Unified Field Theory in a Nutshell

Uniﬁed Field Theory in a Nutshell Elicit Dreams of a Final Theory

G. G. Nyambuya1 1National University of Science and Technology, Faculty of Applied Sciences, Department of Applied Physics, Bulawayo, Republic of Zimbabwe∗ (Dated: July 24, 2014) 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 Herman Weyl [1– 3]’s supposed failed attempt at a unified theory of gravitation and electromagnetism. As is the case with Professor Weyl’s theory, unit vectors in the resulting/proposed theory vary from one point to the next, albeit, in a manner such that they are compelled to yield tensorial affinities. In a separate reading [4], 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 Aµ, 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 four dimensional continuum of spacetime without the need for extra-dimensions.

PACS numbers: 04.60.-m, 95.30.Sf, 11.15.-q, 04.20.Cv

“Imagination will often carry us to worlds that never were. But without it, we go nowhere[5].” – Carl Edward Sagan (1934 − 1996)

INTRODUCTION nouncement of a Higgs-like particle at CERN on July 12, 2012. UCH effort by a great many erudite, notable One of the first major problems beguiling the efforts to M and foremost physicists and mathematicians has finding a unified theory is that each of the two theories gone into the all-noble and all-esoteric search for an all- in question (i.e. QT and the GTR) are considered by a encompassing unified theory of all the forces of Nature; as great many physicists as fundamental physical theories for our own attempts, the present reading marks the third of physics in their own right. By “fundamental physi- such effort. The beguiling problem of how to unify Quan- cal theory” it is understood that these theories are not tum Theory (QT) and Professor Albert Einstein’s Gen- derivable from any other physical theory, they are – in eral Theory of Relativity (GTR) into a consistent and co- Einstein’s terminology, Theories of Principle[65]. Ac- herent unified theoretical framework has not only baffled cepting these theories as fundamental physical theories but eluded the foremost physicists and mathematicians of lands us in torrid philosophical and logical conundrums the past eight decades or so [e.g. Professors – Albert Ein- as this invariably implies the existence of two indepen- stein (1879−1955), Herman Weyl (1885−1955), Theodor dent physical realities – one reality of the very small (QT) Kaluza (1885 − 1954), Erwin Schrödinger(1887 − 1961), and the other of the large (GTR), each governed by sepa- Edward Witten (1951−), Stephen Hawking (1942−), etc; rate Physical Laws. The strong and existing feeling that to mention but a few pre-eminent figures in this all-great there can only be one and only one reality points to one endeavour], and with the failure in recent times to deliver and only one fundamental physical theory, hence, a union its now long overdue promise i.e., failure by the so- fied theory of all of reality is thus needed. Surely, the called most promising theories on this front (e.g. String garment of physical and natural reality must constitute and String Related Theories), the need for a unified the- an undivided whole and it thus obviously makes no sense ory of all he forces of Nature has become particularly to have two distinct fundamental theories for a single urgent and pressing, especially given the most recent an- united garment of reality. 2

How does it come about that physicists have for so Because it has happened and we have experienced it, ev- long been able to succeed in failing to find a unified the- eryone will concur that no amount of applied research ory? Amongst others, we feel – as Dr. Philp Gibbs[66], on the candle (or the wax the making it thereof) would that physicists have held far-too-sacrosanct the two pil- have produced the light bulb – to attain the light bulb, lars modern physics – QT and the GTR; and have in the human mind had to leap-out of its ordinary state of the process, gone so far as to confidently believe that conscientiousness. The above words imply that for us to naturally, a logical fusion of the two ideas must lead to take a leap out of logic and into virgin territory will – at paths that lead straight to the depths of an acceptable the very least – this requires of us to “ . . . imagine events theory that must take humankind to his next level of and conditions of existence that are contrary to the causal understanding of physical and natural reality as we ex- principle . . . ” as we have come to know and experience it. perience it. In his own words about this line of thinking, This means that logic may very well not lead to a the- Dr. Philip Gibbs had this to say[67]: ory with a direct correspondence with experience. For example, no amount of logical reasoning in Newtonian “String theory has had many opportunities to reveal itself as a ‘bridge to nowhere’ but at gravitation would have led mankind into Professor Ein- each turn the road carries on instead and another stein’s brilliant geometric description of gravitation and bridge is crossed. The theory is very tightly con- spacetime, one had to “defy” logic and enter into much strained by the need to be consistent with general higher planes, terrains and farrows of logic and natural relativity and quantum mechanics. The theorists reality. are not making it up. They are [only] following In this reading – given its length, we find no space to the course that logical consistency dictates and it is remarkable that there is anything at all that give a wide literature review or an overview of the history can match the requirements, but of course there of unified field theories. For this, we direct our reader to has to be because the Universe exists with these the excellent review by Emeritus Professor of Physics at features.” Germany’s GöttingenUniversity, Hubert F. M. Goenner; he certainly has done a splendid job on that, he has given What the above words remind us are the wise words by an extensively review of the history of unified theories the great British-German physicist, Professor Max Born from 1918 to 1965 [see Refs. 6, 7]. Therefore, we shall (1882 − 1970). He once said: proceed as planned without giving a historic overview; “Science is not [mere] formal logic. It needs we shall give a brief synopsis of the present reading. the free play of the mind in as great a degree as In §(Motivation), we outline two fundamental rea- any other creative art. It is true that this is a gift sons that call for a revision of Professor Einstein’s GTR. which can hardly be taught, but its growth can be In §(New Weyl Geometry), we give an overview of the encouraged in those who already posses it.” reading [8], were Professor Weyl’s [1] supposedly failed Simple stated, like Dr. Gibbs has said it correctly: the theory is brought back to life. In §(Decomposition present day theorists are trying to construct a new the- of the Metric Tensor), we demonstrate an impor- ory by believing (wrongly or rightly so[68]) that a theory tant part of the unified theory to be developed, namely logically derived from the general theory of relativity and that the metric tensor should be decomposed in to an quantum mechanics will naturally lead to a logically cor- entity that has not ten free parameters, but four. In rect theory which unifies both theories. The folly (per- §(Riemann-Hilbert Space), the spacetime upon with haps) in this approach or way of reasoning is that we are the present theory is build is laid down. In §(Tensorial forgetting the wise words of the great Professor Max Born Affinities), tensorial affinites are proposed in which that “. . . science is not [mere] formal logic . . . it needs the even the desired spacetime is defined. Once the de- free play of the mind in as great a degree as any other sired spacetime is defined, in §(General Field Equa- creative art . . . ”. Logic alone is not enough, nor is intu- tions), the general field equations are written down and ition, we need all these “forces” at play together with a in §(Linear Riemann-Hilbert Space), it is shown that modicum but strong dose of imagination which Einstein as a result of the tensorial nature of the affinities, it said “ . . . is more important than knowledge . . . ”. We is possible to get rid of the non-linear terms of the re- need a leap out of logic to make it into the next virgin sulting curvature tensor, thereby making the theory a territory. Perhaps we must remind the reader of other linear theory. From this linear spacetime with tenso- wise words by one of Germany’s greatest scientists, Pro- rial affinities, in §(Resultant Field Equations), we fessor Karl Ernst Ludwig Marx Planck (1858 − 1947), write down the resulting field equation. Having laid who in his lifetime once said that: down the theory, in §(Stocktaking), we take stock of what has been achieved thus far. In §(Gravitation “The man who cannot occasionally imagine events and conditions of existence that are con- and Gravitomagnetism), we show how one can bring trary to the causal principle as he knows it will the gravitational force in to the fold of the proposed never enrich his science by the addition of a new unified theory. In §(New Geodesic Equation) we idea.” write down an appropriate geodesic equation that em- 3 ploys tensorial affinities. In §(Global Symmetries and accept contrary to the Principle of Relativity that there Yang-Mills Force Fields), we demonstrate that the exists in Nature preferred reference and coordinate sys- proposed unified theory does contain Yang-Mills The- tems because the above Geodesic Law leads us to formu- ory and finally in §(Discussion), (Conclusion) and lating the equations of motion in preferred reference and (Recommendation), we give a general discussion, the coordinate systems, namely, geodesic coordinate systems conclusion drawn thereof and the recommendations for also know as Gaussian coordinate systems. future works. 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 MOTIVATION tells us how matter curves spacetime, i.e.:

Other than the shear and insatiable human desire to be part of such a noble endeavour to finding an all encom- 1 8πG R − Rg = T + Λg , (2) passing unified theory of the forces of Nature; if any, what µν 2 µν c4 µν µν are the real and tangible reasons requiring us to revise Professor Einstein’s GTR? As to ourself – and as may where Rµν is the contracted Riemann curvature tensor be the case with the majority of physicists; we find two and Tµν = ρvµvν + pgµν is the stress and energy tensor compelling reasons. The first has to do with the clearly where ρ is the density of matter, p is the pressure and vµ implied need for the unity of the GTR and QM in the the four velocity, G is Newtons universal constant of grav- simultaneous case of extremely massive and extremely itation, c the speed of light and Λ is the controversial and small objects. The second has to do with a well known so-called cosmological constant term ad hoc-ly added by and largely ignored internal inconsistency of the GTR. Einstein so as to stop the Universe from expanding (Ein- On the first note, QM describes matter at the smallest stein 1917). Einstein was motivated to include the cos- scale-length as exhibiting both point-particle and wave- mological constant because of the strong influence from like properties. The GTR describes at the largest scale- the astronomical wisdom of his day that the Universe length the same matter as mere point-particles. There appeared to be static and thus was assumed to be so. is no wave-like description of matter in the GTR, none Besides this, the cosmological constant fulfilled Mach’s whatsoever. However, after exhausting their nuclear fuel Principle (Mach 1893), a principle that had inspired Ein- which is believed to hold them against the inward tyranny stein to search for the GTR and he thus thought that the of the gravitational force, massive luminous objects such GTR will have this naturally embedded in it – to his dis- as stars – whose macroscopic properties are described satisfaction, the GTR did not exactly fullfil this in the very well by the GTR; these objects can undergo col- manner Einstein had envisaged. Mach’s principle forbids lapse in which case their spatial size can go down to the existence of a truly empty space and at the sametime scale-lengths that require us to use QM to describe them. supposes that the inertia of an object is due to the induc- Under such conditions, it becomes inescapable that one tion effect(s) of the totality of all-matter in the Universe. would need a Quantum Theory of Gravity (QTG) in- Because the field equation (2) is a tensor, it does not order to describe the physics in-and-around such objects. suffer the same problem suffered by the geodesic equa- For this reason, the need for a QTG is not only clearly tion (1). Unlike (1), the field equation (2) needs not be evident, but is a dire need and necessity. formulated exclusively in Gaussian coordinate systems. On the second note – in our view, one of the major Gaussian coordinate systems are those coordinate sys- problems that the GTR faces within its own internal tems such that gµν,σ = 0. It can be shown for example structure of logic is that it is based on pure Riemann that given a flat space-time in which say the rectangu- geometry i.e. a geometry that is well known to violate lar coordinate system (where gµν,σ = 0 holds) are used the Principle of Equivalence at the affine level because to begin with; where [in the rectangular coordinate sys- the affine connections are not tensors. If pure Rieman- tem] the affinities vanish identically in this system and nian geometry is to be the true geometry to describe the changing the coordinate system to spherical, the affini- natural World, then, no Laws of Physics should exist at ties do not vanish. This is a serious desideratum, akin the affine level of Riemann geometry. However, this is to the Newton-Maxwell conundrum prior to Professor not so, since the Geodesic Law: Einstein’s STR i.e., a conundrum of how to reconcile or comprehend the apparent contradiction of the predic- tion of Professor Maxwell’s theory that demanded that d2xλ dxµ dxν λ the speed of light be a universal and absolute speed and 2 − Γµν = 0, (1) ds ds ds the Galilean philosophy of relativity that there is no such that describes the path and motion of particles in space- thing as a universal and absolute speed in the Universe. time emerges at the affine level. Thus accepting Riemann Given for example, that the affinities represent forces geometry as a true geometry of Nature means we must as is the case in the GTR, this means a particle could be 4 made to pass from existence into non-existence (or vis- versa) by simply changing the coordinate system. This g 7−→ eχg (a) on its own violates the Laws of Logic and the need for µν µν , (4) Nature to preserve Her independent reality devoid of A 7−→ A + ∂ χ (b) magic. Clearly, the only way out of this conundrum is µ µ µ to seek, as Professor Einstein, Professor Schrödinger etc where χ = χ(x) is some arbitrary well behaved, smooth, have done; a theory in which the affinities have a tensor continuous and differentiable function. Given that Pro- form, hence in the present approach, the first and most fessor Weyl knew very well that Maxwellian electrody- important guide is to seek tensorial affinities. Einstein, namics is described by a four vector such that the entire Schrödinger etc have made attempts along these lines Maxwellian electrodynamics is invariant under the trans- only to fail. The reason for their failure may perhaps formation (4 b), without wasting much time, the great stem from the fact that theirs was a pure mathematical Professor Weyl seized the golden moment and identified exercise to try to find a set of tensorial affinities from Aµ with the Maxwellian four vector potential of electro- within the framework of the classical spacetime of Rie- dynamics. In his unified theory, Professor Weyl’s quest mannian geometry. It should be said that their failure was based on the subtle idea to treat directions in space- [Professor Einstein, Professor Schrödinger etc] has not time on an equal footing with length [see e.g., 9], that is been a total failure as their (failed) work lead to signifi- to say, if the angle of the unit vector changed from one cant advances and better insights into the nature of the point to the other under parallel transport, why should problem at hand. In-fact, the present work does springs the length not change too? asked Professor Weyl. Ulti- from these so-called failed attempts. Our line of thought mately, this subtle idea bought in electromagnetism into stem directly from their work. the same fold as gravitation. Professor Weyl’s proposed theory led to a rescaling of the fundamental metric tensor via the gauge trans- χ NEW WEYL GEOMETRY formation, gµν 7−→ e gµν . Professor Weyl held that, this rescaling should have no effect on physics – he was As already stated, the present UFT is built on a modified wrong. Professor Einstein initially loved the idea – alas, the devil was in the detail; he noted that the line ele- theory of Professor Weyl [1]’s supposedly failed theory of 2 µ ν ment ds = gµν dx dx would also be rescaled according 1918. This modification is presented in [8]. In the present 2 χ 2 section, we merely give a succinct summary of this work to ds 7−→ e ds . Since ds can be made to serve as a [8]. In [8], we rail against the common approach to find- measuring rod or clock, the agile Professor Einstein was ing a unified theory; we revisit the now forgotten ‘litter’ quick to note that this would mean that certain absolute in the dustbins of physics and mathematics history where quantities, such as the spacing of atomic spectral lines we make the modest endeavour to resuscitate Professor and the Compton wavelength of an Electron for exam- Herman [1–3]’s supposed failed attempted at a unified ple, would change arbitrarily and thus have to depend on theory of gravitation and electromagnetism. their prehistory. With this, Professor Einstein delivered the lethal and venomous blow to Professor Weyl’s the- In his attempt at a unified theory, Professor Weyl re- ory and concluded that it must therefore be non-physical alised that if he supplemented the Christophel three sym- – despite its grandeur and beauty, it had no correspon- bols of Riemann geometry i.e. Γλ , with a tensorial affine µν dence nor bearing with physical and natural reality as we W λ , he obtained a theory with a four vector field (A ) µν µ have come to experience. with the desired gauge transformational properties of the As aforestarted – in the reading [8], we put forward a electromagnetic four vector field. The Weyl tensorial con- New Weyl Geometry (NWG) – a geometry upon which nection W λ is given by: µν the present theory is founded. This NWG changes the foundation stone of the Weyl geometry. The metric of the

λ λ λ λ original Weyl geometry is only conformal at the instance Wµν = δµAν + δν Aµ − gµν A , (3) of a gauge transformation – i.e., the term eχ multiplies the metric only when performing a gauge transformation. µ where gµν is the metric tensor and δ ν is the Kronecker- In the NWG [8], the metric is intrinsically and inherently Delta function. The resultant affine connection, Γ¯λ , µν conformal. That is to say, if we letg ¯µν be the metric of of Weyl’s geometry is the sum of the Christophel three χ the NWG, then,g ¯µν = %gµν where % = e is the confor- λ λ symbol Γµν and the Weyl tensorial affine Wµν , i.e. mal term. In the NWG, the conformal term now plays a ¯λ λ λ Γµν = Γµν − Wµν . The transformational properties decisive role in that it is responsible for the attainment λ ¯λ of Γµν and Γµν are the same, thus insofar as this aspect of tensorial affinities. This NWG is completely free of of the affines are concerned, the two are equivalent. The Professor Einstein’s lethal and venomous criticism. ¯λ affine connection Γµν is splendidly invariant under the The NWG is exactly the same geometry is one two following gauge transformations: all-important differences. The first of which is that 5 the we replace the metric of Riemann geometry gµν spacetime version of the Dirac equations presented in the withg ˜µν ; the metric is now intrinsically and inherently reading [16] stands-out over other attempts in that the conformal. The second is that this NWG employs method used in arriving at these curved spacetime Dirac tensorial affinities and this brings about a fundamental equations [16] is exactly the same as that used by Profes- difference in that one obtains from this a geometric such sor [17, 18]. As will be demonstrated shortly, this method that vectors maintain not only their length but their used in [16] appears to us as the most straight forward angle as-well upon parallel transport. In §(Tensorial and logical manner in which to arrive a curved space- Affinities), we will provide the details of how in time version of the Dirac equation. All that has been the NWG one obtains tensorial affinities from the done in [16] is to decompose the general metric gµν in a conformal terms of the metric (¯gµν = %gµν ). In the manner that allows us to apply Professor Dirac [17, 18]’s next section, we will present an important idea that prescription at arriving at the Dirac equation. is vital to the present unification, namely that the Riemann metric gµν can be decomposed in such a manner that is now contains four and not ten free parameters. Professor Dirac [17, 18]’s original equation is ar- rived at from the Einstein momentum-energy equation µ ν 2 4 ηµν p p = m0c where ηµν is the usual Minkowski µ metric, (p , m0) are the four momentum and rest mass DECOMPOSITION OF THE METRIC TENSOR of the particle in question respectively and c is the usual speed of light in a vacuum. In curved spacetime, we know µ ν 2 4 In the present section we justify one of the major ideas very well that the equation ηµν p p = m0c is given by µ ν 2 4 of the present unified theory. This is the idea that the gµν p p = m0c where gµν is the general metric of a metric tensor can – logically, be decomposed in a manner curved spacetime. If a curved spacetime version of the that allows us to write down an equivalent Dirac equation Dirac equation is to be derived, it must be derived from µ ν 2 4 in curved spacetime. The resulting decomposed metric the fundamental equation gµν p p = m0c in the same is central to the final unified theory, it is so central that way the flat spacetime Dirac equation is derived from the µ ν 2 4 without it, one can not envisage obtaining the required fundamental equation ηµν p p = m0c . Professor Dirac correspondence of the present theory with experience. derived his equation by taking the ‘square-root’ of the µ ν 2 4 That said, there are several curved spacetime versions equation ηµν p p = m0c . It is a fundamental mathe- of the Dirac equation [see e.g., 2, 3, 10–15]. In our mod- matical fact that a two rank tensor (such as the metric est view; save for the introduction of a seemingly mys- tensor gµν ) can be written as a sum of the product of a terious four vector potential Aµ, what makes the curved vector Aµ, i.e.:

1 n o 1 n o g(a) = A γ(a),A γ(a) = γ(a), γ(a) A A = σ(a)A A , (5) µν 2 µ µ ν ν 2 µ ν µ ν µν µ ν

(a) where σµν are 4 × 4 matrices such that (a) 1 n (a) (a)o σµν = 2 γµ , γν and γ-matrices[69] are de-  1, then (λ = 0) : Flat Spacetime. ﬁnes such that:  a = 2, then (λ = +1) : Positively Curved Spacetime.  3, then (λ = −1) : Negatively Curved Spacetime. (7)   I2 0 (a) The index “a” is not an active index as are the Greek in- γ0 =   , dices. This index labels a particular curvature of space- 0 −I2 time i.e. whether spacetime is ﬂat[70], positive or neg- √ (6) atively curved. Written in full, the three metric tensors  λ 2 k  2λI2 i 1 + λ σ (1) (2) (3) gµν , gµν and gµν are given by: γ(a) = 1 , k 2  √  λ 2 k −i 1 + λ σ −2λI2   A0A0 0 0 0 k where I2 is the 2×2 identity matrix, σ is the usual 2×2 h (1)i  0 −A1A1 0 0  gµν =   , (8) Pauli matrices and the 0’s are 2 × 2 null matrices and  0 0 −A2A2 0  a = (1, 2, 3) such that for: 0 0 0 −A3A3 6

0 0 k k where now ∂(s) = ∂ and ∂(s) = s∂ : where k = (1, 2, 3) and s = 0, ±1, ±2, ±3, . . . etc [19, 20]. In equa-  A A A A A A A A  0 0 0 1 0 2 0 3 tion (12), the vector A has completely disappeared from h i A A −A A A A A A µ g(2) =  1 0 1 1 1 2 1 3  , (9) our midst thus drastically simplifying the resultant equa- µν  A A A A −A A A A   2 0 2 1 2 2 2 3  tion in the process. A A A A A A −A A 3 0 3 1 3 2 3 3 What we have done in the present section is to demonstrate that the metric tensor is susceptible to decompo- and the metric g(3) is related to g(2) by the transforma- µν µν sition into a tensor field describable by four unique fields tion Aµ 7−→ iAµ, that is to say, we have to replace Aµ (3) that form a relativistic four vector. In the GTR, the with iAµ in (9) to obtain gµν , i.e.: 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 −A0A0 −A0A1 −A0A2 −A0A3 unified field theory that we hope to build. Despite the h (3)i  −A1A0 A1A1 −A1A2 −A1A3  gµν =   . (10) obvious similarities, the decomposition of the metric that  −A2A0 −A2A1 A2A2 −A2A3  we employ in the proposed unified theory is different from −A3A0 −A3A1 −A3A2 A3A3, the decomposition carried out in the present section. 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 RIEMANN-HILBERT SPACE 1 n (a) (a)o the writing of gµν as gµν = Aµγµ ,Aν γν is to be 2 The proposed all-encompassing Unified Field Theory accepted as a legitimate mathematical fact for as long (UFT) of all the forces of Nature is built on a space- as g is a tensor. Since A is a vector and the γ- µν µ time that we have coined the Riemann-Hilbert Space- matrices are all constant matrices, g is a tensor. There- µν time (RHS). Hereafter, when-ever we refer to the UFT, fore, it follows that the equation g pµpν = m2c4 can µν 0 we mean the UFT presented herein. In the sections that 1 n (a) (a)o µ ν 2 4 now be written as 2 Aµγµ ,Aν γν p p = m0c . As follow, we shall succinctly describe the major highlights clearly demonstrated in [16], if we are to have the equa- of RHS and in so doing, we are in actual fact describ- µ ν 2 4 tion gµν p p = m0c written in the decomposed form ing the UFT itself. This spacetime has been dumbed the 1 n (a) (a)o µ ν 2 4 RHS because it is a hybrid Riemann space which has in it Aµγµ ,Aν γν p p = m c , and one where to fol- 2 0 embedded Hilbert-objects. By Hilbert-objects, we mean low Professor [17, 18]’s original derivation method, they the quantum mechanical wavefunction. The RHS is a will arrive at the three curved spacetime Dirac equations, modified Weyl space which Professor Einstein ruthlessly namely: short-down when Professor Weyl tried to put forward his unified theory of gravitation and electromagnetism. h (a) µ i i~Aµγµ ∂ − m0c ψ = 0. (11) Line Element and Position Vector as 4 × 4 Objects It is not a difficult exercise to show that multiplication of (11) from the left hand-side by the conjugate operator † In the RHS, the entire structure of Riemann geometry is h µ µ† i i~A γ(a)∂µ − m0c leads us to the Klein-Gordon equa- inherited in its complete form and is modified by replac- µ ν 2 2 µ µ tion gµν ∂ ∂ ψ = (m0c /~) ψ provided ∂µA = ∂ Aµ = ing the metric with a new somewhat conformal metric µ µ 0. The condition ∂µA = ∂ Aµ = 0, should be taken as g¯µν which is such thatg ¯µν = %gµν , where % is a 4 × 4 a gauge condition restricting this four vector. matrix and the line element is such that: As it stands, equation (11) would be a horrible equation insofar as its solutions are concerned because the 2 µ ν ds =g ¯µν dx dx . (13) vector Aµ is expected to be a function of space and time (a) (a) (a) i.e. A = A (r, t). Other than a numerical solution, µ µ where xµ is the position of a particle on this spacetime there is no foreseeable way to obtain an exact solution (a) and unlike in the ordinary space we are used to, this is if that is the case. However, we found a way round µ the problem; we fortunately realised that this vector can position vector (i.e., x(a)) is a 4 × 4 matrix, that is to µ µ µ actually be used to arrive at a general spin Dirac equa- say, x(a) = γ(a)x and the metric gµν is such that µ tion thereby drastically simplifying the equation so that gµν = AµAν while x is the usual zero rank position it now is given by: vector that we are used to. The vector Aµ is a 4 × 4 objects. This vector Aµ is assumed to be:

h (a) µ i i~γµ ∂(s) − m0c ψ = 0, (12) (1). Comprised of real elements. 7

† (à) (2). Hermitian i.e. Aµ = Aµ. The position vector Xµ of a point on this spacetime (à) (à) 2 continuum is such that Xµ = xµeµ , so that ds¯ = (à) (à)µ (`) (`) Sixteen Representations of the Metric Xµ X . Because ψ = ψ(r, t) and Aµ = Aµ (r, t), (à) it is clear that eµ can not have a fixed value, hence the RHS has variable unit vectors. The direction and Now, we will add an extra index-` to the metric gµν . We † † magnitude of these unit vectors on the RHS depend on will do this by noticing that gµν = AµAν = AµI4Aν . µ We know that we can always find a set of sixteen their position x . This is a property that is unlike any 4 × 4 hermitian matricesγ ˜` where (` = 0, 1, 2, 3,... 15), other geometry except the so-called failed Weyl geometry † ` [1–3]. The criticism of the great Professor Einstein on such thatγ ˜ γ˜ = I4. Given this, it follows that the so-called failed Weyl geometry does not apply to the g = (˜γÀ )†(˜γÀ ). If we write A(`) =γ ˜À , then, µν µ ν µ ν present RHS. ` † ` (`) (`) (`) we can write gµν = (˜γ Aµ) (˜γ Aν ) as gµν = Aµ Aν . We must hasten to say that, whosoever seeks to dis- Therefore, from here-on, the metric shall be written with credit and bring down the present theory on the same the index-` and this index dose not label components of basis as the great Professor Einstein did with Professor the metric tensor but a representation of the of it. The Weyl’s theory on the grounds that the line element ds¯2 sixteenγ ˜-matrices are given in the appendix. is not invariant under a transformation of the coordi- (`) (`) Notice that we have placed the index-` in Aµ and gµν nate or frame of reference, hence, certain absolute quan- in brackets. The reason for this is that the µν-indices la- tities, such as the spacing of atomic spectral lines and the (`) (`) bel the components of the tensors Aµ and gµν while ` la- Compton wavelength of an Electron for example, would bels representations of these same tensors. We have done change arbitrarily and thus have to depend on their pre- this to distinguish components of the tensor from their history; then, the rebuttal to this line of thought is that representations. This notation we shall use throughout the proper time of any physical system is here not given this reading from here-on. There reader must take note by ds¯2, but by ds2. One must without fail remember that of this. ds¯2 = ρds2; ds2 is an invariant physical quantity while Now, when finally we derive the Dirac equation in the ds¯2 is not. Therefore, Professor Einstein’s agile criticism, separate reading [4], it will be seen that the different rep- while valid, it does not hold ‘any water’ nor ground in (`) resentations of the metric gµν lead to Universes that have the present scheme of things. different charge conjugation, parity and time reversal symmetries. Unlike in the Dirac theory where all these symmetries are upheld, some of the `-representations vi- Some Remarks olate these symmetries. This presents us with a sce- (`) nario and golden opportunity to possible explain why An important point to note is that the metric gµν and not (`) our present Universe seems to have a preponderance of g¯µν plays on the RHS the same role it plays on Riemann matter over antimatter. space i.e. the role of lowering and raising of indices. This, the reader must take note of, because it is what we shall (`)α use. Further, unlike on the Riemann space, g α ≡ 1, Unit Vector (`)α and not, g α ≡ 4. As is demonstrated in [4], the field %, is actually Any space or spacetime must be endowed with unit vec- the Dirac probability density function and is such that tors at each and every point of the continuum. The unit † (à) % = ψ ψ where ψ is the Dirac four component wave- vectors on the RHS are variable, that is to say, if eµ is function. So, the field % represent the material aspect of the unit vector, then: (`) (`) matter while the gµν via the four vector field Aµ , represent the force field aspect of matter. It is the object, (à) ` (a) %, that is the embedded Hilbert object we refereed to in eµ = Aµγµ ψ, (14) the opening paragraph of the present section. where ψ = ψ(r, t) is a set of 4 × 4 matrices i.e.: Now, in the next section i.e. in §(Tensorial Affini- ties), we will move to demonstrate that one can at-   tain tensorial affiens. The attainment of tensorial affini- ψ00 ψ01 ψ02 ψ03 ties will allow us in §(General Field Equations) to  ψ10 ψ11 ψ12 ψ13  ψ =   . (15) write down a general field equation for the RHS. In the  ψ20 ψ21 ψ22 ψ23  §(Linear Riemann-Hilbert Space), we use the ten- ψ30 ψ31 ψ32 ψ33 sorial nature of the affinities to get reed of the non- In the reading [4], it is shown how one can naturally liner aspect of the RHS. Having obtained non-linear field harness from (14) the curved spacetime Dirac equations equations on the RHS, we show in §(Resultant Field presented in [16]. Equations) that these field equations lead us to a fa- 8 miliar territory of Yang-Mills (YM) field equations and Thus – at any rate, Qµ can not be a vector. For this ob- conservation laws. The resulting field equations as they ject (Qµ) to transform as required in (19), the conformal stand appear at face-value to have the gravitational force term % must transform as follows: % 7−→ eφ%, such that: outside of the whole scheme, leading to the feeling or no- tion that the present UFT is only a unified theory of the α ∂x α nuclear forces. We argue via a gravitomagnetic approach φ = = δ 0 . (20) ∂xα0 α in §(Gravitation and Gravitomagnetism) that the gravitational force is very much part of the whole scheme. In this way – and this way alone; we attain the long sort Further, in §(New Geodesic Equation), we show that tensorial affinities. So, the object % is there so that we one can derive the familiar geodesic equation of motion attain this desire for tensorial affinities. with the Lorentz-force term for both the electrical and It should be noted that nothing from within the RHS gravitational forces. requires that the affinities be tensors. This is a require- ment imposed from outside by us because the RHS allows us to make such a choice without violating any of TENSORIAL AFFINITIES its marvellous structural properties. This reminds us of the words of Professor Einstein when he said that when If we replace the metric, gµν , in Riemann geometry with constructing a theory, one must put themselves in the (`) (`) (`) a new metric,g ¯µν , which is such that,g ¯µν = %gµν , and position of God[71] and ask themselves the deepest and (`) maintain that,g ¯µν;α ≡ 0, so that Professor Einstein’s most enduring questions about physical and natural re- criticism of Professor Weyl’s theory is invalidated, then ality, such as: ¯(`)λ as shown in [21], the resulting affine connection, Γ µν , If I were God; (`)λ (`)λ (`)λ is such that, Γ¯ µν = Γ µν − W µν , where: how would I have created the Universe? Did God have a choice in creating the Universe?

(`)λ α α (`) α W µν = δµ Qν + δν Qµ − gµν Q . (16) To these words of Professor Einstein, we add the following words: In this new affine (16), the object, Qµ, is no longer a vector, it is such that, Qµ = ∂µ ln %. Since Qµ, is no If He (God) did have a choice, longer a vector, the first and most important of all the what is that choice and; improvements to be made on the RHS is that the role why that choice and not any other? of the conformal object % now takes a new decisive and pivotal role. It is now required of it that the Weyl affine Our pedestrian feeling on the issue of the Mind of God, (`)λ connection, W µν , must forcefully be constrained such His thoughts and choices is that all the constraints – more that, at the end of the day – when all is said and done, so the gauge conditions that we find necessary for our and for all conditions of existence, the affine connection, theories; the constraints that we impose on fundamen- (`)α Γ¯ µν , is, for better or worse, a tensor! Given that the tal physical theories, these are most certainly God’s sub- (`)α tle thoughts which He employed in making the Universe. three Christophel symbol, Γ µν , naturally transforms as: 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 Cre- α0 µ ν α0 2 α (`)α0 ∂x ∂x ∂x (`)α ∂x ∂ x ator Himself? This once again reminds us of the other of Γ 0 0 = Γ + , (17) µ ν ∂xα ∂xµ0 ∂xν0 µν ∂xα ∂xµ0 ∂xν0 Professor Einstein’s wise words: α it follows that if, Wµν , were to transform as the Christof- “I want to know how God created this World. fel symbol, i.e.: I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts; α0 µ ν α0 2 α (`)α0 ∂x ∂x ∂x (`)α ∂x ∂ x the rest are details.” W 0 0 = W + , µ ν ∂xα ∂xµ0 ∂xν0 µν ∂xα ∂xµ0 ∂xν0 (18) Perhaps, the Mind of God insofar as the architecture, de- ¯α the affine, Γµν , will be a tensor as per desideratum. If the sign and building of the World is concerned, this might α Weyl connection, Wµν , is to transform as demanded in very well be hidden and revealed in the Universal World (18), then – invariably; the object, Qµ, must transform Geometry upon which the Universe is created and His as: thoughts in doing so (i.e., creating the World), in the gauge conditions and exogenous impositions on this Uni- ∂xµ ∂2xα versal World Geometry; i.e., impositions that allow us Q 0 = Q + . (19) µ ∂xµ0 µ ∂xµ0 ∂xα0 to obtain exactly the conditions and laws of the present 9

World. That is to say, are not these gauge conditions a then, % will have the desired transformational properties sacrosanct signature of God’s sleight of hand and mind such that % 7−→ eφ%, where φ is given by (20) and the ob- in creating a Universe of His choice? We leave the reader ject Qµ transforms as given in (19). We shall forthwith to ponder and ex-cogitate on this and the other deeper and hereafter take % to be given by (22). What (22) really questions here raised above about physical and natural means is that the wave-function has here been expressed reality. as a function of the variables of the geometry. It was one Now, back to the real World, the palatable World of of Professor Einstein’s intuitive requirements for any uni- ponderable facts. In the present section, we have demon- fied theory that the wave-function ought to emerge as a strated that one can achieve on the RHS tensorial affini- part of the geometrical description of spacetime. In hav- ties. In the next section, we shall now use this to de- ing the wave-function as geometrical description, Profes- duce the major field equation for matter and energy of sor Einstein hoped to reed QM of its seemingly inherent the RHS. From this equation, we are hopeful that all probabilistic nature. of reality – insofar and fundamental physical forces are Be that it may, we – at any rate, do not believe the concerned; must be harnessed forthwith. Before that, we geometrical description of ψ given in (22) will reed QM of need to do two thing, the first is ask, what kind of func- its seemingly intrinsic and inherent probabilistic nature. tion % will lead to the desired transformation % 7−→ eφ% To address this or to demonstrate the probabilistic nature and second, what gauge conditions if any can we deduce of QM, this requires on its own, a fresh new monograph (`α) about the field Fµν from our experience with Profes- – thus, we shall not go any deeper than we have done sor Maxwell’s electrodynamics? We will start with the here. Further, because gµν is a 4 × 4 object, it follows latter. that % is also a 4 × 4 object too. If % is a 4 × 4 object, it Professor Maxwell’s electrodynamic field tensor has follows that, ψ can not be a 4 × 1 object as is the case in ν the property of being traceless i.e. F ν = 0. This prop- the Dirac wave-function; it has to be a 4×4 array just as (`α)µ is the case with g . Actually, it will become much more erty we will carry over to the present field tensor F ν , µν as we realise that we need this condition in-order to ar- clear in §(Four Dimensional Yang-Mills Theory) rive at the desired equation, that is, we shall assume the that the wavefunction ψ will have to be a 4 × 4 object (`α)µ if we are to attain a four dimensional Yang-Mills theory following as a gauge condition for F ν : on the RHS.

(`α)ν F ν = 0. (21) GENERAL FIELD EQUATIONS This gauge condition will lead is to the another gauge condition that we will present in (47). Riemann geometry is built on the idea of parallel trans- Other than the traceless property, Professor Maxwell’s port of vectors along a given path. A good description field tensor also has the anti-symmetry property of parallel transport is perhaps that by Professor John Fµν = − Fνµ. We shall not be generalizing this to (`α) Baez[72] . Following him [i.e. Professor John Baez]; say the present field tensor Fµν , as so doing requires that one starts at the north pole holding a javelin that points (`) (`) (`) (`) Aµ ∂αAν ≡ −Aν ∂αAµ , and this leads to physically horizontally in some direction, and they carry this javelin meaningless equations as this requires that the vector to the equator always keeping the javelin pointing ‘in as (`) Aµ , be a constant always. Clearly, having the vector same a direction as possible’, subject to the constraint (`) Aµ as a constant vector is not only meaningless, but that it points horizontally, i.e., tangent to the Earth. In rank nonsense for a UFT as the present as this will lead this scenario, we see that the idea is that we are taking to the trivial equation 0 = 0. ‘space’ to be the 2-dimensional surface of the Earth and Now, if the function % will lead to the desired transfor- the javelin is the ‘little arrow’ or ‘tangent vector’, which mation of the affinities, then, we have to ask the question, must remain tangent to ‘space’. After marching down to ‘What must % be a function of in-order to have the de- the equator and making a 90◦ turn at the equator and sired properties?’ Latter, the function % is expected to be then marching along the equation until some-point along identified with the wave-function of the particle in ques- the equator where another 90◦ turn toward the north tion. If this function can be expressed in-terms of the pole is made thus marching back up to the north pole, field functions or variables, then, it would fit very well always keeping the javelin pointing horizontally and ‘in into the philosophy of Occam’s razor of having the min- as same a direction as possible’. “Obviously”, because imal possible variables in a theory. It is not difficult to the surface of the Earth is curved, by the time one gets work it out that if: 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 Z xµ(t) ! 1 µ (`) ν end point. % = exp ∂ gµν dx , (22) µ 2 x (t∗) Parallel transport is an operation that takes a tangent 10

respectively]. As shown in Figure (1), if say we have a vector vλ and we parallel transport it along a closed circuit ABCD in the order A 7−→ B then B 7−→ C then C 7−→ D and then ﬁnally D 7−→ A, if the space in question has a non-zero curvature, upon arrival at its original location, the vector is no-longer equal to the original vector. The changes of this vector along these paths are:

λ λ ν µ dvAB = −Γµν (x)v (x)da , λ λ ν µ dvBC = −Γµν (x + da)v (x + da)da , λ λ ν µ (23) dvCD = +Γµν (x + db)v (x + da)da , FIG. (1): Parallel Transport: The vector V is parallel λ λ ν µ transported in a closed circuit. Upon arrival at its original dvDA = +Γµν (x)v db , position, the vector is not equal to the original vector and this is a result of the curvature of the space in question. λ µ where, Γµν , and, v , are evaluated at the location indi- cated in the parenthesis and the vector, daµ, is the vector along AB and likewise, the vector, dbµ, is the vector vector and moves it along a path in space without turning λ λ along, BC. Collecting these terms (i.e. dvAB + dvBC + it (relative to the space) or changing its length akin to λ λ λ dvCD +dvDA), yields the overall change (dv ) suffered by, the a person that carries a javelin as described above. vλ, i.e.: 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 ∂ Γλ vν ∂ Γλ vν dvλ = µν daαdbµ − µν daβdbµ, (24) and final vector after the parallel transport operation are ∂xα ∂xβ not coincident and the change in this can be computed as will be done below and for this exposition, one can and by differentiating the quantities in the brackets, this visit any good book on the GTR [cf. Refs. 22, p.137-8, further reduces to:

λ λ ν λ λ σ ν α λ ν λ δ σ ν β dv = Γµν,αv − Γµν Γσαv da db − Γµν,βv − ΓµδΓσβv da db , (25)

µ λ α λ and using the identities, da Γµν,σ ≡ da Γαν,σ, one ar- original position in such a way that its direction is no rives at: longer the same as its initial direction. For a moment, let us shy-away from the abstract World of mathematics and pause a perdurable question to the dvλ = Γλ − Γλ + Γλ Γδ − Γλ Γδ vµdaν dbα. µν,α µα,ν δα µν δν µα reader, a question about the real World. Suppose one is (26) in a freely falling laboratory and this laboratory moves This can be written compactly as: in a gravitational field in a closed circuit such that the laboratory leaves a given point and latter it returns to dvλ = Rλ vµdaν dbα, (27) the same-point and throughout its path at all points it µαν is in free-fall. The best scenario is a laboratory orbiting λ λ λ λ δ λ δ a central massive, body. If in this laboratory we have where, Rµαν = Γµν,α − Γµα,ν + ΓδαΓµν − Γδν Γµα, is the Riemann curvature tensor. The above result is the im- a stationery object – do we (or does one) expect that portant reason why for instructive purposes we have gone after a complete orbit this object will have its motion through all the above calculation, namely to find (via this altered? Or, does one expect that an object (inside the exposition) the mathematical relationship that informs laboratory) that – say, has a specific momentum (relative us of the change that occurs for a any given vector after to the laboratory) will after a complete circuit alter its parallel transport. In Riemann geometry, the affinities momentum without any external force being applied to are not tensors and this leads to vectors altering their the free-falling system? direction as they are parallel transported. A vector par- If this did happen, then Newton’s first Law of Motion allel transported along a closed circuit will return to its that defines inertia systems of reference is violated and 11 it would mean that there is no such thing as an inertial becausev ¯µ 6= 0, da¯ν 6= 0 and d¯bα 6= 0. The fact that, ¯λ ¯λ system of reference; actually this renders the Principle of Rµαν = 0, does not necessarily mean that, Γµν = 0. Equivalence obsolete. Surely, something must be wrong Equation (30) is a major field equation of the present the- because the Principle of Equivalence can not be found in ory. In comparison to the GTR, it has the same status as this wanting-state. We argue the reader to carefully go Professor Einstein’s field equation (2). This is the equa- through the above argument to convince themselves of tion that describes all the known (and unknown) forces its correctness (or its incorrectness thereof). Whatever that exist or can ever exist on the RHS. This seemingly conclusion the reader will reach, it does not affect the fi- colossal claim that (30) describes all of the known and nal thesis being advanced namely that the affinities must unknown forces, we shall substantiate in the subsequent be tensors. If they disagree with the above, it really does sections. In the next section, we shall take advantage of not matter as long as they agree that tonsorial affinities the tensorial nature of the affinities and choose a World preserve both the angle and the magnitude of a vector geometry that is linear in the curvature tensor. As will be under parallel transport. demonstrated in the section which follows thereafter, this In the above, we say this renders the Principle of choice – of a World geometry linear in the curvature ten- Equivalence obsolete because for a system in free-fall sor; appears to have a very strong correspondence with like the laboratory above, according to this Principle of the physical and natural reality of the World we live in. Equivalence; it is an inertial system throughout its jour- ney thus we do not expect an object in an inertial system to alter its momentum without a force being applied to LINEAR RIEMANN-HILBERT SPACE it. The non-preservation of angles during parallel transport in Riemann geometry is in violation of the Principle Given that we have attained a geometry with tensorial of Equivalence if it is understood that parallel transport affines, now is our time, this is our moment of exhalation, takes place in a geodesic system of reference i.e., inertial a moment to reap the sweetest fruits of our hard labour systems of reference. i.e., it is time to take the fullest advantage of the tensorial Naturally, we expect that for an observer inside the nature of the affinities. We now have the mathematical laboratory, they should observe a zero net change in the and physical prerogative, legitimacy and liberty to choose momentum. This, in the context of parallel transport of a spacetime where the non-linear terms vanish identically ¯λ vectors, means that, such a spacetime will parallel trans- i.e., a spacetime such that, Γµν 6= 0, and: port vectors (in free-falling frames) in a manner such that after a complete circuit the parallel transported vector ¯σ ¯λ and the original vector, will still have the same mag- Γµν Γσα ≡ 0. (31) λ nitude and direction i.e., dv = 0. Actually, this means Clearly and without any doubt, this fact that we have that throughout its parallel transport, the magnitude and chosen a spacetime that is governed by the constraint direction of the vector must be preserved. Riemann ge- (31), this means that in a single and triumphant moment ometry does not preserve the angles but only the length of joy, we have just reed ourself of the monstrous and of the vector under parallel transport. The only way to troublesome non-linear terms in the Riemann tensor (29) have both the angles and the length preserved is if the because with this beautiful and elegant choice (31), they affinities are tensors and the curvature tensor of such a [non-linear terms] now vanish identically to become but spacetime will be identically equal to zero. We have al- footnotes of history. This means (30) now becomes: ready discovered a geometry whose affinities are tensors. Since the RHS is obtained from the Riemann via the ¯λ ¯λ ¯λ transformation, gµν 7−→ g¯µν , all we need to do now is to Rµαν = Γµν,α − Γµα,ν = 0. (32) make the transformation: Γα 7−→ Γ¯α , so that: µν µν ¯σ ¯ From (31), if we set λ = α, we will have, Γµν Γσ ≡ 0, ¯ ¯α where Γσ = Γσα. Now raising the index µ in the equa- dv¯λ = R¯λ v¯µda¯ν d¯bα, (28) ¯σ ¯ µαν tion, Γµν Γσ ≡ 0, and contracting it with ν, we will have: where: σ Γ¯ Γ¯σ ≡ 0. (33) Linear terms z }| { Now that we have a theory linear in the curvature ten- ¯λ ¯λ ¯λ ¯λ ¯δ ¯λ ¯δ Rµαν = Γµν,α − Γµα,ν + ΓδαΓµν − Γδν Γµα . (29) sor, a theory in which the non-linear terms vanish, we | {z } non−Linear terms can use this to separate the Weyl terms from the Rie- mann terms. Weyl terms are here those terms associ- The fact that, dv¯λ = 0, implies that: ated with the affine vector Qµ and the Riemann terms ` ` are those terms associated with gµν = AµAν . From the ¯λ ¯α α α Rµαν = 0, (30) affine Γµν = Γµν − Wµν , it follows that: 12

We can derive other field equations. We know that the Riemann curvature tensor satisfies the first Bianchi ¯α ¯α ¯α α α Rµσν = Γµν,σ − Γµσ,ν = Rµσν − Tµσν , (34) identity: where: α α α Rµσν + Rνµσ + Rσνµ ≡ 0. (38) Rα (A ) = Γα (A ) − Γα (A ) , (35) µσν µ µν,σ µ µσ,ν µ From this first Bianachi identity and as as-well from (37), is the linear Riemann curvature tensor, and: it follows that:

α α α α α α Tµσν (%) = Wµν,σ (%) − Wµσ,ν (%) , (36) Tµσν + Tνµσ + Tσνµ ≡ 0. (39) is a curvature tensor that we shall call the Weyl curvature The identity (39) leads to an important identity (which, tensor. This Weyl curvature tensor is defined in terms of for our purposes we shall call it a gauge condition). This the material field %, it must represent the material field. identity (gauge condition) is obtained from (39) by con- Now, from (34) and (32), it follows that: tracting the index α with σ, that is to say, if for (39) we set α = σ, one is led to:

α α Rµσν = Tµσν . (37) Qα [g − g ] ≡ 0. (40) At this point – if it turns out the this theory proves to be µν,α µα,ν a correct description of physical and natural reality; we When we arrive at (57), it shall become clear that the have no doubt that if Professor Einstein where watching gauge condition (40) is necessary in-order for us to ar- from above or from wherever in intestacies of spacetime, rive at the familiar Maxwell-Proca equations of electro- he must be smiling endlessly because his life long endeav- dynamics. our was to derive[73] the material tensor from pure ge- Now, we are are ready to derive our last set of field ometry and not to insert it by the sleight of the mind and equations. We know that Riemann curvature tensor sat- hand as he did with his gravitational field equation. As isfies the second Bianchi identity: his internationally acclaimed scientific biographer Abra- ham Pais graphically put it – Professor Einstein would α α α look at his equation (2) and greatly admire the left hand Rδµσ,ν + Rδνµ,σ + Rδσν,µ ≡ 0. (41) side which he would compare with marble – it was beautiful, pure and was a marvel sight for the artist as it From this second Bianachi identity and as as-well from was constructed from pure geometry; but when he [Pro- (37), it follows that: fessor Einstein] would look at the right hand side of this same equation; with equal passion, he was visited by deep T α + T α + T α ≡ 0. (42) melancholy sighs and feelings which made him loath the δµσ,ν δνµ,σ δσν,µ right hand side of his equation – he would compare this In the next section, we shall explore the equations (37), right handside with wood – it was ugly; with no end in- (38), (39), (41) and (42) and from these equations, we sight, Professor Einstein would moan time and again. shall see that one is able to obtain field equations that His ultimate pine and goal thereof was to turn wood we are already familar with. 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 death on April 13, 1955, he RESULTANT FIELD EQUATIONS asked for a pen and his notes so that he could continue to work on the unified theory that he was working on at Before we go into the details of the present section, the time. Without an iota of doubt, if what is before us we must mention that, from here-on, we will now proves itself to have a correspondence with physical and bring back the `-representation index into our equa- (`)α natural reality, then we can safely say we have achieved tions. We know that the affine Γ µν , is defined such one of Einstein’s goals to attaining the ‘Elicit Dream of a (`)α 1 (`)ασ h (`) (`) (`) i that, Γ µν = g gσµ,ν + gνσ,µ − gµν,σ , and Final Theory’ by deriving the material tensor from pure 2 (`) geometry – wood has finally been turned into marble! that the metric, gµν , has herein been define such that, (`) (`) (`) This we are certain has been achieved in the present UFT. gµν = Aµ Aν . Thus, substituting the metric into the The only question is, “Does the theory correspond with affine and then differentiating this metric as required by physical and natural reality?” This we leave for the reader the differentials appearing in the affine, we obtain the to be judge. following: 13

 Term (I) Term (II) Term (III) Term (IV) Term (V) Term (VI) z }| { z }| { z }| { z }| { z }| { z }| { (`)α 1 (`)ασ  (`) (`) (`) (`) (`) (`) (`) (`) (`) (`) (`) (`)  Γ = g A A + A A + A A + A A − A A − A A  . (43) µν 2 σ µ,ν µ σ,ν σ ν,µ ν σ,µ µ ν,σ ν µ,σ | {z } | {z } | {z } (`) (`) (`) gσµ,ν gνσ,µ gµν,σ

Now, if we set: Danish mathematician and physicist – Professor Ludvig Valentin Lorenz (1829 − 1891), in 1867 [23]. For many non−Abelian Term Abelian Term years (including present day literature), this gauge has z }| { z }| { −1 (and is) mistakenly been called the Lorentz gauge, after (`) (`) (`) (`) (`) (`) Fµν = Aµ,ν − Aν,µ + A Aµ Aν,, (44) the great Dutch physicist – Professor Hendrik Antoon Lorentz (1853 − 1928), who also used it in his long paper then, it follows from (43) and (44) that: in 1892 [24]. Fortunately for Professor Ludvig Valentin Lorenz, the story of the misattribution has recently been 1 h i (`δ)α (`)ασ (`) (`δ) (`) (`) told in detail by cf. Jackson & Okun and Nevels & Γ µν = g Aµ Fσν + Aν Fσµ . (45) 2 Chang-Seok [25, 26]. In (44), the reader must take notice of the fact that the For the present UFT – rather naturally than by the Greek index- has now joined the index-` as part of the (`) sleight of hand – the Lorenz gauge condition has been representation indices in the superscript of Fµν , and be- found necessary in-order that there is symmetry in the cause of this, we have added the ‘dummy’ Greek indices, Greek indices of the resulting mathematical objects of α (`) (`) δ and , to Γ , because of field tensors, Fσν , and, Fσν . (`δ) (`δ) µν the theory such as tensors e.g. for Rµν = Rνµ to be Note that (δ 6= ); this is an important fact that needs symmetric under the assumption that g(`) = A(`)A(`), to be remembered the reader. µν µ ν it is absolutely necessary that the Lorenz gauge should Now, moving on; in the expression of the affine con- h i hold [cf. Ref. 27, p.33]. If particles are to have a non- nection, Γ(`δ)α = 1 g(`)ασ A(`)F (`δ) + A(`)F (`) , the µν 2 µ σν ν σµ zero mass in the proposed UFT, we find it necessary that (`) (`δ) term, Aµ F σν , is the sum of the Terms (II, III & we further modify the Lorenz gauge condition, so that it (`) (`) now reads: V) in (43), while, Aν F σµ, is a sum of the Terms (I, IV & VI) of the same equation. We will now write, (`δ)α 1 (`)ασ h (`) (`δ) (`) (`)i Γ µν = 2 g Aµ Fσν + Aν Fσµ , as: µ (`) ∂ A = κI4, (48) 1 1 µ Γ(`δ)α = A(`)F (`δ)α + A(`)F (`)α . (46) µν 2 µ ν 2 ν µ Now that we have written the affine in terms of the field, where κ is a constant non-zero inherent and intrinsic (`δ) Fµν , we are now ready to write down the source coupled physical quantity which is to be associated with the par- and source free field equations. Already, the agile can ticle in question and as will become clear in §(Source (`δ) already see the writing on the wall, that the field, Fµν , Coupled Field Equations), κ is the mass of the parti- as given in (44), is to be identified latter with the YM - cle. As long as κ is a constant, the modified Lorenz gauge force field. (48) serves the same purpose as the original Lorenz gauge µ (`) Before closing this section, we will write down a gauge ∂ Aµ = 0. condition that flows from the gauge condition (21) i.e. (`)ν F ν ≡ 0. If (21) is to hold true, then, from (44), it follows that: Sources Free Field Equations (`)ν (`) A ∂ν Aµ ≡ 0. (47) This implied gauge condition (47) will prove necessary Now, we are going to deduce the source free field equa- in-order for us to arrive at the crucial equation (52). tions from the first Bianachi identities (38). We know (`δ) (`) h (`δ)λ (`δ)λ i that Rαµσν = g Γ¯ µν,σ − Γ¯ µσ,ν , and written Modified Lorenz Gauge αλ (`δ) in an expanded form in terms of the field tensor Fµν , (`) The Lorenz gauge condition (here represented by the re- its derivatives and in terms of the vector field Aµ , we µ (`) will have: lation ∂ Aµ = 0) was introduced for the first time by the 14

 Term 1 Term 2 Term 3 Term 4   Term 5 Term 6 Term 7 Term 8  z }| { z }| { z }| { z }| { z }| { z }| { z }| { z }| { (`δ) 1  (`) (`δ) (`) (`δ) (`) (`) (`) (`)  1  (`) (`δ) (`) (`δ) (`) (`) (`) (`)  Rαµσν = Aµ,σFαν + Aµ Fαν,σ + Aν,σFαµ + Aν Fαµ,σ − Aµ,ν Fασ − Aµ Fασ,ν − Aσ,ν Fαµ + Aσ Fαµ,ν  . (49) 2   2  

(`δ) Inserting Rαµσν into the second Bianachi identity (38), Source Coupled Field Equations (`) (`δ) (`) (`δ) one obtains a sum of terms, Aµ,σFαν and Aµ Fαν,σ cyclic in the indices µσν. The equation or identity We shall now proceed to derive the source coupled ﬁeld (`δ) (`δ) (`δ) Rαµσν + Rανµσ + Rασνµ ≡ 0, will only make sense if: equations. This, we shall do by use of equation (37) by contracting the indices α and σ, that is, we set, α = σ, so that the new equation is: (`) (`δ) (`) (`δ) (`) (`δ) Aµ Fαν,σ + Aσ Fαµ,ν + Aν Fασ,µ ≡ 0, (50) and: (`δ) (`) Rµν = Tµν . (56)

(`) (`δ) (`) (aδ) (`) (`δ) (`) Aµ,σFαν + Aν,µFασ + Aσ,ν Fαµ ≡ 0. (51) First, we shall compute, Tµν . It should not be difficult to deduce that: If we multiply (51) throughout by A`λ and then contract the indices λ and µ (i.e. set λ = µ), we will have: (`) α h (`) (`) i 2 (`) Tµν = −∂µQν − Q gµν,α − gµα,ν + κ gµν , (57) (`) (`)µ (`δ) (`) (`)µ (`δ) Aν,µA Fασ + Aσ,ν A Fαµ ≡ 0. (52) where: Now, if we multiply (52) throughout by A(`)λ and then contract the indices λ and σ (i.e. set λ = σ), we will 2 α κ = −∂ Qα = − ln %, (58) have: µ 2 where = ∂ ∂µ. Soon, it shall become clear that κ is a mass term – albeit, a Proca mass term. A(`) A(`)µ A(`)σF (`δ) ≡ 0. (53) ν,µ ασ Now, from the gauge condition (40) i.e., the condition α h (`) (`) i therefore: that Q gµν,α − gµα,ν ≡ 0, if follows that (57) can now be written as: (`) (`)µ Aν,µA = 0 (a) and/or . (54) (`) 2 (`) T = −∂µQν + κ g . (59) (`)σ (`δ) µν µν A Fασ = 0 (b) (`) We shall assume that only (54b) holds true and The tensor Tµν is symmetric in its Greek indices. The (`) `µ term ∂µQν , may at face value lead one to think that this Aν,µA 6= 0. Because of the gauge condition (47), if (`) `µ term ∂µQν is not symmetric in the Greek indices, thus Aν,µA = 0, the resulting equations do not make sense leading to an apparent contradiction. However, one must at all. We shall thus take (54b) as a sacrosanct gauge bare in that Qµ = ∂µ ln %, thus we can write (59) as: condition. This gauge condition (54b) will prove necessary in deriving the sourced field equation when we arrive at (68). (`) 2 (`) Tµν = −∂µ∂ν ln % + κ gµν , (60) Now, from (51), we will derive the source free field (`)λ (`) equations. If we multiply (51) throughout by A and in which case it becomes clear that Tµν is symmetric in then contract the indices λ and µ (i.e. set λ = µ), we its Greek indices. will have: µν It is not difficult to deduce that the scalar T = g Tµν , is such that: (`δ) (`δ) (`δ) Fµν,α + Fαµ,ν + Fνα,µ = 0. (55) (`) (`)µν (`) α 2 In the framework of Maxwellian electrodynamics, the T = g Tµν = −2∂ Qα = 2κ . (61) above set of equations (55) gives the source free field (`δ) equations. Now we proceed to compute Rµν . In the case of, (`δ) (`δ)α (`δ)α Rµν = Γ µν,α − Γ µα,ν , we will have: 15

1 h i 1 h i R(`δ) = A(`)∂ F (`µ)α + A(`)∂ F (`ν)α − A(`)∂ F (`ν)α + F (`µ)α ∂ A(`) + F `ν)α ∂ A(`) − F (`µ)α ∂ A(`) µν 2 µ α ν ν α µ α ν µ 2 ν α µ µ α ν µ ν α (62)

(`) (`)µν (`δ) and the scalar R = g Rµν , is such that: Now, the source coupled ﬁeld equations shall be derived from the following equation: 1 h i R(`) = + A(`)σ∂ F (`µ)α + F (`ν)α ∂ A(`)σ . (63) 2 α σ σ α

Since R(`) = T (`), and T (`) ≡ 2κ2, it follows that: (`)µ (`δ) `µ (`) A Rµν = A Tµν . (66) (`) (`)µν (`δ) 2 R = g Rµν = 2κ , (64) therefore:

(`)µ (`δ) (`)µ (`) Therefore, we need to compute A Rµν and A Tµν . (`)σ (`µ)α (`ν)α (`)σ 2 (`)µ (`δ) A ∂αF σ + F σ∂αA ≡ 4κ . (65) For this eﬀort, we shall begin with A Rµν , i.e.:

1 h n o n oi A(`)µR = ∂ F (`α)α + F (`ν)α + A(`)µ F (`ν)α ∂ A(`) − F (`µ)α ∂ A(`) . (67) µν 2 α ν ν µ α ν µ ν α Now, in the representation indices, let us set α = ν for the Greek indices. This means that the above will reduce to:

1 h i A(`)µR(`δ) = ∂ F (`α)α + A(`)µF (`ν)α ∂ A(`) − ∂ A(`) . (68) µν α ν 2 µ α ν ν α

`µ (`α) (`)µ (`) From the gauge condition (54b) i.e. A Fµν ≡ 0, it foregoing, we are led to the term, A Tµν , now being follows that (68) will reduce to: given by:

(`)µ (`δ) (`α)α A R = ∂αF . (69) (`)µ (`) (`) 2 (`) µν ν A Tµν = Jν + κ Aν . (72) This already looks very familiar – doesn’t it? Now, putting everything together i.e. from (66), (69) Now, let us proceed to calculate the left handside of (`)µ (`) and (72), it follows that: (66) i.e. A Tµν . From (59), it follows that:

µ (`) (`) 2 (`) (`)µ (`) (`)µ 2 (`) ∂ Fµν = Jν + κ Aν . (73) A Tµν = −A ∂µ∂ν ln % + κ Aν (70) From the definition of % as given in (22), it is not difficult In the framework of electrodynamics, not only have to show that: we derived the source coupled equations of Professor Maxwell [28], but the Maxwell-Proca equations of electrodynamics first proposed by the great Romanian physi- (`)µ (`) A ∂µ∂ν ln % = Aµ . (71) cist, Professor Alexandru Proca [29–31], as an equation representing massive photons. This same equation was By inference from Maxwellian electrodynamics, we know first explored by him in the subsequent years [32–38] and that, Aµ = Jµ, where Jµ is the four electrodynamic is today universally accepted as an equation represent- (`) current. By way of analogy, the term, Aµ , must be a ing massive gauge fields. In comparison to the Professor (`) (`) four current as-well i.e., Aµ = −Jµ ; the minus sign Proca’s theory; as equation (73) stands, it is now clear has here been inserted for convenience. Accepting the that κ must indeed be a mass term. It should be stated 16 clearly that, besides the fact that this equation yields re- already know from experience, we have required a total sults that resonant with experience; the fundamental jus- of five gauge conditions and these gauge conditions are: tification of Professor Proca’s modification does not exist and the present effort may very-well justify this seemingly (1).The first gauge condition is the Lorentz gauge µ (`) ad hoc modification by Professor Proca. ∂ Aµ ≡ κI4. This gauge condition is necessary for (`δ) (`δ) Thus, in (55) and (73), we have attained the desired symmetry purposes e.g. for Rµν = Rνµ [cf. Ref. 27, set of equations that enable us to obtain – as a bare p.33]. minimum – all of Professor Maxwell’s equations. Prima facie, the gravitational force appears to be absent from (2).The second gauge condition (21) has been imported the present scheme. We will tackle this issue soon in the from Maxwellian electrodynamics and applied to the present theory i.e. F (`)ν ≡ 0. This gauge is necessary section on “Gravitation and Gravitomagnetism”. ν in-order to get reed of terms that do not appear in the In the next subsection, we will have a look at the second usual equations of both Maxwellian and YM-Dynamics. Bianachi identities (41) and (42), from where we deduce (`)ν (`) some conservation laws. (3).The third gauge condition (47) i.e. A ∂µAν ≡ 0; is a consequence of the second gauge condition stated above. This gauge condition proves to be necessary in-order Second Bianachi Identity Equations to get reed of terms that do not appear in the usual equations of both Maxwellian and YM-Dynamics.

We are now going to use the second Bianachi identities (`)µ (`) (4).The fourth gauge condition (54) is, A Fµν ≡ 0; like (41) and (42) to demonstrate the conservation of the ﬁeld the other gauge conditions, this gauge condition proves (`δ) (`) tensors, Rµν and Tµν ; consequentially, we will be able to be necessary in-order to get reed of terms that do not to harvest from this, the law of conservation of current. appear in the usual equations of both Maxwellian and We know from (41) that: YM-Dynamics.

The field equations that we have produced seem to (`δ)α (`δ)α (`δ)α R µσν,β + R µβσ,ν + R µνβ,σ ≡ 0, (74) have nothing to do with gravitation but the nuclear forces. This obviously gives one the feeling that gravi- and contracting the indices α and σ, we will have: tation has now been shut out of the unification scheme i.e. all the other forces have willingly submitted to ge- 1 ometrization and gravitation has been shut out somehow. ∂µ R(`δ) − R(`)g(`) = ∂µR(`δ) = 0. (75) µν 2 µν µν In the first two instalments [27, 39, 40] leading to the present version of the UFT, we noted this and we had to We know also know from (42) that: find some rather draconian means to fit gravitation into the scheme.

(`)α (`)α (`)α T µσν,δ + T µδσ,ν + T µνδ,σ ≡ 0, (76) GRAVITATION AND GRAVITOMAGNETISM and as above, contracting the indices α and σ, we will have: Professor JoséHera [41] formulated – in our view; a very important Existence Theorem that states that, given any space and time-dependent localized scalar and vec- µ (`) 1 (`) (`) ∂ T − T g = Qν = 0. (77) µν 2 µν tor sources satisfying the continuity equation – as is the case with electromagnetism; there exists in general, two (`δ) What the above equations imply is that the tensors Rµν retarded vector fields (X, Y ) that satisfy a set of four (`) field equations that are similar in nature and form to and Tµν , are conserved quantities. Maxwell’s equations. By applying the theorem to the usual electrical charge and current densities, the two re- STOCKTAKING tarded fields are identified with the electric (E) and magnetic (B) fields and the associated field equations with Though we are not done yet, it is perhaps time to take Maxwell’s equations i.e. (X := E, and Y := B). In a stock of what we have achieved thus far so that we bring nutshell, what Professor JoséHera [41] proved is that, if some clarity and as-well so that the length of the road % is the charge density and J is the associated current ahead may be made known and furthermore, so that corresponding to this charge, i.e.: the reader does not develop fatigue. Firstly, in-order to generate the desired field equations (i.e. 73 and 55) ∂% = −∇ · J, (78) which surely have a resemblance with equations we that ∂t 17 then, there must exist two corresponding fields, X and (1850 − 1925) reconsidered Professor Maxwell’s imagi- Y , that satisfy the following set of equations: native and brilliant but forgotten thoughts [43, 44]. De- spite its brilliance and fit of imagination, both Profes- sor Maxwell [28] and Dr. Oliver Heaviside [43, 44]’s ∇ · X = α%, (79) work was speculative in nature, with no real justification from the then known fundamental principles of either physics or logic but rather from the intuitive power of human reason and imagination. Only in recent times, ∇ · Y = 0, (80) has a justifiable fundamental physical basis for Professor Maxwell and Dr. Heaviside’s speculation (hereafter, the Maxwell-Heaviside Gravitomagnetic Theory) been her- ∂Y alded [41, 42, 45]. Presently, gravitomagnetism is ac- ∇ × X + γ = 0, (81) ∂t cepted as first order approximation of Professor Ein- stein’s GTR [cf. 46, 47]. Writing in his book “Causality, Electromagnetic In- β ∂X duction and Gravitation”, Professor Oleg D. Jefimenko ∇ × Y − = βJ, (82) α ∂t (1922 − 2009) revived [45] this almost forgotten if not abandoned line of thought which most physicists now where α, β, γ are arbitrary positive constants and are viewed as a pseudo-science because of lack of a real fun- 2 related to the speed of light c by the equation α = βγc . damental physical and theoretical justification save for In the case of electricity and magnetism, if X and Y are the sheer power and wit of human intuition and imag- identified with the electric and magnetic fields respec- ination. Taking the work of Professor Jefimenko one tively, then, we will have Professor Maxwell’s classical important and crucial step further, Professor JoséHera equations for electrodynamics – in which case α = 1/ε0, [41] formulated an important Existence Theorem stated β = µ0 and γ = 1. Clearly, this axiomatic approach of above. It is this theorem that we now find to be not deriving Maxwell’s field equations strongly suggests that only pivotal but crucial in justifying the way in which electric charge conservation – and nothing else; can be we are now going to introduce gravitation on the realm, considered to be the most fundamental assumption un- structure and edifice of the RHS. derlying Maxwell’s equations of Electrodynamics. Professor JoséHera [41] did not consider gravitation in In the case of electricity, if X and Y are identified his work. However, this work can be extended not only to with the electric and magnetic fields respectively, then we gravitation but to any field whose charge and correspond- have Professor Maxwell’s classical equations for electro- ing current obeys the continuity equation. Assuming – dynamics.This axiomatic approach of deriving Maxwell’s as already stated above; that mass is identified with the field equations strongly suggests that electric charge con- gravitational charge of the body in question – as is the servation – and nothing else; can be considered to be case in physics; and as-well assuming the conservation of the most fundamental assumption underlying Maxwell’s mass and momentum, which in-turn means the conser- equations of Electrodynamics. vation of gravitational charge, then, there must exist two If – as is widely believed; the force of gravity is con- vector fields (call them g and B ) that satisfy a set of four veyed by mass, then, if the associated current (momen- g Maxwell-type equations. These four equations describe tum) of this mass is conserved as given in (78), then, what is known as gravitomagnetism. Gravitomagnetism there must exist the two fields associated with mass and is nothing but a formal analogy of Maxwell’s equations momentum. One of the fields is obviously the gravita- and the field equations of gravitation. tional field g. The other must be a magnetic like force. Let us give this field the symbol Bg and let us call this the The problem with the issue of gravitomagnetism is gravitational magnetic field and for short the g-magnetic that some have taken this to mean that the generated field. What this means is that there must exist a simi- g-magnetic field must have an effect on electrically-laden lar set of Professor Maxwell’s equations for gravitation. bodies – no! This field, Bg, only couples to mass just as This phenomenon is typically called gravitomagnetism. the magnetic field generated by electric currents couples The first to consider the possibility of a formal anal- to electrically-laden bodies. Who then are to introduce ogy between gravitation and electromagnetism is Pro- gravitational into the present scheme? If the gravita- fessor Maxwell himself, in his landmarking treatise on tional phenomenon does take part in Maxwell-Heaviside “A Dynamical Theory of the Electromagnetic Field” [28]. gravitomagnetism, then, there must exist a gravitomag- ˜(`) After failing to justify (to himself) the implied negative netic four vector, Aµ , just as there exist an electromag- energies associated with the gravitational field, Professor netic four vector. A straight forward way to introduce (`) (`) Maxwell abandoned this line of thought [cf. 42]. Twenty the vector, A˜µ , would be to split the vector, Aµ , into eight years had to pass before Dr. Oliver Heaviside the sum of two vectors, the electromagnetic four vector, 18

(`) ˜(`) Abµ , and the gravitomagnetic four vector, Aµ , i.e.:  µ ˜(`α) ˜(`) 2 ˜(`)  ∂ Fµν,α = Jν +κ ˜ Aν (`) (`) ˜(`)  Aµ = baAbµ +a ˜Aµ , (83) . (92)  (`α) (`α) (`α)  F˜µν,α + F˜αµ,ν + F˜να,µ = 0 where ba anda ˜ are constants of superposition. The vectors (`) (`) ˜(`) Aµ , Abµ and Aµ , are such that: Therefore, herein, the same equations governing the electrical phenomenon govern the gravitational phenomenon in such a way that both are driven by two independent A(`)A(`)µ = Ab(`)Ab(`)µ = A˜(`)A˜(`)µ = 1(84) µ µ µ vectors which are sadly related to each other only by ad- hence: dition and the resulting mathematics thereof. The two sit side-by-side with no intimacy whatsoever. However, in §(Siamese Unification of Gravitation with Elec- † † 2 2 ba ba +a ˜ a˜ = |ba| + |a˜| = 1, (85) tricity), 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. ba · a˜ = ba · a˜ = 0. (86) Therefore, the Riemann metric g(`), is such that: µν NEW GEODESIC EQUATION

(`) 2 (`) 2 (`) Let us now address the problem raised in gµν = |ba| gbµν + |a˜| g˜µν , (87) §(Motivation), i.e., the problem of the geodesic (`) (`) (`) (`) ˜(`) ˜(`) law namely that it is neither invariant nor covariant where gbµν = Abµ Abν andg ˜µν = Aµ Aν . The RHS metric i.e.,g ¯µν , is therefore given by: 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, (`) 2 (`) 2 (`) g¯µν = % |ba| gbµν + % |a˜| g˜µν , (88) one needs to formulate this equation first in a Gaussian coordinate system and thereafter make a transformation and in-turn, the curvature tensor is given by: to a coordinate system of their choice. As already said in §(Motivation), this geodesic equation is in conflict with the very principle upon which the GTR is founded, R¯(`δ) = |a|2 R(`δ) + |a˜|2 R˜(`δ) = 0. (89) αµσν b bαµσν αµσν namely the Principle of Relativity, which requires that one should be free to formulate the geodesic equation In the complete absence of electromagnetism (i.e. ba = 0), the RHS is expected to maintain all of its properties, of motion in any legitimate coordinate system of their for this to be so, we must have for the gravitomagnetic choice without having to be preferentially constrained to (`δ) start from a Gaussian coordinate system. The geodesic curvature tensor R˜ = 0, at all times whether or not αµσν law equation (1) is derived from the Lagrangian: electromagnetism exists or not. This invariably means (`δ) that R˜αµσν = 0, at all times and for all conditions of existence, thus, it follows from the foregoing that the dxµ dxν L = g , (93) field equation for RHS where both electromagnetism and µν ds ds gravitomagnetism exist, we must have: by inserting this into the Lagrangian equation of motion, namely: ¯(`δ) (`δ) ˜(`δ) Rαµσν = Rbαµσν = Rαµσν = 0. (90) Therefore, the resulting field equations for the electro- d ∂L ∂L − = 0, (94) magnetic sector of the RHS will be: ds ∂x˙ µ ∂xµ

and thereafter making the necessary logical algebraic op-  µ (`α) (`) 2 (`)  ∂ Fbµν,α = Jbν + κb Abν erations and manipulations. In the present, one must , (91) realise that the metricg ¯µν is that which is given rise to  (`α) (`α) (`α)  Fbµν,α + Fbαµ,ν + Fb να,µ = 0 by a massive electrically laden particle whose mass and electrical charge are say M and Q, respectively. Thus, while for the gravitomagnetic sector, the ﬁeld equations we have both a gravitational and electrical ﬁeld, hence, will be: our geodesic equation of motion must be able to describe 19

(`) the trajectory of a massive electrically laden test parti- where γb =κq/m ` i, and likewise, for Le , we propose: cle. Given that the RHS’s metric is,g ¯µν = %gµν , one might think – that, as is the case with the GTR; the dxµ dxν Lagrangian: Le(`) = γ%g(`) , (97) e eµν ds ds

µ ν where γ = mg/mi, and the quantityκ ` is expected to be (`) (`) dx dx e L =g ¯µν , (95) a universal and fundamental physical ‘constant’ with the ds ds units of mass per unit electrical charge, it is a constant will yield an appropriate equation of motion that agrees that makes γe a dimensionless parameter. with experience in describing the trajectory of a massive Now, inserting L(`) = Lb(`) + Le(`), into (94), and upon electrically laden test particle in the vicinity of a massive making the correct algebraic operations and manipula- electrical charged parent body. In the next few para- tions, one is lead to the following geodesic equation of graphs, we shall shed some light on this issue whereby, motion: we come up with a Lagrangian that leads to the appropriate equation that is in tandem with physical and natural d2xα (`δ)α dxµ dxν reality. − Γ¯˘ = 0, (98) ds2 µν ds ds In its bare form, equation (1) has no natural provision for describing the trajectory of an electrically laden test where: particle. If the parent body, i.e., the body about which the electrically laden test particle describes its motion; is (`δ)α (`δ)α ¯˘(`δ)α ¯ ¯ also carrying an electric charge, then, if equation (1) is to Γ µν = γeΓe µν + γbΓb µν , (99) describe the path of the electrically laden test particle, it (equation1) will have to be modify outside of the confines is a tensorial affine for as long as the γ’s are all scalars. (`δ)α (`δ)α of the GTR. What we want – or more appropriately, what ¯ ¯ The tensor affinities Γe µν and Γb µν , are such that: is expected of a Universal World Geometry such as the RHS, is that the geodesic equation of motion for the RHS (`δ)α must have within the confines of the RHS, the provision ¯ (`δ)α (`)α Γe µν = Γe µν + Wf µν , for trajectories of electrically laden test particle. If at all, how are we to attain such a geodesic equa- where : tion capable of describing the trajectory of an electrically laden test particle in the vicinity of a massive electrically h i Γ(`δ)α = g(`)ασ g(`) + g(`) − g(`) charged parent body? From (88), it is clear that the RHS e µν e eσµ,ν eνσ,µ eµν,σ (100) (`) metric,g ¯µν , is split between the electrical and gravita- 1 h (`) (`δ)α (`) (`)α i (`) 2 (`) 2 (`) = Aeµ Fe ν + Aeν Fe µ , tional components i.e.:g ¯µν = % |ba| gbµν + % |a˜| g˜µν . It 2 is logical that the electrical charge, q; of the test parti- (`)α α α (`) α cle must couple to the electrical component of the metric Wf µν = δµ Qν + δν Qµ − gµν Q , (`) e g¯µν , and likewise, the gravitational charge, mg; of the (`δ)α test particle must couple to the electrical component of and like wise, we have Γb µν , such that: (`) g¯µν . The suggestion coming from the fact that the met- (`δ)α (`) ¯ (`δ)α (`)α ricg ¯µν has two separate components of the electrical Γb µν = Γb µν + Wc µν , and the gravitational, is that we must have two separate Lagrangians for the electrically laden test particle; where : Lagrangians that are conjoined by simple addition. Let h i these Lagrangians be: L(`), for the electrical component, (`δ)α (`)ασ (`) (`) (`) b Γb µν = gb gbσµ,ν + gbνσ,µ − gbµν,σ (101) and Le(`), for the gravitational component, so that the (`) (`) (`) (`) resulting Lagrangian L , is such that L = Lb + Le . 1 h ` (`δ)α ` (`)α i = AbµFb ν + Abν Fb µ , If mi, is the inertia mass of the electrically laden test 2 particle, then, we find the following as the appropriate (`)α α α (`) α Lagrangians that result in a geodesic equation of motion Wc µν = δµ Qν + δν Qµ − gbµν Q . that yields equations of motion that match with what we (`δ)α know from experience, i.e., for Lb(`), we propose: ¯˘ Now, since the affine Γ µν , is a tensor, the problem of the geodesic equation highlighted in §(Motivation) dxµ dxν is solved if-and-only-if this affine tensor yields equations Lb(`) = γ%g(`) , (96) b bµν ds ds of motion which correspond with experience. To that 20 end, we shall now move to show – as a way of justification of the potency and latent power of equation (98) d2xα (`δ)α dxν – that this equation, does indeed produce an equation − Γ¯˘ = 0, (102) that yields – in the absence of gravitation, the usual ds2 0ν ds Lorentz equation of motion for a test particle in an am- bient electromagnetic field. In the presence if an am- an assuming the following gauge conditions: bient gravitational field, this same equation yields the usual Newtonian gravitational equation of motion – al- (`) α beit, with the all-important difference that new gravito- Aα v ≡ 0, α (103) magneitc Lorentz terms surface; these new gravitomag- Qαv ≡ 0, neitc Lorentz terms are just as in the case of the electrical force. then, (102) reduces to: Therefore, to first order approximation, (98) becomes:

EMD 2 µ z }| { 2 d x mg ν (`δ)µ κq` ν (`)µ mg +κq ` µ d ln % 2 = v Fe ν + v Fb ν + v 2 2 . (104) dτ mi mi mi c dt | {z } | {z } GMD Quantum Force

Without the Gravitomagneto-Dynamics (GMD) compo- U(1) × SU(2)] of [49–51] as-well as Quantum Chromody- nent, the above equation is the acceptable Lorentz force namics (QCD) which obeys SU(3) symmetry and is the law. The µ = (1, 2, 3) components of the GMD compo- theory of the Strong force. Thus, YM-Theory forms the nent give the usual equation of motion for a particle in a basis of the physicist’s current understanding of particle gravitational field with it supplemented a corresponding physics which is commonly refereed to as the Standard gravitomagnetic Lorentz term. Model (SM). The question that we wish to ask and answer in this section is whether or not the present theory has within its fabric the ability to describe YM-Theory. Fgm 1 ∂Ae = −∇Φg + . (105) The answer is yes, it has the ability to describe Yang- mg c ∂t Mills theory. We will not give a full demonstration of The new term on the far right of (104) we have coined it this, but only a glimpse of it. The exhibition of the full the quantum force as it depends on the quantum proba- demonstration will be conducted in the accompanying and up-coming monograph. bility amplitude %(a). We shall keep to our mission, that is, simple write down the equations and leave the explo- ration for latter to be done by ourself and others who will see light in the work here presented. In the subsequent Electromagnetism section, we shall now daringly move to make a proposal, a proposal that would bring some form of intimacy be- Against the desideratum of the searching mind and very tween the gravitational and electrical phenomenon. much to its chagrin, the natural picture emergent from the present UFT is one were the Universe is exclusively GLOBAL SYMMETRIES AND YANG-MILLS divided into two parts where one part is gravitational and FORCE FIELDS the other electrical. In a separate reading [52], we have proposed, more by the sleight of hand rather than from In all its noble endeavours as presently understood, Yang- the naturalness of the theory a truce-like union between Mills (YM) theory [48] – which was born out of the these gravitational and electrical components of the Uni- speculative work of two future Professors of physics, verse. For the present section, we shall only consider the Chen-Ning Franklin Yang (b.1922) and Robert Laurence electrical component with the intent to demonstrate the Mills (1927 − 1999); this theory seeks to describe the existence of YM-Theory in the present UFT. behaviour of elementary particles using non-Abelian Lie First we must realise that there are four fields in the (`δ) (`0) Groups and it is at the nimbus and core of the unifica- representation Fbµν : δ = 0, 1, 2, 3; and these are Fbµν , (`1) (`2) (`3) tion theory of the Electromagnetic and Weak force [i.e., Fbµν , Fbµν and Fbµν . Depending on whether or not 21

(`δ) the non-Abelian component of Fbµν vanish, this field is rent SM, albeit, with the important difference that it either Abelian or non-Abelian. We know that the elec- is now a four dimensional YM-Theory. For this, let us (`) tromagnetic field is Abelian. How are we to generate generally consider the fields (ψ, Aµ ) that transform ac- this naturally from the theory? First, we note that the cording to some representation of a certain Lie Group (`) (`) electromagnetic vector potential Aµ obeys, not only the G. That is, we shall consider the elements (ψ, Aµ ) to µ (`) (`) Lorentz gauge [∂ Aµ ≡ 0], but the Coulomb gauge as- comprise a group. Because (ψ, Aµ ) are 4 × 4 matrices, k (`) well i.e. ∂ Ak ≡ 0 : k = 1, 2, 3. This invariably means this means that, for every element of the Group G, we 0 (`) (`) will have some 4 × 4 unitary matrices Uj; these matri- that [∂ A0 ≡ 0], that is, the component A0 has no ex- (`) (`) ces U satisfy the same multiplication rules as the corre- plicit time dependence. If like A the components A j 0 k sponding elements of G. Further, because the elements have also no explicit time dependence, then, from the (`) (`δ) (`0) (ψ, A ) are 4 × 4 matrices, they must submit to some definition of F as given in (44), it follows that F µ bµν bµν decomposition were the wavefunction ψ and the scalar must be the electromagnetic field since for this field, the (`) non-Abelian component is forced to vanish by the ex- and vector product Aµ can be written as the direct sum (`) of subspaces of the Group G as follows: plicit time independence of Aµ . The remainder of the (`1) (`2) (`3) fields [i.e., Fbµν , Fbµν , Fbµν ] are then supposed to be the non-Abeliean YM-Fields. X ψ = τjψjτj, (106) What is interesting in all this is that, the electromag- j netic 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 (`) X (`) Aµ = τjAjµ τj, (107) own four vector field. In the scheme and core of the j spirit of unification, this is considered to be the sought for unification were all the forces emerge from a singe where the objects τj are 4 × 4 matrices and are known field, they are different manifestations of the same field. as the generators of the Group G and they satisfy the (`1) (`2) (`3) The question now is, do the fields Fbµν , Fbµν and Fbµν following Lie Algebra[74]: obey SU(2) or SU(3) gauge invariance? If they do, then we have SU(2) and SU(3) YM-Theory on the RHS. In  the next subsection, we will show that any one of these [τi, τj] = iijkτk (a)   fields – depending on the configuration thereof; can be  shown to obey SU(2), albeit, SU(2) on a four dimensional {τi, τj} = 0 for (i 6= j) (b) , (108)  space. 2  τj = I (c) 

Four Dimensional Yang-Mills Theory where ijk is a set of structural constants peculiar to the Lie Group under consideration. The generators of the We are now going to demonstrate that under certain con- group are deﬁned by the representation appropriate for ditions of existence on the RHS, there exists YM-Theory the gauge Group G under consideration. Substituting – a YM-Theory that is exactly as that found in the cur- (106) and (107) into (14), we obtain:

    (`a) X (`) (a) X X (`) (a) X (`a) eµ =  τjAjµ τj γµ  τjψjτj = τjAjµ γµ τj = ejµ , (109) j j j j

where:

X X (`) (`a) (`) (`) ` ` (a) g¯µν = %jAjµAjµ = gjµν , (111) ejµ = τjAjµ γµ τj. (110) j j (`) Just like (ψ, Aµ ), the metric that emerges from (109) is decomposable and in its decomposed form, it is given by: where: 22

necessary condition not naturally found in the domains and confines of the theory, a condition (`) ` ` g¯jµν = %jAjµAjν . (112) that would make the theory beautifully match with experience? Would you not impose this It is very important to note at this point that if the wave- condition by assuming it be a vital but (an function ψ were a 4×1 object and not a 4×4 object as is as-yet) naturally unjustified ingredient of the the case in the present, then, it would have been impos- theory? sible to decompose the metric in a manner that we have just done in (111). As will become clear shortly, this de- Rather sceptically, fixation is largely viewed by the better composition is a sine qua non in-order for the attainment majority of researchers as some form of ‘cheating’ so as of YM-Theory on the RHS. to arrive at the correct or desired answer. If anything, Now we come to the most crucial part on our ‘sojourn we are of the benevolent view that what one must ask to demonstrate’ the existence of YM-Theory on the RHS. after fixing a theory to match with experience is: Some may consider and ultimately accuse the step that we are about to take as a deliberate and artificial insem- ination of a condition into the theory so that the theory What – if any; does this fixation tell us about the Laws of Physics in relation to physical and matches with experience. We see no problem with fixing natural reality? a theory so as to match with experience. Like Professor Einstein’s GTR, most people would demand of a theory Before answering this question ourself, let us write down to be constrained by its internal and natural logic in- the ansatz condition that we feel leads us to field equa- order to match with physical and natural reality. We ask tions that have a sure correspondence with the equations you the reader: that we already known from experience – i.e., the equa- Insofar as adoption is concerned, what would tions of YM-Theory as they appear in the current SM. you do if you realised that there is an absolutely We find that if the following condition:

−1 ! (`) (`) (`) X X (`) (`) ik (`) (`) Ajδ Ajµ Ajν,δ ≡ κδ jikAiµ Akν = κδj Aiµ Akν , (113) i k

is adopted (or assumed) and is taken as a fundamen- In form and structure, this field is the usual YM-Field as tal gauge condition were κδ are the associated coupling we know it – albeit – a YM-Field whose jµν-entries or (`δ) elements are 4 × 4 matrices. constants corresponding to the fields Fjµν , then, we will obtain all of YM-Theory on the RHS – albeit – a four Now, if all the physics of the World is embodied in the dimensional YM-Theory which is much richer in content field equations (73) and (55), then, the following trans- and meaning than the currently obtaining YM-Theory. formations will leave all of the physics of the World in- In (113) above (and hereafter), the raised ik-indices in variant: ik the structural constants j sum-up together with the ik-indices in the product A(`)A(`), just as is the case in  iµ kν ψj 7−→ Ujψj (a)  the Einstein summation convention. (`) (`) 1 ik (`)  A 7−→ A + ∂µθj + j A θk (b) , Notice that the ik-indices of in the middle term of jµ jµ κδ iµ (`) (`) κ2 2 ik (`)  J 7−→ J − ∂µθj − κ A θk (c)  (113) are not raised (i.e., jik) while in the last part of jµ jµ κδ j iµ ik (115) this equation, they have been raised i.e. j . We have raised to indicate summation in the usual Einstein sum- where: mation convention. It is this raising of the ik-indices that enabled us to do away with the double summation signs appearing in the middle part of (113). Uj(x) = exp [τjθj(x)] , (116) Now, with this gauge condition (113) given, it is a hermitian unitary matrix which is such that U †U = follows that the field (44) will now be given by j j I, and θ = θ (x) is some function that depends on xµ. F (aδ) = P F (aδ), where: k k µν j jµν While the transformations (115) will leave all of the physics of the World invariant, the transformation (115 (`δ) (`) (`) ik (`) (`) Fjµν = Ajµ,ν − Ajν,µ + κδj Aiµ Akν . (114) c) will alter the geometry of the sub-particles and as-well 23 that of the particles themselves. Now, in relation to the laughs if not scoff at it, that perhaps in all the twenty question that we paused earlier about the fixation of the years (1994 − 2014) of the relentless labour that we have theory to match with experience i.e. the question “What put up in-order to attain the present unified theory, He this fixing tells us about the Laws of Physics in relation has led us up the garden path on a sure blind alley. Be to physical and natural reality?”, we now realise that this that it may, we surely must – for the sack of posterity; be fixation is telling us that one can alter the geometry (via bold enough as to express our true heartfelt confidence the alteration of the metric gµν ) of a particle without that the New Universal World Geometry (the RHS) here affecting the resulting physics thereof. This alteration discovered most certainly describes our World or will in that leaves the entire physics of the World invariant is the immediate future prove to do exactly that. If expe- that embodied in the arcane and beautiful mathematics rience proves otherwise – then, without an iota or dot of of Lie Groups. This therefore suggests that particles may doubt; at least some elements of the present endeavour very will be undergoing a constant random changes in will surely percolate significantly into the Final Univer- their geometries in-accordance with the mathematics of sal World Geometry to be discovered in the future by Lie Groups in such a way that from a physics stand- others. point, one can never notice this constant random change The theory is built on what is supposed to be Professor in the geometry of the particle. Weyl’s fail idea [1] of adding a conformal term to the Rie- Thus, to justify the fixation (113) here-made without mann metric. Unlike Professor Weyl, we have demanded any prior knowledge that Nature employs YM-Theory in of this conformal term of yield for us tensorial affinities. exactly the manner it does as we know, one would have Coupled with the idea of decomposing the metric into a to ask themselves the ansatz-type question: product of a four vector, we have shown that one can ob- What condition exists – if any; and is there tain all the equations associated with Yang-Mills Theory, to be imposed upon the RHS in such a manner albeit, the resulting objects are 4×4 matrices. It appears that one can somewhat in an arbitrary, willy-nilly that gravitation may very well be explained by the very and willy-wally fashion, alter the geometry of a same mathematics that governs the nuclear. Why the particle without affecting the resulting physics same mathematics applicable to the nuclear forces may thereof? apply to gravitation, the nuclear and gravitational forces The answer would be the imposition of the gauge con- seem to have no unification. dition (113); it is a condition which for better and/or We should mention here that in building the present ` UFT, we have not made reference to the literature on for worse, is to be adhered to by the vector fields, Aµ. Whence, if – truly; the RHS is the Fundamental and unified field theories in the intervening yeas 1918 − 2014 Universal Geometry of the World, then, according to all and the reason for this being that we have have had to go that we know at present, it would appear as though YM- back to 1918 and built our theory from that time. In the Theory as presently understood, comes about as an ex- end, the final equations strongly appear to resemble cur- ogenous imposition and not a natural occurrence on the rently accepted equations describing the nuclear forces. RHS. Thus, we see here that to Professor Einstein’s all- Because of this, we feel that the present work – if proved daring question of whether or not the Good Lord had a correct, may be pointing to the fact that, in our effort choice in making the Universe, He [God] very well might to arriving at the final theory, we may very well have have had a choice because the gauge condition (113) is a taken the ‘wrong’ theoretical route because our theories choice that seems to have been tailor made by Whosoever in building the theory of nuclear force has been predom- created the World. If anything, this condition (113) is a inated shaped by results from experiments rather than sure finger print of the Divine and Sacrosanct Tailor of from theory. From a theoretical standpoint, it appears the Universe. It is possible that if we knew much more we have been trying to glue together general relativity than we know now, this condition (113) may turn-out and quantum mechanics. In that effort to glue these two to be a natural occurrence on the RHS, in which case, theories, we have held them to be sacrosanct leading us it would invariably mean that God may not have had a to seeking a self consistent manner to glue them while choice in creating a Universe governed by the mathemat- preserving their independent realities. ics of YM-Theory or Lie Groups. At present, we see no Basing on our true heartfelt confidence, we must say way of naturally harnessing it [condition (113)] from the that, what is before us so strongly seems to point to the RHS except by way of the sleight of hand were we have fact that the whole of physical phenomenon may now had to equanimously and gently impose it from outside. be derived from a single Universal World Geometry that obeys the Universal World Law beholden to the principle presaged in Nyambuya (2010), namely that: DISCUSSION Generalized Principle of Equivalence: We cannot possibly know whether or not the Good Lord Physical Laws have the same form in all equiva- looks down from above at our present endeavour and lent reference systems independently of the coor- 24

dinate system used to express them and the com- is the same force which brings to us the rays of plete physical state or physical description of an light and heat which make life possible upon this event emerging from these laws in the respective planet.” reference systems must remain absolutely and independently unaltered – i.e., invariant and con- Professor Einstein said these words when he was describ- gruent; by the transition to a new coordinate sys- ing his then proposed final theory on distant parallelism tem. [53] that latter proved to have no resemblance with physical and natural reality as we know it. We hearty, sincerely and in all probity believe that we It strongly appears that we have been able to achieve have herein shown that it is possible to describe all the one thing that Professor Einstein sought in a unified the- known forces of Nature on a four dimensional geometric ory, i.e., he envisioned the material field ψ being part theory that needs not the addition of extra dimensions as and parcel of the fabric of spacetime. On this, i.e. his is the case with string and string related theories which vision of the final theory, Professor Einstein is quoted as today predominately stand on the beacon and chief ech- having said that the left handside of his equation is “like elons of unification as the most favoured/promising the- marble and the right handside is like wood” and that he ories. Although a lot of work is still to be done as men- found wood so ugly that his dream was to turn wood tioned in the penultimate of the introduction section that into marble. These feelings of Professor Einstein against two volumes each spanning about 200 pages are at an his own GTR are better summed up in his own words advanced stage of preparation, in our most humble and in a letter to Father Georges Lematre (1894 − 1966) the modest opinion, we believe that the present attempt is Belgian Roman Catholic Priest on September 26, 1947: a significant contribution to the endeavour of the unification program of physics. We are of the strong opinion “I have found it very ugly that the field equation that the reader should be their own judge. We shall state should be composed of two logically independent a few main points that lead us to believe that the present terms which are connected by addition. About attempt is a significant contribution to the endeavour of justification of such feelings concerning logical the unification program of physics. simplicity is to difficult to argue. I can not help First, we know that in Riemann geometry and as-well to feel [sorry] and I am unable to believe that such in the GTR, the metric tensor is described by 10 differ- an ugly thing should be realised in Nature.” ent potentials and in-turn these potentials describe the We consider that our ideas of a UFT have here been gravitational field. On the RHS, the metric tensor is formulated. What lies ahead is to determine the precise described by just 4 different potentials and not 10 and nature of the theory via a rigours mathematical study these potentials describe both the gravitational, electro- against the background of our experience. The theory magnetic and nuclear and sub-nuclear fields. At the very certainly contains tensor equations and the quantum the- least, this surely is a paradigm shift which is a much ory of the Standard Model is a part of it. more simpler model of spacetime and matter that the We could like to close with the words from the great most promising models! The RHS on which the present Professor Einstein: theory is founded, is different from Riemann spacetime in that the unit vectors of the RHS are variable at all “In light of knowledge attained, the happy points on this continuum. This property of the RHS that achievement seems almost a matter of course, it has variable unit vectors make it fall into the category and any intelligent student can grasp it with- of Weyl’s brilliant but failed geometry, but the RHS is out too much trouble. But the years of anxious different because the affinities are tensors; the meaning searching in the dark, with their intense longing, of which is that vectors do not change their length and their alterations of confidence and exhaustion and angles under parallel transport. the final emergence into the light – only those who We note that while the gravitational force has here have experienced it can understand it.” been brought under the same roof with the nuclear forces, it is not unified with these forces in a manner Pro- In addition to the depth of Professor Einstein’s words fessor Michael Faraday had hoped, i.e., it being inter- which are sure born out of true experience, we must has- convertible to other forces just as the electronic and mag- ten to add that in the pursuit of new knowledge, one netic forces are inter-convertible into one another. It is aught to have the necessary ‘iron spine’ to put up, and not even what Professor Einstein had envisaged in his ultimately stand in the way of rebuke and discourage- 1929 interview with the journal Nature when he said[75] ment from seemingly established professors (and eminent “Now, but only now, we know that the force which personalities), who, on their usual course or discourse, moves Electrons in their ellipses about the nuclei vehemently reject the new in-favour of maintaining the of the atom is the same force which moves our old, for it is about the only thing they have come to Earth in its annual course about the Sun, and it know and are thus only prepared to have that of old – 25 and that alone; as the only true and purest wisdom of has herein been achieved via the attainment of the the World. The foremost duty to all of humanity – i.e., tensorial affinities. The resulting geodesic equation past, present and posterity; of the young, curious and has Lorentz terms for both the electrical and gravita- the searching mind, is to vehicle and usher in new ideas tional phenomenon. Besides, it has some extra new terms that need to be explored. This may very well procured from the lofty vineyard of the purest and finest explain some of the gravitational anomalies in the Solar of their thoughts. Most at times, this rebuke and dis- system and as-well that of the flat rotation curves of spi- couragement springs from pure prejudice. Whatever the ral galaxies which has lead to the darkmatter hypothesis. reason or reasons may be, the young, curious, zestful and searching mind must never tire, yield, give-up nor give- (5). Gravitomagnetism which emerges from Professor in, but continue un-diverted on its course, destiny and Einstein’s GTR as a first order approximation emerges path – leading to newer depths; knowing that persistence in the present via Professor JoséHera [41]’s existence theory as an exact physical phenomenon. The emergent – sure – yields its rewards, when one refuses to quit! gravitomagnetic phenomenon is exactly that which was envisaged by Professor Maxwell [28] and Dr. Heaviside [43, 44] many score years ago. CONCLUSION (6). Professor’s Einstein’s gravitational waves which are – Assuming the correctness or acceptability of the ideas to first order approximation; vibrations of the Riemann metric g and were predicted in 1916 at the inception of presented herein, we hereby make the following conclu- µν his GTR and have so far escaped direct detection by the sion: finest instruments ever built; these gravitational waves may – sadly – not exist if the present theory stands the (1). It is our feeling that a unified field theory of all the ruthless test of experience. Instead, Maxwell-Heaviside forces of Nature may very well have been achieved in Gravitomagnetic waves analogous to electromagnetic the present instalment. Despite the plethora of work waves is what may exist, should the present prove that lies ahead, we can – in the meantime hope that our worthwhile. readers will see light in what we have presented and that they may take part in the completion of the theory. We (7). The present theory is free from singularities for as (`) are certain that the completion of this theory can not long as % and Aµ are non-singular. Actually, the be the effort of a single mind, but that of many minds (`) present theory very readily provides for % and Aµ to working together. be non-singular for all conditions of experience and existence. If this is the case, then, Professor Einstein’s (2). The electrical and gravitational phenomenon have black-holes which have so far escaped direct detection, herein not been unified in a manner as was initially may not exist. Yes, regions of extreme curvature are envisaged by a great many researchers e.g. notably predicted to exist by the theory, but this will not lead to Professor Faraday and Einstein; were a reciprocal action singular black-holes, but to regions of extreme curvature. and interplay between the two fields has been highly anticipated. No, the two fields here seat harmoniously (8). All the particles in Nature are predicted by the present side-by-side in a quasi-independent manner on the theory to have a non-zero mass. This includes the RHS. We should say that, we also expected the final photon and the neutrino – these must have mass, just theory to give a reciprocal action and interplay between as is the case with all other particles in Nature. This the electrical and gravitational phenomenon, but alas! aswell includes all gauge bosons. Nature has Her own ways; this is not the case here and we can not force it to happen but rather accept what (9). It should be stated clearly that, besides the fact that is on the table. If anything, we can only measure the Professor Proca’s theory of massive gauge fields yields present theory against the background of our experi- results that are in complete resonance with experience to see if it has any bearing and correspondence ence; the fundamental theoretical justification of this with physical and natural reality has we come to know it. modification of Maxwellian Electrodynamics does not exist – actually no clearly visible effort in the available (3). A tensor affine theory that includes all the known force literature has been made to seek the fundamental fields of the World has here been achieved on a four theoretical justification for the Proca theory. The dimensional continuum of spacetime without the need present effort may very-well be the required justification for extra dimensions. In relation to present efforts that for this brilliant but seemingly ad hoc modification of employ extra dimension (i.e., string and string-related Professor Proca. If this is the case that the present theories) and stand at the highest echelons of promise is a fundamental theoretical justification for the Proca as the most promising unified theories, the attainment theory, then, the present effort would be a significant here of a unified theory which includes all the forces of step forward in the understanding of the fundamental Nature on a four dimensional continuum of spacetime, theoretical origins of the Proca theory. is but a tremendous simplification. If correct, it is sure a great leap forward. (10). As demonstrated in the reading [4], the present unified theory yields not only the Dirac equation, but the general (4). A geodesic equation that does not suffer the set back spin Dirac equations in curved spacetime together with of needing preferred coordinate and reference systems 26

the fundamental theoretical origins of the Dirac equation APPENDIX: THE γ˜-MATRICES as a condition for smoothness of the RHS. We consider this to be one of the most immediate and most outstand- ing achievements of the present theory. This we believe must give credence to the theory that, perhaps, it most 0 0 I2 1,2,3 0 σk certainly contains in it, a grain and an element of truth γ˜ = , γ˜ = (117) I2 0 σk 0 and is thus worthy of one’s valuable time to consider as a candidate unified theory. I 0 σ 0 γ˜4 = 2 , γ˜5,6,7 = k (118) 0 I2 0 σk RECOMMENDATIONS 1 I I 1 σ σ γ˜8 = √ 2 2 , γ˜9,10,11 = √ k k Once again, assuming the correctness or acceptability of 2 I2 −I2 2 σk −σk the ideas presented herein, we hereby put forward the (119) following recommendations: 12 1 I2 −I2 13,14,15 1 σk −σk (1). In-order to obtain the full quantized version of the γ˜ = √ , γ˜ = √ 2 −I2 −I2 2 −σk −σk present UFT, there is need to apply to it [present (120) 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. Electronic address: [email protected], [email protected] (2). There is need to consider gravitomagnetism as a fully [1] H. K. H. Weyl, Preuss. Akad. Wiss 26, 465 (1918). fledge science and not a first approximation of general [2] H. K. H. Weyl, Z. Phys. 56, 330 (1927). relativity. We anticipate that such as approach must [3] H. K. H. Weyl, Proc. Nat. Acad. Sci. 15, 323 (1927). resolve some of the existing gravitational anomalies such [4] G. G. Nyambuya, http://www.vixra.org 1, 1 (2014). that the secular drift of the Earth-Moon system [54, 55] [5] C. E. Sagan, Cosmos (Random House, New York, 1980), from the Sun and the drift of the Moon from the Earth p. 4, 1st ed., ISBN 0394502949. [56, 57] and as-well the observed flyby anomalies [58, 59] [6] H. F. M. Goenner, Living Reviews in Relativity 7 (2004). and even darkmatter [60–63]. Actually, we have made [7] H. F. M. Gönner,Living Reviews in Relativity 17 (2014), preliminary moves toward this end and, it appears to us URL http://www.livingreviews.org/lrr-2014-5. that this may help solve some of these anomalies. [8] G. G. Nyambuya, J. Mod. Phys. 4, 1 (2014). [9] A. Afriat (2008), pp.1-17. (arXiv:0804.2947v1 ). (3). Further, there is need to consider the RHS with in [10] A. D. Alhaidari and A. Jellal (2014), pp.1-8. mind the task of putting it on a firm much more clear (arXiv:1106.2236v3 ). mathematical footing. [11] M. Arminjon and F. Reifler, Brazilian Journal of Physics 43, 64 (2013). (4). Furthermore, there is end to consider the geodesic [12] M. Arminjon and F. Reifler, Brazilian Journal of Physics equation to see what is meaning is with regard to 40, 242 (2010), arXiv:0807.0570. electromagnetism, gravitation and quantum mechanics. [13] M. D. Pollock, (Moscow, Russia) Acta Physica Polonica B. 41, 1827 (2010). (5). Finally, we can not at this point say what kind of [14] M. Arminjon, Foundations of Physics 38, 1020 (2008). picture the present UFT projects of the Universe, [15] V. A. Fock, Z. Phys. 57, 261 (1929). however, one thing appear clear, its picture seem to has [16] G. G. Nyambuya, Foundations of Physics 37, 665 (2008). signatures of the present world. There is thus need to [17] P. A. M. Dirac, Proc. Roy. Soc. (London) A117, 610 put the theory on a much firmer footing by checking all (1928a). the mathematics here laid down. [18] P. A. M. Dirac, Proc. Roy. Soc. (London) A118, 351 (1928b). [19] G. G. Nyambuya, Apeiron 16, 516531 (2009). Acknowledgments: We are grateful to the National [20] G. G. Nyambuya, J. Mod. Phys. 4, 1050 (2013). University of Science & Technology (NUST)s Research [21] G. G. Nyambuya, In Review 1, 1 (2013), URL http: & Innovation Department and Research Board for their //vixra.org/abs/1304.0006. unremitting support rendered toward our research en- [22] H. Stephani, Relativity: An Introduction to Special and deavours; of particular mention, Prof. Dr. Mundy, Dr. General Relativity (Cambridge University Press, New York, USA, 2004), 3rd ed. P. Makoni and Prof. Y. S. Naiks unwavering support. [23] H. A. Lorentz, Arch. Nèerl.Sci. 25, 287 (1892). This paper is dedicated to my mother Setmore Nyambuya [24] L. Lorenz, Philos. Mag. 34, 363552 (1867), reprinted in and to the memory of departed father Nicholas Nyam- H. A. Lorentz, Collected Papers (Martinus Nijhoff, the buya (1947 − 1999). Hague, 1936) Vol. II, pp. 164-343. 27

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