framework which is found to have powerful implications in modern physics.
Many mathematicians and physicists wrongly perceived the quaternions and octonions as respectively describing 4-dimensional and 8-dimensional spacetime (having both one-dimensional time), which is inappropriate.

Penrose² regarded Hamilton's 22 year-devotion in his life in attempting to develop the quaternion calculus resulted in relative failure. On the contrary, we regard the Brougham Bridge's stone carved with the Hamilton fundamental equation would become a momentous milestone of the application of the hypercomplex calculus on the geometry of multidimensional time in both macroscopic and microscopic realms.

References:

1.    Sokolnikoff, L.S: "Tensor Analysis," Wiley Toppan, Second Edition, New York, 1964,   p. 205.

2.    Penrose, R.: "The Road to Reality," Vintage Books, London, 2005, p. 201

Symmetry and Symmetry Breaking

The asymmetry and its associated diversity that we observe today was the result of symmetry breakings which occurred in the early stage of the cosmos. In the beginning, ^a), the conditions were very different from those prevailing today, they were symmetric. The spatial dimensions as we know today did not yet exist; all dimensions were inherently temporal. However, as those temporal dimensions were yet undivided, there was no past, present ^b) and future.

At those conditions, the energy ^c) was unstable and tended to break into its positive and negative components. When it happened, the associated 4-spacetime (cosmos) ^d) was split into two parts creating an interface (3-hypersurface) in between the two. The dimensions across the interface transformed into spatial; leaving the dimensions outside it remained intact ^e).  Space, therefore, was born.

The two opposing energies ^f) perpetually generated sort of 4-lights (quantum fields) piercing through the interface (space) inducing secondary 3-(classical) fields which permeated and propagated across the interface (Figure-1A). As the quantum fields hit the interface, the strongest of them (Higgs fields) generated bright sparks which immediately disappeared as the opposite fields annihilated them (Figure-1B).

 The fundamental particles as we know are in reality nothing but these quantum-sparks which perpetually appear and disappear at the interface. As those quantum fields hit the entire surface of the interface and penetrate it only a short distance (across through the thickness of the space), they seem to us (who live in such interface/3-space) as eternal, omnipresent and invisible objects that can create and annihilate quantum particles.

Minkowski ^{1, g)} brilliantly fused the space and time into its undifferentiated state and brought back the spacetime into its original condition. However, then, something wrong happened. Instead of bringing the spacetime back into its symmetrical condition, Einstein²assumed that such unification did not make the temporal and spatial dimensions equivalent. Einstein failed to recognize that the asymmetry as we see today was the result of the spacetime symmetry breaking. This blunder has hampered the progress of physics for more than one hundred years now.

On the discovery of the four-dimensional spacetime, Einstein³ commented: “The non-mathematician is seized by a mysterious shuddering when he hears of four-dimensional things, by a feeling not unlike that awakened by thoughts of the occult.” No wonder, even after one hundred years of experience dealing with such spacetime, physicists are still bewildered and fail to recognize that their chaotic spacetime model does not represent the post symmetry breaking we observe today.

Supersymmetry Breaking and Multidimensional Worlds

The symmetry breaking of the 4-spacetime as we previously described was only one of the long series of successive symmetry breakings. It was the last of the long chain of a successive splitting of a higher-dimensional spacetime into its lower-dimensional parts.

To make it clear, let’s take the ambient 10-spacetime as a start. As this 10-spacetime broke its supersymmetry, a 9-hypersurface came into being along with its associated temporal dimension, t₇.  The latter, in turn, was split creating a smaller 8-hypersurface and its associated time, t₆ and so forth. This series of splits continued resulting in successive creations of the spacetimes in descending order of their dimensions and ended when the 3-space came into being along with its associated time t₁. A total of seven worlds ^h) have successively come into being with their own individual time, t_i, light and its respective speed, c_i, Planck constant, h_i, and “gravitational” constant, G_i.  

We can depict those seven worlds in term of their relative dimensionality (Figure-2.) or pictorially described as concentric spheres whose dimensions are larger outwards, in which the innermost layer is the 3-space with all of its solar system, stars, galaxies and super-galaxies (Figure-3A).

It is worthy to note that this picture may clarify the exact physical meaning of the ancient cosmology. For hundreds of years, people had wrongly considered this configuration as the geocentric cosmology in which the earth was at the center of the universe (Figure-3B). Even now, modern physicists fail to properly grasp the multidimensionality of the seven heavens described in the ancient cosmology ⁴.

Notes:

a.    It is the relative beginning, not the beginning of time.

b.   The notation of spacetime given for the cosmos at its original state is misleading as it gives the impression as it was asymmetrical from the beginning. It would be more appropriate if we use the notation world, cosmos or more technically [metric] manifold.

c. Energy in its entirety (4-energy); to avoid misunderstanding it would be more appropriate if we use the ancient notation: eon or simply eon.  The energy as we know is merely its superficial property (3-energy).

d.   There was no space, as space and the present time are different aspects of the same thing.

e.  This symmetry breaking is analogous to the phenomenon which occurs in the separation of two immiscible liquids, such as oil and water. In the body of the liquids, the cohesive forces are symmetric exerting equally in all directions. At the interface, however, such symmetry is broken because of unbalanced force exerting at the interface. As the system is in equilibrium,  the potential energy known as interfacial tension counter the net unbalance force. In terms of coordinate geometry, we may say that the interfacial tension differentiates the dimensions across the interface ("superficial" dimensions) from those of the original.

f.     The relativistic energy is composed of two opposite components as expressed in E² = m²c⁴ + p²c²

g.    Minkowski died one year only after the discovery, leaving confusion on his discovered object (spacetime)’s structure.

h.   The ancients called such worlds seven heavens.

References:

1.    Einstein, A. et al.: " The Principle of Relativity," Dover Publications, Inc., New York, 1952, p. 75.

2.    Einstein, A.: " The Meaning of Relativity," Princeton University Press, Fifth Edition, New Jersey, 1954, p. 31

3.    Einstein, A.: "Relativity," Crown Publishers Inc., Fifteenth Edition, New York, 1952, p.55

4.    Hawking, S.: "A Brief History of Time," Bantam Books, London, 1989, p. 3.