_{1}

^{*}

The present reading is part of our on-going attempt at the foremost endeavour of physics since man began to comprehend the heavens and the earth. We present a much more improved
Unified Field Theory of all the forces of
Nature
i.
e. the gravitational, the electromagnetic, the weak and the strong nuclear forces. The proposed theory is a radical improvement of Professor Hermann Weyl’s supposedly failed attempt at a unified theory of gravitation and electromagnetism. As is the case with Professor Weyl’s theory, unit vectors in the proposed theory vary from one point to the next,
albeit, in a manner such that they are—for better or for worse; compelled to yield tensorial affinities. In a separate reading, the Dirac equation is shown to emerge as part of the description of the these variable unit vectors. The nuclear force fields—
i.
e., electromagnetic, weak and the strong— together with the gravitational force field are seen to be described by a four-vector field
A_{μ}, which forms part of the body of the variable unit vectors and hence the metric of spacetime. The resulting theory very strongly appears to be a logically consistent and coherent unification of classical and quantum physics and at the same time a grand unity of all the forces of
Nature. Unlike most unification theories, the present proposal is unique in that it achieves unification on a fourdimensional continuum of spacetime without the need for extra-dimensions.

Much effort by a great many erudite, notable and foremost physicists and mathematicians has gone into the all- noble and all-esoteric search for an all-encompassing unified theory of all the forces of Nature; as for our own attempts, the present reading marks the third such effort. The beguiling problem of how to unify Quantum Theory (QT) and Professor Albert Einstein’s General Theory of Relativity (GTR) into a consistent and coherent unified theoretical framework has not only baffled but eluded the foremost physicists and mathematicians of the past eight decades or so [e.g. Professor—Albert Einstein (1879-1955), Hermann Weyl (1885-1955), Theodor Kaluza (1885-1954), Erwin Schrödinger (1887-1961), Edward Witten (1951-), Stephen Hawking (1942-), etc.; to mention but a few pre-eminent figures in this all-great endeavour], and with the failure in recent times to deliver on its now long overdue promise i.e., failure by the so-called most promising theories on this front (e.g. String and String Related Theories), the need for a unified theory of all the forces of Nature has become particularly urgent and pressing, especially given the most recent announcement of a Higgs-like particle at CERN coincidentally on the occasion of the United States of America’s 236^{th} independence from colonial Britain on the 4^{th} of July 2012. Surely, a universally acceptable form of order is urgently needed in the “House of Physics”.

One of the first major problems beguiling the efforts to finding a unified theory is that each of the two theories in question (i.e. QT and the GTR) is considered by a great many physicists as fundamental physical theories of physics in their own right. By “fundamental physical theory” it is understood that these theories are not derivable from any other physical theory, they are—in Professor Einstein’s terminology, Theories of Principle^{1}. Accepting these theories as fundamental physical theories lands us in torrid philosophical and logical conundrums as this invariably implies the existence of two independent physical realities—one reality of the very small (QT) and the other of the large (GTR), each governed by separate Physical Laws. The strong and existing feeling that there can only be one and only one reality that points to one and only one fundamental physical theory, hence, a unified theory of all of reality is thus needed. Surely, the garment of physical and natural reality must constitute an undivided whole and it thus obviously makes no sense to have two distinct fundamental theories for a single united garment of reality.

Thus far, the attainment of a unified field theory has proved to be elusive, it is thus reasonable to ask “How does it come about that physicists have for so long been able to succeed in failing to find a unified theory?” Amongst others, we feel—as Dr. Philp Gibbs^{2}, that physicists have held far-too-sacrosanct these two giant pillars of modern physics—i.e., QT and the GTR; and they have in the process, gone so far as to confidently believe that naturally, a logical fusion of the two ideas must lead to paths that lead straight to the depths of an acceptable theory that must take humankind to his next level of understanding of physical and natural reality as we experience it. In his own words about this line of thinking, Dr. Philip Gibbs had this to say [

“String theory has had many opportunities to reveal itself as a ‘bridge to nowhere’ but at each turn the road carries on instead and another bridge is crossed. The theory is very tightly constrained by the need to be consistent with General Relativity and Quantum Mechanics. The theorists are not making it up. They are [only] following the course that logical consistency dictates and it is remarkable that there is anything at all that can match the requirements, but of course there has to be because the Universe exists with these features.”

What the above words remind us are the wise words by the great British-German physicist, Professor Max Born (1882-1970). He once said:

“Science is not [mere] formal logic. It needs the free play of the mind in as great a degree as any other creative art. It is true that this is a gift which can hardly be taught, but its growth can be encouraged in those who already posses it.”

Simple stated, like Dr. Gibbs has correctly said it: the present day theorists are trying to construct a new theory by believing (wrongly or rightly so^{3}) that a theory logically derived from the general theory of relativity and quantum mechanics will naturally lead to a logically correct theory which unifies both theories. For example, one of the co-founders of string theory—the pre-eminent Japanese-American physicist, Professor Miko Kaku, in one of his popular television shows, he frankly admits that the traditional way of doing science [i.e., of observation a hypothesis a theorisation a prediction a falsification] is thrown out the window when it comes to string theory; in this field, one has to relay on intelligent guess work as they march forward in their arduous search for a theory that is logically consistent with what already exists and is accepted by the majority of scientist as a theory representative of the present Universe.

The folly (perhaps) in this approach or way of reasoning is that we are forgetting the wise words of the great Professor Max Born that “science is not [mere] formal logic it needs the free play of the mind in as great a degree as any other creative art”. Logic alone is not enough, nor is intuition, we need all these “forces” at play together with a modicum but strong dose of imagination which Professor Einstein said “is more important than knowledge”. We need a leap out of logic to make it into the next virgin territory. Perhaps we must remind the reader of other wise words by one of Germany’s greatest scientists, Professor Karl Ernst Ludwig Marx Planck (1858-1947), who in his lifetime once said that:

“The man who cannot occasionally imagine events and conditions of existence that are contrary to the causal principle as he knows it will never enrich his science by the addition of a new idea.”

Because it has happened and we have experienced it, everyone will concur that no amount of applied research on the candle (or the wax the making it thereof) would have produced the light bulb—to attain the light bulb, the human mind had to leap-out of its ordinary state of conscientiousness. The above words imply that for us to take a leap out of logic and into virgin territory will—at the very least—this requires of us to “imagine events and conditions of existence that are contrary to the causal principle” as we have come to know and experience it. This means that logic may very well not lead to a theory with a direct correspondence with experience. For example, no amount of logical reasoning in Newtonian gravitation would have led mankind into Professor Einstein’s brilliant geometric description of gravitation and spacetime, one had to “defy” logic and enter into much higher planes, terrains and farrows of logic and natural reality.

In this reading—given its length, we find no space to give a wide literature review or an overview of the history of unified field theories. For this, we direct our reader to the excellent review by Emeritus Professor of Physics at Germany’s Göttingen University, Hubert F. M. Goenner; in the readings [

In § (2), we outline two fundamental reasons that call for a revision of Professor Einstein’s GTR. In § (3), we give an overview of the reading [

Other than the shear and insatiable human desire to be part of such a noble endeavour to finding an all encompassing unified theory of the forces of Nature; if any, what are the real and tangible reasons requiring us to revise Professor Einstein’s GTR? As to ourselves—and as may be the case with the majority of physicists; we find two compelling reasons. The first has to do with the clearly implied need for the unity of the GTR and QM in the simultaneous case of extremely massive and extremely small objects. The second has to do with a well known and largely ignored internal inconsistency of the GTR.

On the first note, QM describes matter at the smallest scale-length as exhibiting both point-particle and wave- like properties. The GTR describes at the largest scale-length the same matter as mere point-particles. There is no wave-like description of matter in the GTR, none whatsoever. However, after exhausting their nuclear fuel which is believed to hold them against the inward tyranny of the gravitational force, massive luminous objects such as stars—whose macroscopic properties are described very well by the GTR; these objects can undergo collapse in which case their spatial size can go down to scale-lengths that require us to use QM to describe them. Under such conditions, it becomes inescapable that one would need a Quantum Theory of Gravity (QTG) in-order to describe the physics in-and-around such objects. For this reason, the need for a QTG is not only clearly evident, but is a dire need and necessity.

On the second note—in our view, one of the major problems that the GTR faces within its own internal structure of logic is that it is based on pure Riemann geometry i.e. a geometry that is well known to violate the Principle of Equivalence at the affine level because the affine connections are not tensors. If pure Riemannian geometry is to be the true geometry to describe the natural World, then, no Laws of Physics should exist at the affine level of Riemann geometry. However, this is not so, since the Geodesic Law:

that describes the path and motion of particles in spacetime emerges at the affine level. Thus accepting Riemann geometry as a true geometry of Nature means we must accept contrary to the Principle of Relativity that there exists in Nature preferred reference and coordinate systems because the above Geodesic Law leads us to formulating the equations of motion in preferred reference and coordinate systems, namely, geodesic coordinate systems also know as Gaussian coordinate systems.

Einstein’s GTR is based on two Equations (1): the geodesic Equation (1) which determines how particles on in the curved spacetime and (2): the field equation which tells us how matter curves spacetime, i.e.:

where

Gaussian coordinate systems are those coordinate systems such that

Given for example, that the affinities represent forces as is the case in the GTR, this means a particle could be made to pass from existence into non-existence (or vis-versa) by simply changing the coordinate system. This on its own violates the Laws of Logic and the need for Nature to preserve Her independent reality devoid of magic. Clearly, the only way out of this conundrum is to seek, as Professors Einstein, Schrödinger etc have done; a theory in which the affinities have a tensor form, hence in the present approach, the first and most important guide is to seek tensorial affinities. Professor Einstein, Schrödinger etc. have made attempts along these lines only to fail. The reason for their failure may perhaps stem from the fact that theirs was a pure mathematical exercise to try to find a set of tensorial affinities from within the framework of the classical spacetime of Riemannian geometry. It should be said that their failure [Professor Einstein, Schrödinger etc.] has not been a total failure as their (failed) work led to significant advances and better insights into the nature of the problem at hand. In-fact, the present work does spring from these so-called failed attempts. Our line of thought stems directly from their line of thought. Thus, we can never say their [Professor Einstein, Schrödinger etc.] effort was in vein.

As already stated, the present UFT is built on a modified theory of Professor Weyl [

In his attempt at a unified theory, Professor Weyl realised that if he supplemented the Christophel three symbols of Riemann geometry i.e.

where

where

Professor Weyl’s proposed theory led to a rescaling of the fundamental metric tensor via the gauge transformation,

As aforestated—in the reading [

The NWG is exactly the same geometry is one two all-important differences. The first of which is that the we replace the metric of Riemann geometry

In the present section we justify one of the major ideas of the present unified theory. This is the idea that the metric tensor can—logically, be decomposed in a manner that allows us to write down an equivalent Dirac equation in curved spacetime. The resulting decomposed metric is central to the final unified theory, it is so central that without it, one can not envisage obtaining the required correspondence of the present theory with experience.

That said, there are several curved spacetime versions of the Dirac equation (cf. [

Professor Dirac [

where

γ-matrices^{4} are 4 × 4 matrices defined such that:

where

The index “a” is not an active index as are the Greek indices. This index labels a particular curvature of spacetime i.e. whether spacetime is flat^{5}, positive or negatively curved. Written in full, the three metric tensors

and the metric

Especially for a scientist and/or mathematician, there is little if anything they can do but accept facts as they

stand and present them-self. So doing, this means that the writing of

accepted as a legitimate mathematical fact for as long as

now be written as

low Professor [

It is not a difficult exercise to show that multiplication of (11) from the left hand-side by the conjugate opera-

tor

As it stands, Equation (11) would be a horrible equation insofar as its solutions are concerned because the vector

where now

In Equation (12), the vector

What we have done in the present section is to demonstrate that the metric tensor is susceptible to decomposition into a tensor field describable by four unique fields that form a relativistic four vector. In the GTR, the metric tensor exists as a compound mathematical object comprising ten unique fields. We shall take a leaf from the present exercise whereby we shall import it into the unified field theory that we hope to build. Despite the obvious similarities, the decomposition of the metric that we employ in the proposed unified theory is different from the decomposition carried out in the present section.

The to be proposed all-encompassing Unified Field Theory (UFT) of all the forces of Nature is built on a spacetime that we have coined the Riemann-Hilbert Spacetime (RHS). Hereafter, when-ever we refer to the UFT, we mean the UFT presented herein. In the sections that follow, we shall succinctly describe the major highlights of RHS and in so doing, we are in actual fact describing the UFT itself. This spacetime has been dumbed the RHS because it is a hybrid Riemann space which has in it embedded Hilbert-objects. By Hilbert-objects, we mean the quantum mechanical wavefunction. The RHS is a modified Weyl space which Professor Einstein ruthlessly short-down when Professor Weyl tried to put forward his unified theory of gravitation and electromagnetism.

In the RHS, the entire structure of Riemann geometry is inherited in its complete form and is modified by replacing the metric with a new somewhat conformal metric

where

1) Comprised of real elements.

2) Hermitian i.e.

Now, we will add an extra index-

If we write

the metric shall be written with the index-

Notice that we have placed the index-

Now, when finally we derive the Dirac equation in the separate future reading, it will be seen that the different representations of the metric

Any space or spacetime must be endowed with unit vectors at each and every point of the continuum. The unit vectors on the RHS are variable, that is to say, if

where

In future reading where the Dirac equation will be derived from the soils of the present theory, it is shown how one can naturally harness from (15) the curved spacetime Dirac equations presented in [

The position vector

We must hasten to say that, whosoever seeks to discredit and bring down the present theory on the same basis as the great Professor Einstein did with Professor Weyl’s theory on the grounds that the line element

An important point to note is that the metric

As will be demonstrated in a future reading where the Dirac equation will be derived from the soils of the present theory, the field

Now, in the next section i.e. in § (6), we will move to demonstrate that one can attain tensorial affiens. The attainment of tensorial affinities will allow us in § (7) to write down a general field equation for the RHS. In the § (8), we use the tensorial nature of the affinities to get reed of the non-liner aspect of the RHS. Having obtained non-linear field equations on the RHS, we show in § (9) that these field equations lead us to a familiar territory of Yang-Mills (YM) field equations and conservation laws. The resulting field equations as they stand appear at face-value to have the gravitational force outside of the whole scheme, leading to the feeling or notion that the present UFT is only a unified theory of the nuclear forces. We argue via a gravitomagnetic approach in § (11) that the gravitational force is very much part of the whole scheme. Further, in § (12), we show that one can derive the familiar geodesic equation of motion with the Lorentz-force term for both the electrical and gravitational forces.

If we replace the metric,

In this new affine (17), the object,

then, it clearly follows that if,

the affine,

Thus—at any rate,

In this way—and this way alone; we attain the long sort tensorial affinities. So, the object

It should be noted that nothing from within the RHS requires that the affinities be tensors. This is a requirement imposed exogenously by us because the RHS allows us to make such a choice without violating any of its marvellous structural properties. This reminds us of the words of Professor Einstein when he said that when constructing a theory, one must put themselves in the position of God^{6} and ask themselves the deepest and most enduring questions about physical and natural reality, such as:

“If I were God; how would I have created the Universe? Did God have a choice in creating the Universe?”

To these words of Professor Einstein, we add the following words:

“If He (God) did have a choice, what is that choice and; why that choice and not any other?”

Our pedestrian feeling on the issue of the Mind of God, His thoughts and choices is that all the constraints— more so the gauge conditions that we find necessary for our theories; the constraints that we impose on fundamental physical theories, these are most certainly God’s subtle thoughts which He employed in making the Universe. These exogenous impositions are there to attain a particular structure of the World. If any, who could have imposed these conditions from outside except the Creator Himself? This once again reminds us of the other of Professor Einstein’s wise words:

“I want to know how God created this World. I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts; the rest are details.”

Perhaps, the Mind of God insofar as the architecture, design and building of the World is concerned, this might very well be hidden and revealed in the Universal World Geometry upon which the Universe is created and His thoughts in doing so (i.e., creating the World), in the gauge conditions and exogenous impositions on this Universal World Geometry; i.e., impositions that allow us to obtain exactly the conditions and laws of the present World. That is to say, are not these gauge conditions a sacrosanct signature of God’s sleight of hand and mind in creating a Universe of His choice? We leave the reader to ponder and ex-cogitate on this and the other deeper questions here raised above about physical and natural reality.

Now, back to the real World, the palatable World of ponderable facts. In the present section, we have demonstrated that one can achieve on the RHS tensorial affinities. In the next section, we shall now use this to deduce the major field equations for matter and energy of the RHS. From this equation, we are hopeful that all of reality—insofar and fundamental physical forces are concerned; must be harnessed forthwith. Before that, we need to do two things, the first is ask, what

Professor Maxwell’s electrodynamic field tensor has the property of being traceless i.e.

This gauge condition will lead us to the another gauge condition that we will present in (48). Other than the traceless property, Professor Maxwell’s field tensor also has the anti-symmetry property

not be generalizing this to the present field tensor

this leads to physically meaningless equations as this requires that the vector

Now, if the function

then,

Be that it may, we—at any rate, do not believe the geometrical description of

Riemann geometry is built on the idea of parallel transport of vectors along a given path. A good description of parallel transport is perhaps that by Professor John Baez^{7}. Following him [i.e. Professor John Baez]; say one starts at the north pole holding a javelin that points horizontally in some direction, and they carry this javelin to the equator always keeping the javelin pointing “in as same a direction as possible”, subject to the constraint that it points horizontally, i.e., tangent to the Earth. In this scenario, we see that the idea is that we are taking “space” to be the 2-dimensional surface of the Earth and the javelin is the “little arrow” or “tangent vector”, which must remain tangent to “space”. After marching down to the equator and making a 90˚ turn at the equator and then marching along the equation until some-point along the equator where another 90˚ turn toward the north pole is made thus marching back up to the north pole, always keeping the javelin pointing horizontally and “in as same a direction as possible”. “Obviously”, because the surface of the Earth is curved, by the time one gets back to the north pole, the javelin will be pointing in a different direction. The javelin is said to have been parallel transported from its initial starting point to the final end point in one cycle of a closed circuit.

Parallel transport is an operation that takes a tangent vector and moves it along a path in space without turning it (relative to the space) or changing its length akin to the a person that carries a javelin as described above. In flat space, we can say that the transported vector is parallel to the original vector at every point along the path. In curved space as described above, the original and final vector after the parallel transport operation are not coincident and the change in this can be computed as will be done below and for this exposition, one can visit any good book on the GTR (cf. [

As shown in

where,

and by differentiating the quantities in the brackets, this further reduces to:

and using the identities,

This can be written compactly as:

where,

For a moment, let us shy-away from the abstract World of mathematics and pause a perdurable question to the reader, a question about the real World. Suppose one is in a freely falling laboratory and this laboratory moves in a gravitational field in a closed circuit such that the laboratory leaves a given point and latter it returns to the same-point and throughout its path at all points it is in free-fall. The best scenario is a laboratory orbiting a central massive, body. If in this laboratory we have a stationery object—do we (or does one) expect that after a complete orbit this object will have its motion altered? Or, does one expect that an object (inside the laboratory) that—say, has a specific momentum (relative to the laboratory) will after a complete circuit alter its momentum without any external force being applied to the free-falling system?

If this did happen, then Newton's first Law of Motion that defines inertia systems of reference is violated and it would mean that there is no such thing as an inertial system of reference; actually this renders the Principle of Equivalence obsolete. Surely, something must be wrong because the Principle of Equivalence can not be found in this wanting-state. We argue the reader to carefully go through the above argument to convince themselves of its correctness (or its incorrectness thereof). Whatever conclusion the reader will reach, it does not affect the final thesis being advanced namely that the affinities must be tensors. If they disagree with the above, it really does not matter as long as they agree that tonsorial affinities preserve both the angle and the magnitude of a vector under parallel transport.

In the above, we say this renders the Principle of Equivalence obsolete because for a system in free-fall like the laboratory above, according to this Principle of Equivalence; it is an inertial system throughout its journey thus we do not expect an object in an inertial system to alter its momentum without a force being applied to it. The non-preservation of angles during parallel transport in Riemann geometry is in violation of the Principle of Equivalence if it is understood that parallel transport takes place in a geodesic system of reference i.e., inertial systems of reference.

Naturally, we expect that for an observer inside the laboratory, they should observe a zero net change in the momentum. This, in the context of parallel transport of vectors, means that, such a spacetime will parallel transport vectors (in free-falling frames) in a manner such that after a complete circuit the parallel transported vector and the original vector, will still have the same magnitude and direction i.e.,

where:

The fact that,

because

Given that we have attained a geometry with tensorial affines, now is our time, this is our moment of exhalation, a moment to reap the sweetest fruits of our hard labour i.e., it is time to take the fullest advantage of the tensorial nature of the affinities. We now have the mathematical and physical prerogative, legitimacy and liberty to choose a spacetime where the non-linear terms vanish identically i.e., a spacetime such that,

Clearly and without any doubt, this fact that we have chosen a spacetime that is governed by the constraint (32), this means that in a single and triumphant moment of joy, we have just reed ourselves of the monstrous and troublesome non-linear terms in the Riemann tensor (30) because with this beautiful and elegant choice (32), they [non-linear terms] now vanish identically to become but footnotes of history. This means (31) now becomes:

From (32), if we set

Now that we have a theory linear in the curvature tensor, a theory in which the non-linear terms vanish, we can use this to separate the Weyl terms from the Riemann terms. Weyl terms are here those terms associated with the affine vector

where:

is the linear Riemann curvature tensor, and:

is a curvature tensor that we shall call the Weyl curvature tensor. This Weyl curvature tensor is defined in terms of the material field

Now, from (33) and (35), it follows that:

At this point—if it turns out the this theory proves to be a correct description of physical and natural reality; we have no doubt that if Professor Einstein where watching from above or from wherever in the intestacies of spacetime, he must be smiling endlessly because his life long endeavour was to derive^{8} the material tensor from pure geometry and not to insert it by the sleight of the mind and hand as he did with his gravitational field Equation (2). As his internationally acclaimed scientific biographer Abraham Pais graphically put it—Professor Einstein would look at his Equation (2) and greatly admire the left hand side which he would compare with marble—it was beautiful, pure and was a marvel sight for the artist as it was constructed from pure geometry; but when he [Professor Einstein] would look at the right hand side of this same equation; with equal passion, he was visited by deep melancholy sighs and feelings which made him loath the right hand side of his equation—he would compare this right handside with wood—it was ugly; with no end insight, Professor Einstein would moan time and again until his last breath on 13 April 1955.

His ultimate pine and goal thereof was to turn wood into marble, which meant deriving the material field from pure geometry. Professor Einstein wanted to find the final theory, this he pursued to the very end of life to a point that while on his death bed on April 13, 1955, instead of worrying about the imminent end of his fruitful life, he asked for a pen and his notes so that he could continue to work on the unified theory that he was working on at the time. It is sad to say that Professor Einstein never laid a fertile egg on this front—i.e., the front of unification.

Be that it may, it is without an iota of doubt that we say that, if what is before us proves itself to have a correspondence with physical and natural reality, then we can safely say we have achieved one of Professor Einstein’s goals to attaining the “Elicit Dream of a Final Theory” by deriving the material tensor from pure geometry—wood, one way or the other, has finally been turned into marble! This we are certain has been achieved in the present UFT. The only question is, “Does the theory correspond with physical and natural reality?” This we leave in hand of the reader so that they may be their own judge on the matter.

Now, we shall derive other field equations. We know that the Riemann curvature tensor satisfies the first Bianchi identity:

From this first Bianachi identity and as as-well from (38), it follows that:

The identity (40) leads to an important identity (which, for our purposes we shall call it a gauge condition). This identity (gauge condition) is obtained from (40) by contracting the index

When we arrive at (58), it shall become clear that the gauge condition (41) is necessary in-order for us to arrive at the familiar Maxwell-Proca equations of Electrodynamics.

Now, we are are ready to derive our last set of field equations. We know that Riemann curvature tensor satisfies the second Bianchi identity:

From this second Bianachi identity and as as-well from (38), it follows that:

In the next section, we shall explore the Equations (38), (39), (40), (42) and (43) and from these equations, we shall see that one is able to obtain field equations that we are already familar with.

Before we go into the details of the present section, we must mention that, from here-on, we will now bring back the

Thus, substituting this metric into the affine connection and then differentiating it as required by the differentials appearing in the affine, we obtain the following:

Now, if we set:

then, it follows from (44) and (45) that:

In (45), the reader must take notice of the fact that the Greek index-

Now, moving on; in the expression of the affine connection,

Now that we have written the affine in terms of the field,

Before closing this section, we will write down a gauge condition that flows from the gauge condition (22) i.e.

This implied gauge condition (48) will prove necessary in-order for us to arrive at the crucial equation (53).

The Lorenz gauge condition (here represented by the relation

For the present UFT—rather naturally than by the sleight of hand—the Lorenz gauge condition has been found necessary in-order that there is symmetry in the Greek indices of the resulting mathematical objects of the theory such as tensors e.g. for

where

Now, we are going to deduce the source free field equations from the first Bianachi identities (39). We know

that

derivatives and in terms of the vector field

Inserting

and:

We will take (51) and (52) as general field equations. If we multiply (52) throughout by

Further, if we multiply (53) throughout by

therefore:

We shall assume that only (55b) holds true and

Now, from (52), we will derive the source free field equations. If we multiply (52) throughout by

In the framework of Maxwellian Electrodynamics, the above set of Equations (56) gives the exact source free field equations. It therefore follows that we have derived the first set of Maxwell’s equations, namely, the source free equations. This—to any searching mind, especially after a twenty year arduous journey of relentless searching; is certainly a reassuring development. One can take a breath of fresh air and walk straight to look out the window and hope to see a pristine blue-sky.

We shall now proceed to derive the source coupled field equations. This, we shall do by use of Equation (38) by contracting the indices

First, we shall compute,

where:

where

The tensor

in which case it becomes clear that

Now we proceed to compute

and the scalar

Since

therefore:

Now, the source coupled field equations shall be derived from the following equation:

Therefore, we need to compute

Now, in the representation indices, let us set

From the gauge condition (55b) i.e.

This already looks very familiar—doesn’t it?

Now, let us proceed to calculate the left handside of (67) i.e.

From the definition of

By inference from Maxwellian Electrodynamics, we know that,

Now, putting everything together i.e. from (67), (70) and (73), it follows that:

In the framework of Electrodynamics, not only have we derived the source coupled equations of Professor Maxwell [

Thus, in Equations (56) and (74), we have attained the desired set of equations that enable us to obtain—as a bare minimum—all of Professor Maxwell’s equations. Prima facie, the gravitational force appears to be absent from the present scheme. We will tackle this issue soon in the § (11). In the next subsection, we will have a look at the second Bianachi identities (42) and (43), from where we deduce some conservation laws.

We are now going to use the second Bianachi identities (42) and (43) to demonstrate the existence of a conserved four current from the field tensors,

For this, we are going to use Equation (18) that has been derived in the reading [

As shown therein, this equation is necessary for there to exist massive long range and long lived photons. If this equation holds, it therefore follows that:

Now, we can proceed to show the existence of a conserved four current from the field tensors,

and contracting the indices

We know also know from (43) that:

and as above, contracting the indices

What the above equations imply is that the tensors

where

Though we are not done yet, it is perhaps time to take stock of what we have achieved thus far so that we bring some clarity and as-well so that the length of the road ahead may be made known and furthermore, so that the reader does not develop fatigue. Firstly, in-order to generate the desired field equations (i.e. 74 and 56) which surely have a resemblance with equations we that already know from experience, we have required a total of five gauge conditions and these gauge conditions are:

1) The first gauge condition is the Lorentz gauge

2) The second gauge condition (22) has been imported from Maxwellian Electrodynamics and applied to the present theory i.e.

3) The third gauge condition (48) i.e.

4) The fourth gauge condition (55) is,

The field equations that we have produced seem to have nothing to do with gravitation but the nuclear forces. This obviously gives one the feeling that gravitation has now been shut out of the unification scheme i.e. all the other forces have willingly submitted to geometrization and gravitation has been shut out somehow. In the first two instalments [

Professor José Hera [

then, there must exist two corresponding fields, X and Y, that satisfy the following set of equations:

where

If—as is widely believed; the force of gravity is conveyed by mass, then, if the associated current (momentum) of this mass is conserved as given in (83), then, there must exist the two fields associated with mass and momentum. One of the fields is obviously the gravitational field g. The other must be a magnetic like force. Let us give this field the symbol B_{g} and let us call this the gravitational magnetic field and for short the g-magnetic field. What this means is that there must exist a similar set of Professor Maxwell's equations for gravitation. This phenomenon is typically called gravitomagnetism.

The first to consider the possibility of a formal analogy between gravitation and electromagnetism is Professor Maxwell himself, in his landmarking treatise on “A Dynamical Theory of the Electromagnetic Field” [

Writing in his book “Causality, Electromagnetic Induction and Gravitation”, Professor Oleg D. Jefimenko (1922-2009) revived [

Professor José Hera [_{g}) that satisfy a set of four Maxwell-type equations. These four equations describe what is known as gravitomagnetism. Gravitomagnetism is nothing but a formal analogy between Maxwell’s equations and the field equations of gravitation.

The problem with the issue of gravitomagnetism is that some have taken this to mean that the generated g-magnetic field must have an effect on electrically-laden bodies—no! This field, B_{g}, only couples to mass just as the magnetic field generated by electric currents couples to electrically-laden bodies. How then are we to introduce gravitational into the present scheme? If the gravitational phenomenon does take part in Maxwell-Hea- viside gravitomagnetism, then, there must exist a gravitomagnetic four vector,

where

where in general

hence:

and:

Therefore, the Riemann metric

where

and in-turn, the curvature tensor is given by:

In the complete absence of electromagnetism (i.e.

Therefore, the resulting field equations for the electromagnetic sector of the RHS will be:

while for the gravitomagnetic sector, the field equations will be:

Therefore, herein, the same equations governing the electrical phenomenon govern the gravitational phenomenon in such a way that both are driven by two independent vectors which are sadly related to each other only by addition and the resulting mathematics thereof. The two sit side-by-side with no intimacy whatsoever. However, in a future reading, we shall propose a truce whereby we slip in a component of the gravitational magnetism into the electromagnetic phenomenon. Before that, let us derive first the geodesic equation of motion.

Let us now address the problem raised in § (2), i.e., the problem of the geodesic law namely that it is neither invariant nor covariant under a change of the system of coordinates and/or change in the reference system. As currently obtaining in the GTR, in order to derive the equation of motion, one needs to formulate this equation first in a Gaussian coordinate system and thereafter make a transformation to a coordinate system of their choice. As already said in § (2), this geodesic equation is in conflict with the very principle upon which the GTR is founded, namely the Principle of Relativity, which requires that one should be free to formulate the geodesic equation of motion in any legitimate coordinate system of their choice without having to be preferentially constrained to start from a Gaussian coordinate system. The geodesic law Equation (1) is derived from the Lagrangian:

by inserting this into the Lagrangian equation of motion, namely:

and thereafter making the necessary logical algebraic operations and manipulations. In the present, one must realise that the metric

will yield an appropriate equation of motion that agrees with experience in describing the trajectory of a massive electrically laden test particle in the vicinity of a massive electrical charged parent body. In the next few paragraphs, we shall shed some light on this issue whereby, we come up with a Lagrangian that leads to the appropriate equation that is in tandem with physical and natural reality.

In its bare form, Equation (1) has no natural provision for describing the trajectory of an electrically laden test particle. If the parent body, i.e., the body about which the electrically laden test particle describes its motion; is also carrying an electric charge, then, if Equation (1) is to describe the path of the electrically laden test particle, it (Equation (1)) will have to be modify outside of the confines of the GTR. What we want—or more appropriately, what is expected of a Universal World Geometry such as the RHS, is that the geodesic equation of motion for the RHS must have within the confines of the RHS, the provision for trajectories of electrically laden test particles.

If at all, how are we to attain such a geodesic equation capable of describing the trajectory of an electrically laden test particle in the vicinity of a massive electrically charged parent body? From (94), it is clear that the RHS metric,

The suggestion coming from the fact that the metric

where

where

Now, inserting

where:

is a tensorial affine for as long as the γ’s are all scalars. The tensor affinities

and like wise, we have

Now, since the affine

Therefore, to first order approximation, (104) becomes:

an assuming the following gauge conditions:

then, (108) reduces to:

Without the Gravitomagneto-Dynamics (GMD) component, the above equation is the acceptable Lorentz force law. The

The new term on the far right of (110) we have coined it the quantum force as it depends on the quantum probability amplitude

In all its noble endeavours as presently understood, Yang-Mills (YM) theory [

Against the desideratum of the searching mind and very much to its chagrin, the natural picture emergent from the present UFT is one were the Universe is exclusively divided into two parts where one part is gravitational and the other electrical. In a separate future reading which tries to find intimacy between the electrical and gravitational phenomenon, we have proposed, more by the sleight of hand rather than from the naturalness of the theory a truce-like union between these gravitational and electrical components of the Universe. For the present section, we shall only consider the electrical component with the intent to demonstrate the existence of YM- Theory in the present UFT.

First we must realise that there are four fields in the representation F; and these are

non-Abelian. If

then, from the definition of

What is interesting in all this is that, the electromagnetic field together with the resulting YM-Fields, these fields are all derived from the same vector field. This is unlike in the current SM were every field requires its own four vector field. In the scheme and core of the spirit of unification, this is considered to be the sought for unification were all the forces emerge from a singe field, they are different manifestations of the same field. The question now is, do the fields

We are now going to demonstrate that under certain conditions of existence on the RHS, there exists YM- Theory—a YM-Theory that is exactly as that found in the current SM, albeit, with the important difference that it is now a four dimensional YM-Theory. For this, let us generally consider the fields

where the q’s are real or complex numbers which are constraint by the usual wave-function normalization condition, namely

where in the ordinary picture of QM, one can view the q’s in the same way one views the probability amplitudes

in quantum superposition i.e. in-ordinary QM, we have

and _{j} are probability amplitudes. The objects ^{9}:

where

where:

Just like

where:

It is very important to note at this point that if the wavefunction

Now we come to the most crucial part on our “sojourn to demonstrate” the existence of YM-Theory on the RHS. Some may consider and ultimately accuse the step that we are about to take as a deliberate and artificial insemination of a condition into the theory so that the theory matches with experience. We see no problem with fixing a theory so as to match with experience. Like Professor Einstein’s GTR, most people would demand of a theory to be constrained by its internal and natural logic in-order to match with physical and natural reality. We ask you the reader:

Insofar as adoption is concerned, what would you do if you realised that there is an absolutely necessary condition not naturally found in the domains and confines of your theory, a condition that would make the theory beautifully match with experience? Would you not impose this condition by assuming it be a vital but (an as-yet) naturally unjustified ingredient of the theory?

Rather sceptically, fixation is largely viewed by the better majority of researchers as some form of “cheating” so as to arrive at the correct or desired answer. If anything, we are of the benevolent view that what one must ask after fixing a theory to match with experience is the following question:

What—if any; does this fixation tell us about the Laws of Physics in relation to physical and natural reality?

Before answering this question ourselves, let us write down the ansatz condition that we feel leads us to field equations that have a sure correspondence with the equations that we already known from experience—i.e., the equations of YM-Theory as they appear in the current SM. We find that if the following condition:

is adopted (or assumed) and is taken as a fundamental gauge condition were

Notice that the ik-indices of

Now, with this gauge condition (120) given, it follows that the field (45) will now be given by

In form and structure, this field is the usual YM-Field as we know it—albeit—a YM-Field whose

Now, substituting (112) and (113) into (98), the resulting field equations:

In (122), we have also written the resulting field equation for the gravitational and electrodynamic field. These equations surely lead to the following corresponding equations for the sub-particles

Now, if at the most fundamental level of reality all the physics of the World is embodied in the field equations (122) and (123) [or (74) and (56)], then, the following transformations will leave all of the physics of the World invariant:

where:

is a hermitian unitary matrix which is such that

While the transformations (124) will leave all of the physics of the World invariant, the transformation (124b) will alter the geometry of the sub-particles and as-well that of the particles themselves. In relation to the question that we paused earlier about the fixation of the theory to match with experience i.e. the question “What this fixing tells us about the Laws of Physics in relation to physical and natural reality?”, we now realise that this fixation is telling us that one can alter the geometry (via the alteration of the metric

Thus, to justify the fixation (120) here-made without any prior knowledge that Nature employs YM-Theory in exactly the manner it does as we know, one would have to ask themselves the ansatz-type question:

What condition exists—if any; and is there to be imposed upon the RHS in such a manner that one can somewhat in an arbitrary, somewhat willy-nilly and willy-wally fashion, alter the geometry of a particle without affecting the resulting physics thereof?

The answer would be the imposition of the gauge condition (120); it is a condition which for better and/or for worse, is to be adhered to by the vector fields,

We cannot possibly know whether or not the Good Lord looks down from above at our present endeavour and laughs if not scoff at it, that perhaps in all the twenty years (1994-2014) of the relentless and unflinching labour that we have put up in-order to attain the present unified theory, He has led us up the garden path on a sure blind alley. Be that it may, we surely must (unflinchingly)—for the sack of posterity; be bold enough as to express our true heartfelt confidence that the New Universal World Geometry (the RHS) here discovered most certainly describes our World or will in the immediate future prove to do exactly that. If experience proves otherwise—then, without an iota or dot of doubt; at least some elements of the present endeavour will surely percolate significantly into the Final Universal World Geometry to be discovered in the future by others.

The present UFT is built on what is supposed to be Professor Weyl’s fail idea [

We should mention here that in building the present UFT, we have not made reference to the literature on unified field theories in the intervening years 1918-2014 and the reason for this being that we have have had to go back to 1918 and built our theory from the debris of Professor Weyl’s supposed failed unification effort. In the end, the final equations strongly appear to resemble currently accepted equations describing the nuclear forces. Because of this, we feel that the present work—if proved correct, may be pointing to the fact that, in our effort to arriving at the final theory, we may very well have taken the “wrong” theoretical route because our theories in building the theory of nuclear force have been predominately shaped by results from experiments rather than from theory. From a theoretical standpoint, it appears we have been trying to glue together general relativity and quantum mechanics using only logical connections and reasoning. In that effort to glue these two great theories together, we have respectively held them to be sacrosanct leading us to seeking a self consistent manner to glue them together while preserving their independent realities.

Basing on our true heartfelt confidence, we must say that, what is before us so strongly seems to point to the fact that the whole of physical phenomenon may now be derived from a single Universal World Geometry that obeys the Universal World Law beholden to the principle presaged in reading [

Generalized Principle of Equivalence: Physical Laws have the same form in all equivalent reference systems independently of the coordinate system used to express them and the complete physical state or physical description of an event emerging from these laws in the respective reference systems must remain absolutely and independently unaltered—i.e., invariant and congruent; by the transition to a new coordinate system.

We hearty, sincerely and in all probity believe that we have herein shown that it is possible to describe all the known forces of Nature on a four dimensional geometric theory that needs not the addition of extra dimensions as is the case with string and string related theories which today predominately stand on the beacon and chief echelons of unification as the most favoured/promising theories. Although a lot of work is still to be done as mentioned in the penultimate of the introduction section that two volumes each spanning about 200 pages are at an advanced stage of preparation, in our most humble and modest opinion, we believe that the present attempt is a significant contribution to the endeavour of the unification program of physics. We are of the strong opinion that the reader should be their own judge. We shall state a few main points that lead us to believe that the present attempt is a significant contribution to the endeavour of the unification program of physics.

First, we know that in Riemann geometry and as-well in the GTR, the metric tensor is described by ten different potentials and in-turn these potentials describe the gravitational field. On the RHS, the metric tensor is described by just four different potentials and not ten and these potentials describe both the gravitational, electromagnetic and nuclear and sub-nuclear fields. At the very least, this surely is a paradigm shift which is a much more simpler model of spacetime and matter that the most promising models! The RHS on which the present theory is founded, is different from Riemann spacetime in that the unit vectors of the RHS are variable at all points on this continuum. This property of the RHS that it has variable unit vectors make it fall into the category of Professor Weyl’s brilliant but failed geometry, but the RHS is different because the affinities are tensors; the meaning of which is that vectors do not change their length and angles under parallel transport.

We note that while the gravitational force has here been brought under the same roof with the nuclear forces, it is not unified with these forces in a manner Professor Michael Faraday had hoped, i.e., it being inter-convert- ible to other forces just as the electronic and magnetic forces are inter-convertible into one another. It is not even what Professor Einstein had envisaged in his 1929 interview with the journal Nature when he said [

“Now, but only now, we know that the force which moves Electrons in their ellipses about the nuclei of the atom is the same force which moves our Earth in its annual course about the Sun, and it is the same force which brings to us the rays of light and heat which make life possible upon this planet.”

Professor Einstein said these words when he was describing his then proposed final theory on distant parallelism [

It strongly appears that we have been able to achieve one thing that Professor Einstein sought in a unified theory, i.e., he envisioned the material field

I have found it very ugly that the field equation should be composed of two logically independent terms which are connected by addition. About justification of such feelings concerning logical simplicity is to difficult to argue. I can not help to feel [sorry] and I am unable to believe that such an ugly thing should be realised in Nature.

If anything, we consider that our ideas of a UFT have here been fully and well formulated. What lies ahead is to determine the precise nature of the theory via a rigours mathematical study against the background of our experience. The theory certainly contains tensor equations and the quantum theory of the Standard Model is a part of it. We could like to close with the words from the great Professor Einstein:

In light of knowledge attained, the happy achievement seems almost a matter of course, and any intelligent student can grasp it without too much trouble. But the years of anxious searching in the dark, with their intense longing, their alterations of confidence and exhaustion and the final emergence into the light—only those who have experienced it can understand it.

In addition to the depth of Professor Einstein’s words which are sure born out of true experience, we must hasten to add that in the pursuit of new knowledge, one ought to have the necessary “iron spine” to put up, and ultimately stand in the way of rebuke and discouragement from seemingly established professors (and eminent personalities), who, on their usual course or discourse, vehemently reject the new in-favour of maintaining the old, for it is about the only thing they have come to know and are thus only prepared to have that of old—and that alone; as the only true and purest wisdom of the World. The foremost duty to all of humanity—i.e., past, present and posterity; of the young, curious and the searching mind, is to vehicle and usher in new ideas procured from the lofty vineyard of the purest and finest of their thoughts. Most at times, this rebuke and discouragement springs from pure prejudice. Whatever the reason or reasons may be, the young, curious, zestful and searching mind must never tire, yield, give-up nor give-in, but continue un-diverted on its course, destiny and path—leading to newer depths; knowing that persistence—sure—yields its rewards, when one refuses to quit!

Assuming the correctness or acceptability of the ideas presented herein, we hereby make the following conclusions:

1) It is our feeling that a unified field theory of all the forces of Nature may very well have been achieved in the present instalment. Despite the plethora of work that lies ahead, we can—in the meantime hope that our readers will see light in what we have presented and that they may take part in the completion of the theory. We are certain that the completion of this theory can not be the effort of a single mind, but that of many minds working together.

2) The electrical and gravitational phenomenon have herein not been unified in a manner as was initially envisaged by a great many researchers e.g. notably Professor Faraday and Einstein; were a reciprocal action and interplay between the two fields has been highly anticipated. No, the two fields here seat harmoniously side-by- side in a quasi-independent manner on the RHS. We should say that, we also expected the final theory to give a reciprocal action and interplay between the electrical and gravitational phenomenon, but alas! Nature has Her own ways; this is not the case here and we can not force it to happen but rather accept what is on the table. If anything, we can only measure the present theory against the background of our experience to see if it has any bearing and correspondence with physical and natural reality as we come to know it.

3) A tensor affine theory that includes all the known force fields of the World has here been achieved on a four-dimensional continuum of spacetime without the need for extra dimensions. In relation to present efforts that employ extra dimension (i.e., string and string-related theories) and stand at the highest echelons of promise as the most promising unified theories, the attainment here of a unified theory which includes all the forces of Nature on a four-dimensional continuum of spacetime, is but a tremendous simplification. If correct, it is sure a great leap forward.

4) A geodesic equation that does not suffer the set back of needing preferred coordinate and reference systems has herein been achieved via the attainment of the tensorial affinities. The resulting geodesic equation has Lorentz terms for both the electrical and gravitational phenomenon. Besides, it has some extra new terms that need to be explored. This may very well explain some of the gravitational anomalies in the Solar system and as-well that of the flat rotation curves of spiral galaxies which has led to the darkmatter hypothesis.

5) Gravitomagnetism which emerges from Professor Einstein’s GTR as a first order approximation emerges in the present via Professor José Hera [

6) Professor Einstein’s gravitational waves which are—to first order approximation; vibrations of the Riemann metric

7) The present theory is free from singularities for as long as

8) All the particles in Nature are predicted by the present theory to have a non-zero mass. This includes the photon and the neutrino—these must have mass, just as is the case with all other particles in Nature. This as well includes all gauge bosons. The idea of massive photons is given attention in the readings [

9) It should be stated clearly that, besides the fact that Professor Proca’s theory of massive gauge fields yields results that are in complete resonance with experience; the fundamental theoretical justification of this modification of Maxwellian Electrodynamics does not exist—actually no clearly visible effort in the available literature has been made to seek the fundamental theoretical justification for the Proca theory. The present effort may very-well be the required justification for this brilliant but seemingly ad hoc modification of Professor Proca. If this is the case that the present theory is surely a fundamental theoretical justification for the Proca theory, then, the present effort would be a significant step forward in the understanding of the fundamental theoretical origins of the Proca theory.

10) As will be demonstrated in a future reading, the present unified theory yields not only the Dirac equation, but the general spin Dirac equations in curved spacetime together with the fundamental theoretical origins of the Dirac equation as a condition for smoothness of the RHS. We consider this to be one of the most immediate and most outstanding achievements of the present theory. This we believe must give credence to the theory that, perhaps, it most certainly contains in it, a grain and an element of truth and is thus worthy of one’s valuable time to consider as a candidate unified theory.

Once again, assuming the correctness or acceptability of the ideas presented herein, we hereby put forward the following recommendations:

1) In-order to obtain the full quantized version of the present UFT, there is need to apply to it [present theory], the already available and well known methods of second quantization. This second quantization introduces the uncertainly principle into the theory. Certainly, this is what is needed for a full quantization of the theory. We are hopeful that this task will be taken up by others.

2) There is need to consider gravitomagnetism as a fully fledge science and not a first approximation of general relativity. We anticipate that such as approach must resolve some of the existing gravitational anomalies such that the secular drift of the Earth-Moon system [

3) There is need to consider the RHS with in mind the task of putting it on a firm much more clear mathematical footing.

4) There is need to consider the geodesic equation to see what is meaning is with regard to electromagnetism, gravitation and quantum mechanics.

5) Finally, we can not at this point say what kind of picture the present UFT projects of the Universe, however, one thing appear clear, its picture seem to has signatures of the present world. There is thus need to put the theory on a much firmer footing by checking all the mathematics here laid down.

We are heftily grateful to the National University of Science & Technology (NUST)’s Research & Innovation Department and Research Board for their unremitting support rendered toward our research endeavours; of particular mention, Prof. Dr. P. Mundy, Dr. P. Makoni, Dr. D. J. Hlatywayo and Prof. Dr. Y. S. Naik’s unwavering support. We are grateful to the insights of Dr. William Straub—his insights have helped refine some portions of the present theory. This reading is dedicated to my mother Setmore Nyambuya and to the memory of my dearly missed departed father Nicholas Nyambuya (04.10.1947-23.09.1999).