1

REALIZATION OF A NEW CONFORMAL SYMMETRY GROUP FOR THE GRAND UNIFIED THEOREM IN PROJECTIVE SPACE-TIME GOEMETRY

By

A.O.E. Animalu The Institute for Basic Research

4A Limpopo Street, FHA, Maitama, Abuja &

Department of Physics and Astronomy University of Nigeria, Nsukka

e-mail: nascience@aol.com

ABSTRACT A review is presented of the construction of various realizations of the conformal group in space-time Minkowskian (pseudo-Euclidean) geometry of special relativity theory, Riemannian (differential) geometry of Einstein’s general relativity theory as well as other subtly different (especially Maduemezia’s de-Sitter group approach to) Riemannian theories of gravitation, and in classical and quantum field theories, in order to bring them within the purview of Oyibo’s Grand Unified Theorem (GUT) based on a realization of Oyibo’s new group of conformal transformations in terms of the invariance of the cross-ratio of the space-time (or energy-momentum) coordinates in projective geometry of space-time (or 4-momentum space) under bilinear (also called homographic) transformations. The implications for the generic equations of the Grand Unified Theorem(GUT) and their characteristic solutions as a viable mathematical framework for a theory of everything is discussed.

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1. STATEMENT OF THE PROBLEM Intensive study of large space-time symmetry groups, like the SO(4,2) conformal (including scale) symmetry group of transformations containing the )4()1,3( TSO o Poincare group, a direct product of the usual SO(3,1) Lorentz group and the )4(T Abelian group of translations in the Minkowskian (pseudo-Euclidean) space-time geometry of special relativity theory, was stimulated in late 1960s as a result of the observation of scaling in the MIT-SLAC deep-inelastic electron-proton scattering experiments at high energies. The reason for this interest, according to MIT Professor Roman Jackiw[1] in his 1970 Physics Today article entitled Introducing Scale Symmetry, was that “when symmetries are present, the solution of any physical problem is simplified, because we can get at the properties of a system without completely solving all the equations that describe the system”. Also in the 1960s, a conformal group formalism has been successfully developed, under the name “dynamical group” methods, for such diverse problems as electromagnetic interactions, the hydrogen atom, and hadron properties[2] and in diverse geometries ranging from metrical (Minowskian and Riemannian) geometries to non-metrical (Projective, also called Descriptive) geometries[3]. It is no wonder then that considerable interest has been aroused by Oyibo’s new characterization of the conformal group of transformations[4], which can be abstracted from the Navier-Stokes momentum equations, in the proof of a Grand Unified Theorem (GUT) for unification of the four basic (gravitational, electromagnetic, weak and strong) forces and possibly a fifth unknown force in nature. In the 1995 paper entitled, Generalized Proof of Einstein’s Theory Using A New Group Theory[4a], referred to in his 2001 published book [4b], a function,

),...,,( 21 pYYYGG = (1.1) is said to be conformally invariant under a given group transformation

),,...,,(: 21 kyyyfYT piik = (1.2) if kT is the group of the transformation and

),...,,(),,...,,(),...,,( 212121 ppip yyyGkyyyFYYYG •= (1.3)

where ),,...,,( 21 kyyyF pi is a function of iy and k the single group parameter. On the basis of this definition of conformal symmetry, Oyibo derived a set of five “generic” equations of the GUT,

).4,3,2,1,0( ,0)()()()( 3210 ==+++ nGGGG znynxntn (1.4)

and found their characteristic solutions in terms of the absolute invariant nη of the subgroup of transformations for the independent coordinate variables characterized by the relation

zyxctZYXT nnnn :::::: = (1.5) in the form:

13

12

11

10 )( ++++ +++= n

nn

nn

nn

nn zgygxgctgη , (1.6) where mng are components of a metric tensor. Since Eq.(1.5) is equivalent to a definition of scale transformation (also called dilatation) in 3-dimensional projective space based on four homogeneous coordinates ),,,( zyxct , such a (scale) transformation is usually included in the characterization of any conformal group of transformations. Consequently, by virtue of Noether’s theorem, 0η could be interpreted[5c] as the

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time-component of the conserved dilation current, λµλµ Θ= xD , where λ

µΘ is the energy-momentum tensor for a classical or quantum field. An outstanding contentious problem with the above definition is the construction of an explicit function G with an explicit relationship between iy and iY as well as physically consistent interpretation of the absolute invariant nη for 1≥n . This problem will be tackled in Sec. 2 as a prelude to a review of other contending definitions of the conformal group of transformations (in Sec. 3), such as the one given by Maduemezia[6a] in his 1999 published article entitled, De-Sitter Group Approach to the Theory of Gravitation [6b] which was based on imbedding curved space-time of n-dimension in a flat (n+m) de-Sitter space, involving the chain of groups beginning from the SO(4,2) conformal group through the SO(4,1) de-Sitter group and the )4()1,3( TSO o Poincare group to the SO(3,1) Lorentz group:

)1,3()4()1,3()1,4()5()1,4()2,4( SOTSOSOTSOSO →→→→ oo , (1.7) where )(nT is an Abelian translation group of translations in n-dimensions. The problem in the review may be stated as two questions [6a] as follows: 1. How does the imbedding of a curved 4-dimensional space with the Schwarzschild

metric in a flat 6-dimensional space (employed in ref.[6b]) fall within the purview of Eqs.(1.1) to (1.3)?

2. Since Einstein’s general relativity theory is neither a “correct” theory of gravity nor a “generalization” of the Special Relativity Theory of light (electromagnetic wave) propagation in vacuum per se and is not compatible with quantum mechanics and the gauge principle, what does Oyibo’s proof of Einstein’s Theory

)( 2mcE = of Special Relativity using a New Group Theory defined as above have to do with unification of gravity with the other three (weak, electromagnetic, and strong) forces in nature vis-à-vis the success and experimental verification of the standard (gauge) model?

The first of these two questions will be answered in Sec. 3, while the second will be answered in Sec. 4 where realizations of the solutions of the GUT equations given in Eq.(1.6) will be discussed. The implications of the answers for the generic equations of GUT and their characteristic solutions as a viable mathematical framework for a “theory of everything” will be discussed and conclusions drawn in Sec. 5.

2. CONSTRUCTION OF SO(4,2) CONFORMAL GROUP IN PROJECTIVE SPACE-TIME AND ENERGY-MOMENTUM SPACE.

2.1 Mathematical Formulation The modern concept of projective geometry[7] is that n-dimensional space projective geometry over the complex number field, Pn(C), is the study of properties of geometrical objects – points, lines, planes, quadric surfaces, etc – which are invariant under the group of projective transformations. For n=3, the full projective group (p. 354 of ref.[7a]) consists of the Euclidean group characteristic of Euclidean geometry, the group of bilinear (also called homographic) transformations, collineations (point-to-point and plane-to-plane transformations), correlations (points-to-planes and lines-to- lines), as well as birational (also called Cremona) transformations which are point-point and one-to-one and are associated with the operation of inversions with respect to a quadric surface. But we shall be concerned primarily with explicit realizations of Eq.(1.3) and SO(4,2) conformal invariance in P3(C) from the subgroup

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of bilinear transformations which preserve the cross-ratios of the four homogeneous coordinates, ),,,(),,,( 3210 xxxxzyxct ≡ , and of the corresponding energy-momentum homogeneous coordinates, ),,,(),,,/( 3210 pppppppcE zyx ≡ , associated with the objects,

22

22

21

20

2 xxxxxxgs −−−== νµµν ; 2

222

21

20

2)( ppppppgmc −−−== νµµν . (2.1)

and the cross-ratios of the four µx ’s and four µp ’s are defined by (p.36 of ref.[7b])

))(())((

),;,(2130

31203210 xxxx

xxxxxxxxG

−−−−

≡ ; ))(())((

),;,(2130

31203210 pppp

ppppppppG

−−−−

≡ . (2.2)

These cross-ratios are invariant under the group of bilinear (also called homographic) transformations defined by

,2221

1211

axa

axaX

+

+=

µ

µµ )0( 21122211 ≠− aaaa ; ,

2221

1211

bpb

bpbP

+

+=

µ

µµ )0( 21122211 ≠− bbbb , (2.3)

which include the scale transformation, 0)( , ≠= ρρ µµ xX ; )0( , ≠= κκ µµ pP , (2.4)

i.e., ),;,(),;,( 32103210 xxxxGXXXXG = ; ),;,(),;,( 32103210 ppppGPPPPG = . (2.5)

Now, we may rewrite one of the two cross-ratios defined in Eq.(2.2),

Λ−=−−−−

≡))(())((

),;,(2130

31203210 xxxx

xxxxxxxxG say, (2.6a)

in the form 0))(())(( 21303120 =−−Λ+−− xxxxxxxx . (2.6b)

which can be interpreted in two ways. On one hand, by completing squares, Eq.(2.6b) can be rewritten as sum of squares

νµµν xxXXXXXXX −≡=+−+Λ−+Λ+ with ,0)()())(1( 212

203

231

202

223

201 , (2.6c)

which (for 1−≠Λ ) is manifestly SO(4,2)- invariant. On the other hand, by virtue of the geometric principle of duality, the object (“quadric”) defined by Eq.(2.6b) can be characterized by its two sets of generators, iu and )1,0( , =ivi , such that if, under the transformation in Eq.(2.3), we have the following induced homographic transformation ( [p. 355 of ref.[7a])

( ) ( )

( ) ,)(

)(,

)()(

, )(

)(,

1

0

2221

1211

3130

2120

2212

211110

1

0

3130

212010

1

0

3130

212010

−−

−Λ−−

≡

−−

−Λ−−→

−−

−Λ−−

uu

AAAA

xxxx

xxxx

FFFF

vv

uu

XXXXXXXX

vvuu

xxxxxxxx

vv

and we select ,12221

1211 =

AAAA

then we have the “left collineation”

−−

−Λ−−

≡

−−

−Λ−−

3130

2120

2212

2111

3130

2120

)()(

)(

)(xxxx

xxxxFFFF

XXXXXXXX

, (2.7a)

which permutes only the v-generators, and may be rewritten in the form ),...,(),,...,(),...,( 303030 xxGxxFXXG

))∗Λ= . (2.7b)

This provides an explicit realization of the transformation law envisaged in Eq.(1.3). And since any general square matrix can be written as a sum of its symmetric and antisymmetric parts, the interest of the GUT lies (as in our 1996 published paper entitled Lie-Santilli Iso-Appraoch to the Unification of Gravity and Electromagnetism[5a]) in realizations of Eq.(2.7b) in which ),...,( 30 xxG

) is symmetric

and ),,...,( 30 ΛxxF antisymmetric, so that ),...,( 30 XXG)

is their “Lax pair”.

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Mathematically, the meaning to be attached to the cross-ratio when one of the numbers, say 0x , is infinite is easily found by replacing 0x by a fraction and putting the denominator of the fraction equal to zero after reduction. Thus, in particular,

,),;,(21

31321 xx

xxxxxG

−−

≡∞ 33 ),1;0,( xxG =∞ . (2.8)

Moreover, by permuting the four numbers 3210 ,,, xxxx in Eq.(2.6a), one would find 241234!4 =×××= cross ratios, but only six of these are independent, namely on

putting KxxxxG =),;,( 3210 , the following (p. 38 of ref.[7b]): KxxxxG =),;,( 3210 ; KxxxxG /1),;,( 2310 = ; KxxxxG −= 1),;,( 3120 (2.9)

)1/(1),;,( 1320 KxxxxG −= ; KxxxxG /11),;,( 2130 −= ; )1/(),;,( 1230 −= KKxxxxG

2.2 Physical Interpretation For physical interpretation, let us evaluate the cross-ratio defined, as in O(4,2) dynamical group methods[2] by the following 4-momentum of a particle:

)sinh,(cosh),,,( 3210 ξξξr

mcpppp ≡ , (2.10a) where ξ

r is characterized in P3(C) as follows,

),1,1,)(/( −= icvξr

1−=i , so that, explicitly,

( ))/sinh(),/sinh(),/sinh(),/cosh(),,,( 3210 cvvcvvcvivcvcmpppp +−≡ (2.10b) and the cross-ratio of the four µp ’s is:

λ≡=−−−−

≡ )/2exp())(())((

),;,(2130

31203210 cv

pppppppp

ippppiG say. (2.11a)

This defines a “ refractive index” ( vcn /≡ ), of the medium in the form: ( )iGcvn ln//1 2

1=≡ . (2.11b) But if we suppose that rv 2 ω= , and put kc =/ω , then by virtue of the relations in Eq.(2.9), we infer that the cross-ratio,

01320 /)(1

11

1),;,( UrUe

ppppiGkr

≡−

=−

≡λ

, (2.12a)

characterizes a Hulthen potential, )(rU , which satisfies Riccati’s nonlinear differential equation (p. 201 of ref.[8], and Appendix B of ref.[5c])

RQUPUdrdU ++= 2 (2.12b)

where, 0U , QP , and R are constants (independent of r). The well-known fact (p. 202 of ref.[8]) that the general integral of Riccati’s equation is a homographic function of the constant of integration is now seen to be consistent with the invariance of the cross-ratio ),;,( 1320 ppppG of the µp ’s under homographic transformations. Indeed, returning to Eq.(2.11b), we may characterize (for propagation of light in a medium) the variation of the refractive index due to the dependence of c and v on the temperature , T say, of the medium by a different ial refractive index given by C.N, Animalu and A.O.E. Animalu[9]

cvcv

dvdc

2

22 −= (2.13a)

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which can also be separated into a pair of coupled differential equations,

( ),22 cvKdTdc −= (Riccati’s equation) ; Kcv

dTdv 2= , (2.13b)

where K is a constant. In addition to the recurrence of Riccati’s equation for c(T) in Eq.(2.13b), we can integrate Eq.(2.13a) exactly to find :

vcvc33

30

22 += , or 3

0323 cvvc += , or 3

032

0 )()(3 tcrtcr += with trv /= , (2.14)

where 30c is a constant. We shall return to this solution in Sec. 4.3.

Furthermore, for motion of a fluid particle in vacuum, we may (keeping c constant in Eq.(2.11b)) characterize the variation of n due to the dependence of v on the time, t, via the Navier-Stokes momentum equation in one dimension (x) plus time (t), given (for xvv = ) by C.N. Animalu[10] (see, Appendix A below):

2

2)(

xvB

xp

tvv

tv x

xx

xx

∂∂++

∂∂−=

∂∂+

∂∂ µρρ

r (2.15)

Thus, by putting

∑∞

=

=≡+∂∂

−0

)~

exp()(k

kkxx xkitdDBxp r

, and ∑∞

=

=0

)~

exp()(k

kkx xkitav (2.16)

Eq.(2.15) becomes

).()~

()()()(~

2)()()( 2

1

10 taktdtatakitataik

dttda

kk

k

kkkk µρρ +−

++ ∑

−

=−

lll (2.17)

Consequently, for 0=k , this states that )(0 tav = satisfies Riccati’s equation

( ) 0)()()~()(~)(00

20

200

0 =−++ tdtaktakidt

tda µ . (2.18)

Since dtda /0 represents acceleration, this equation is a key to the unified force field concept of GUT while (2.12b) is a key to the unified potential field concept in ref.[5c]. It is no wonder then that the five generic equations of the GUT originated from Oyibo’s studies [4c] of the five Navier-Stokes equations.

2.3 Representation of Mass as Conformal Invariant in P3(C) Perhaps by far the most significant aspect of the new group is its use to represent the rest mass of a particle as a conformal invariant of the energy-momentum conservation law, expressed in special relativity theory by the well-known relation

23

22

21

20

2)( ppppmc −−−= (2.19) from which Einstein deduced that, in the “rest frame” defined by,

0 23

22

21 =++ ppp , 2

02)( pmc = (2.20)

mass and energy would be equivalent, i.e., .2mcE = We observe that, since the first of the two relations in Eq.(2.20) characterizes a “point sphere” , i.e., a sphere of zero radius, any 3-momentum coordinates of the form in P3(C),

),,2(),,( 321 iMciMcMcPPP −≡ , )1( −=i (2.21) would lie on the “point sphere”, for all values of the real mass, M. Consequently, the invariance of the cross-ratio

),;,())(())((

))(())((

),;,( 32102130

3120

2130

31203210 PPPPG

PPPPPPPP

pppppppp

ppppG ≡−−−−

=−−−−

≡ (2.22)

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under the group of bilinear transformations defined by

,2221

1211

bpb

bpbP

+

+=

µ

µµ )0( 21122211 ≠− bbbb , (2.23)

implies that, if Einstein’s “law” is to remain invariant, i.e., 00 pmcP ≡= , then

( ))(2exp)2)(()2)((

))(())((),;,( 0

12130

31203210 θθ −≡

−++−≡

−−−−= i

iMMiMmiMMiMm

PPPPPPPPppppG (2.24)

where )/(tan 1 Mm−=θ and )2(tan 10

−=θ . This leads to the logarithmic expression,

( )),;,(log21

32100ppppG

i=− θθ .

which, for small θ , such that Mm / tan =≈ θθ , leads to the result

( )),;,(log21/ 3210 ppppGi

Mm = +. )2(tan 1− . (2.25)

This states that: the rest mass (m) of a particle is an invariant of the group of bilinear transformations of the cross ratio of its (conserved) energy-momentum 4-vector. Therefore mass, energy and momentum should be transformable, one into another. For example, at high energy )( 0 ∞→p , it is possible by virtue of Eq.(2.8) to generate mass from 3-momentum as follows:

( ) .log21),;,(log

21/

21

31321

−−≡∞=

pppp

ipppG

iMm (2.26)

3. CHARACTERIZATION OF THE CONFORMAL GROUP VIA IMBEDDING OF A CURVED SPACE IN A FLAT DE-SITTER SPACE.

3.1 Conformal Group of Transformations in Minkowski Space Because of the novelty of the characterization of the SO(4,2) conformal group in the preceding section via the invariance of the cross-ratio of the four coordinates or four momentum in projective geometry, it is necessary to remind the reader of the usual conformal group of transformations in a metric (Euclidean or pseudo-Euclidean) space with coordinates ),,,(),,,( 3210 zyxctxxxx ≡ . The group of conformal transformations contains, in addition to the SO(3,1) Lorentz group of transformations of the quadratic form,

233

222

211

200 )( zgygxgctgxxg +++≡νµ

µν , with )1,1,1,1()( −−−+= diaggµν (2.10) defined by

;' µννµµ axx +Λ= (3.1)

also the dilatation: ;' µµ ρxx = (3.2)

and the special conformal transformation: );21/()(' 222 xfxfxfxx +++= λ

λµµµ (3.3)

The special conformal transformations may be built up from two inversions (I) and a translation (T) in the form ITI, as follows:

( )22

222

/

/ / /

µµ

µµµµµµ

fxx

fxxfxxxxx

ITI+

+→+→→ (3.4)

The conserved dilatation is a scalar for a single particle, λ

λ pxD = , (3.5a) but a four-vector for a classical or quantum field

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λµλµ Θ= xD , (3.5b)

while the conserved conformal current is a four-vector for a single particle ν

µννµµ pxgxxK )2( 2−= (3.6a) but a tensor for a classical or quantum field

λµµλλµµν Θ−= )2( 2xgxxK (3.6b)

where µνΘ is the symmetric energy-momentum tensor of the field. By virtue of Noether’s theorem, conformal (including scale) symmetry implies the following conservation equations for the dilatation and conformal currents,

0, =Θ≡ µµµµD , 02, =Θ≡ µ

µµνµν xK (3.7) which, accordingly, demand the vanishing of the trace of the energy momentum tensor )0( =Θ µ

µ in this pseudo-Euclidean (Minkowskian) geometry. Bearing in mind the above definition of the usual conformal group of transformations, we now proceed in the rest of this section to bring Einstein’s formulation of his general relativity theory (of gravitation) and the subtly different group-theoretic “Riemannian formulation of gravity”[6b] within the purview of the projective geometric approach described in Sec.2.

3.2 Conformal Symmetry of Gravitational Field Theories. According to Einstein’s general relativity theory of gravitation [11], the field tensor of the gravitational field (outside a source) is a symmetric metric tensor defined by

µνµνµνµν κΘ−=−≡ RgRG 21 . (3.8)

which obeys a covariant conservation equation 0, =νµνG . (3.9)

where µνR is the Ricci’s tensor, and R the curvature scalar given in terms of the Riemann curvature tensor µαβνR for the 4-dimensional (curved) space-time geometry endowed with the line element,

233

222

211

200 )()()()( dzgdygdxgcdtgdxdxg +++≡νµ

µν (3.10) by

,αβµαβνµν gRR = µν

µν gRR = (3.11) Thus, conformal invariance of Einstein’s general relativity theory of gravitation would be valid in this metric (Riemannian space-time) geometry, only if both

0=Θ µµ and 0, =Θ νµν hold for the field, which is evidently inconsistent for a massive

field. For this reason, there is a subtle difference between “Einstein’s general relativity theory of gravitation” and the alternative “Riemannian formulation of gravity”[4b] for a massive particle. In the latter, conformal invariance is achieved by imbedding curved space-time of n-dimension in a flat (n+m) de-Sitter space which involves the chain of groups indicated in Eq.(1.7). It is well-known[6c] that the space-time outside a symmetric mass M equipped with the Schwarzschild metric

9

rc

GMZ

rr

ZZ

g2

22

22

1 ;

sin00000000/10000

)( −=

−−

−=

θ

µν (3.12)

with respect to the spherical coordinate ordering ),,,( ϕθrt may be imbedded in flat 4+2 de-Sitter space with a quadratic form

2625242322212 )()()()()()( dzdzdzdzdzdzds −−−−+≡ (3.13) which is invariant under the conformal group SO(4,2) via the construction

rZ

cGM αα −== 1 ;2

2 (3.14)

θϕθϕθ cos ;sinsin ;cossin

)( ;sin ;cos654

321

rzrzrz

rfztZztZz

===

=== (3.15)

where f(r) satisfies the differential equation,

+=

rrZdrdf αα

4

22

4

1 , i.e.,

+±=≡

rrZdrdf

drdz αα

4

23

41 (3.16)

It is thus evident that the definitions of conformal transformations by Eq.(3.15) for curved (Riemannian) space and by Eq.(3.3) for the conventional .(flat) SO(3,1) Minkowski space are irreconcilably different. In particular, the construction of the imbedding defined by Eq.(3.15) involves three disjointed geometrical objects, namely a circle of radius Z in ),( 21 −zz plane ,

;)()( 2221 Zzz =+ (3.17a) and a sphere of radius r in ),,( 321 −zzz subspace

)()()( 2262524 rzzz =++ (3.17b) as well as a curve, )(3 rfz = , representing a branch of the solution of the differential equation (3.16) in ),,( 321 −zzz subspace. The inclusion of an artificial circle in the ),( 21 −zz plane in such a “Riemannian formulation of gravity” was avoided in Maduemezia’s SO(4,1) de-Sitter group approach[6b], by introducing a fifth coordinate g-axis in the usual line element

,)()()()()()()( 22222222 dgacdtdzdydxcdds +−++=−≡ τ (3.18) via an ansatz

−=

rcGM

g s2

21 . (3.19)

But a singularity persists on a sphere of radius, 2 2cGMr == α defined by Eq.(3.17b) where the circle defined by Eq.(3.17a) shrinks to a “point circle”

0,)()( 2221 =+ zz i.e., 21 izz ±= , )1( −=i . (3.20) These singular features of the “Riemannian formulation of gravity” signal a transition from metrical (Euclidean and Riemannian) geometries to non-metrical (projective) geometry. Therefore, in order to bring the imbedding approach within the purview of the definition of conformal transformation in Sec. 2, we may, by virtue of the geometric principle of duality in projective space (p. 14 of ref.[7b]), reformulate the conformal invariance of gravitational field theories in terms of the dual cross ratio, which we now proceed to define.

10

3.3 Reformulation in terms of the Dual Cross Ratio. The geometric principle of duality between points and planes in P3(C) may be defined in the following way. Consider a geometric object defined by the relation.

0)( 33221100 =+++ zgygxgctg . (3.21a) For fixed 0:0:0:0::: 33221100 ≠gggg , this equation represents a set of points zyxct ::: lying on a plane, and conversely, for fixed 0:0:0:0::: ≠zyxct it represents a set of planes 33221100 ::: gggg passing through the point. This is an algebraic statement of the fact that “points” and “planes” are duals of each other in P3(C). Accordingly, we may define a dual cross-ratio of the four aag ’s as follows:

))(())((

),;,(~

22113300

3311220033221100 gggg

ggggggggG

−−−−

≡ . (3.21b)

where )cos,,,(),,,( 222133221100 θrrZZgggg −−−≡ − from Eq.(3.12) so that

,tan1cos

1

))(cos()cos)((

))(())((

),;,(~

22

2122

221222113300

3311220033221100

θθ

θθ

+=→

+−++−+=

−−−−

≡

−

−

rZrZrZrZ

gggggggg

ggggG

(3.22)

when 0 →Z at the Schwarzschild radius, 22 cGMr = . This means, geometrically, that G

~ is a measure of an “angular distance” between a set of four concurrent planes through a fixed point of P3(C). Moreover, since Λ−= ~),;,(~

33221100 ggggG say, is invariant under the bilinear (homographic) group of transformations of the ααg , we again infer SO(4,2) conformal invariance of the curved (Riemannian) space of the Schwarzschild solution of Einstein’s equation in the form (cf Eq.(2.6c)):

,0)()(~))(~1( 212

203

231

202

223

201 =Π+Π−Π+ΠΛ−Π+ΠΛ+ (3.23)

where, ννµµµν gg −≡Π (no summation of indices). Furthermore, since, as is well-known (p. 49 of ref.[12]), the Schwarzschild’s “solution”,

[ ] 1/)(21)( −−= rrmrg rr , ∫≡r

drrrrm0

2 )(4)( ρπκ , (3.24)

of Einstein’s equation for the gravitational field is governed by Riccati’s equation:

0)1

8(1 2 =−−− rrrr

rr gr

rgrdr

dgπκρ (3.25)

where ),sin()()( 222222

002 ϕθθ ddrdrrgdrrgds rr +++= (3.26)

and πκρ16 is the scalar curvature, ρ being the (curvature) energy density, we have the same level of consistency of Eq.(3.21b) with the homographic properties of solutions of Riccati’s equation indicated in Sec. 2.

3.4 Reformulation in terms of Plucker’s Line Coordinates. For completeness, we turn next to a reformulation in terms of the Plucker’s line coordinates (p. 16 of Ref.[7b]), the line being a self-dual geometric object in P3(C). We recall from Eq.(2.2) that the cross-ratio of the four µx ’s is given by

11

1203

1302

2130

31203210 ))((

))((),;,(

ffff

xxxxxxxx

xxxxG =−−−−

≡ (3.27)

in terms of the νµµν xxf −= , (3.28)

which may be expressed as components of an antisymmetric tensor,

)3,2,1,0,( ),( =−= νµµννµµν exexF , (3.29) i.e.,

( ) ,)(),(),(),,( 302010030201 ExxxxxxFFFr

≡−−−= ( ) HxxxxxxFFF

r≡−−−= )(),(),(),,( 132132311223

where 1:1:1:1::: 3210 ≡eeee defines the so-called “unit point” of P3(C). The six independent components of µνF define Plucker’s coordinates of the line joining µx to µe and obey the identical relation

0. =≡ HEFFrr

σρµνµνσρε , (3.30)

where µνσρε is the usual pseudoscalar metric tensor which is +1 if ( µνρσ ) is an even permutation of (0123) , -1 if an odd permutation, and zero if any index is repeated. Eq.(3.30) is analogous to the transversality of the electric and magnetic field vectors in electromagnetism. Moreover, in terms of the µνF , the Euclidean expression

2sxxg =νµµν , i.e., 22

322

21

20 sxxxx =−−− (3.31a)

can be associated with the corresponding (dual) expression 222 BEFFRS

rr−≡= νρµσ

µνρσ , (3.31b) where νσµρρσµνµνρσ ggggR −= has the structure of the Riemann tensor for a space of

unit constant (including zero) curvature. Geometrically speaking, for 02 =s Eq.(3.31a) is the point-equation of a light-cone whose tangential line-equation is given by Eq.(3.31b) for 02 =S , in accordance with the geometric principle of duality, as shown in Fig. 1 below.

Physically, since Eq.(3.31b) has the structure of the “Lagrangian” of a free electromagnetic field and is manifestly SO(3,3) conformal invariant in the six line coordinates, µνF , it is apparent that such an invariant group of the “Lagrangian” provides complete information about the system.

Fig.1: Dual representation of the light “cone” as a locus of points or an envelope of its tangent lines.

12

We may, therefore, summarize what has been accomplishment in this section by saying that we have brought the conformal invariance of gravitational field theories within the purview of the new group of conformal transformations in terms of the cross-ratios of the four “point” coordinates ( µx ), and the four (dual) “plane” coordinates ( µµg ), and found a mathematical realization of their SO(4,2) conformal invariance in terms of the six “line” coordinates ( µνF ) resembling, in geometrical terms, the antisymmetric tensor of the free electromagnetic field.

4. CONFORMAL INVARIANCE OF THE GUT EQUATIONS AND THEIR SOLUTIONS

4.1 Introduction In view of the foregoing analysis, let us proceed finally in this section to address Maduemezia’s question [6a] as to what Oyibo’s proof of Einstein’s relation )( 2mcE = in Special Relativity using a New group Theory has to do with grand unification of gravity with the other three (weak, electromagnetic and strong) forces in nature, vis-à-vis the success and experimental verification of the standard (gauge) model. We shall first reformulate the conformal invariance of the generic equations (1.4) of the GUT and their characteristic solutions in Eq.(1.6) in terms of cross ratios of the appropriate “coordinates” of the relevant geometric objects in P3(C), and construct explicit realizations of the solutions as a hierarchy of relativities before turning to exhibit a linkage of the GUT with the gauge principle.

4.2 Formulation in Terms of Cross Ratios Having formulated conformal invariance of gravitational field theories in P3(C) in terms of the cross-ratios of the “point” coordinates ( µx ) and the dual “plane” coordinates ( µµg ), it is necessary to take the five generic equations (1.4) of the GUT and their solutions in Eq.(1.6) as given and similarly prove their conformal invariance in terms of the relevant cross ratios. For the four “field” components,

znynxntn GGGG )(:)(:)(:)( 3210 in Eqs.(1.4), the cross-ratio is given by

( ) ( )( )( )( ) 4,3,2,1,0for ,

)()()()(

)()()()())(,)(;)(,)(ˆ

2130

31203210 =

−−

−−≡ n

GGGG

GGGGGGGGG

ynxnzntn

znxnyntnznynxntn (4.1)

and for the solutions in Eq.(1.6) the cross-ratio of the four “point” coordinates, 1

31

21

11

0 ::: ++++ nnnn xxxx , is given by

.4,3,2,1,0for say, ))(())((),;,(ˆ

12

11

13

10

13

11

12

101

31

21

11

0 =Λ−=−−−−≡

++++

++++++++ n

xxxxxxxxxxxxG nnnnn

nnnnnnnn . (4.2)

while the cross ratio of the four dual coordinates, 3210 ::: nnnn gggg , is given by

4,3,2,1,0for ,))(())((

),;,(ˆ2130

31203210 =

−−−−

≡ ngggggggg

ggggGnnnn

nnnnnnnn (4.3)

Consequently, a proof of the conformal invariance of the equations for each 0=n say, follows by rationalizing the expression in Eq.(4.2) to get Eq.(2.6b), i.e.,

0))(())(( 213003120 =−−Λ+−− xxxxxxxx (4.4a) and rewriting it, on completing squares, as sum of squares

νµµν xxXXXXXXX −≡=+−+Λ−+Λ+ with ,0)()())(1( 212

203

231

2020

223

2010 , (4.4b)

which (for 1−≠Λ n ) is manifestly SO(4,2)- invariant.

13

From a geometrical point of view, there is no formal difficulty in extending a similar argument for 0=n to 1≥n . However, inasmuch as there is apparently no restriction, except conformal invariance, on the ,)( 1 tnG …, and µng ,…, the challenge is to construct explicit realizations of the characteristic “solutions” relevant to the problem of grand unification of gravity with the other basic forces in nature. This is where the geometric principle of duality and its implementation in Eq.(2.7a & b) via Lax pairing of symmetric and antisymmetric tensors[5a] come into play, as we now proceed to show.

4.3 Construction of the Solutions via the Lax Pairing Method We have stated in Sec. 2.1 that equations of the type (4.2) can be rationalized in the form,

,0))(())(( 12

11

13

10

13

11

12

10 =−−Λ+−− ++++++++ nnnn

nnnnn xxxxxxxx for 4,3,2,1,0=n (4.5)

which, for 0=n , is equivalent to a quadric defined by Eq.(4.4a) and can be expressed either as sum of squares as in Eq.(4.4b) or, by virtue of the geometric principle of duality, as a matrix equation for a set of generators, 10 :uu ,

0)(

1

01

21

11

31

0

13

11

12

10 =

−−−Λ−−

++++

++++

uu

xxxxxxxx

nnnn

nnn

nn

. (4.6)

In order to reach the Lax pairing concept introduced in connection with the matrix representation in Eq.(2.7a & b), let us consider the problem discussed in ref.[5b] of defining a rectangular hyperbola, 0)())(( 2 =−+−+ ctyxyx , which is analogous to Eq.(4.5) for 0=n by the envelop of its pair of asymptotic line genartors (see, Fig. 2) defined by the linear homogeneous equations:

,02

1 =

+−−

−+ww

yxctctyx

or 0})~

()({ 303 =++ νµνµνµν δσβσ wyctx . (4.7)

This involves a symmetric matrix ( 3σ ), and its Lax pair ( 3~σβ ) with an antisymmetric

matrix ( 0~β ) given by

−

−=

−

=

−

=0110~

,0110~

,10

01303 σββσ . (4.8)

Fig. 2. Rectangular hyperbola as envelope of its asymptotic (tangent) lines .

14

The Lax pairing (transformation) of the “metric” tensor )( 3σ≡g and an antisymmetric tensor ( 0

~β≡f )

fggg =→ ~ , (4.9) is now seen as an explicit realization of Eq.(2.7b) which is required for unification of symmetric (gravitational) and antisymmetric (electromagnetic) tensor field in the framework of the GUT. As a method of constructing realizations of the characteristic solutions of the generic equations of the GUT, let us now consider a rather intriguing case of a symmetric

33× matrix (g) given by the electric charge matrix ) ( qQ of the Gell-Mann Zweig quark triplet (u,d,s) and an antisymmetric 33× matrix (f) (which is non-unitary) that transforms it into the charge matrix ( lQ ) of the corresponding lepton triplet ),,( −− µν e

100

010000

030300

000

00

0000

030300

000

31

31

32

lQfQf q ≡

−−=

−

−

−

+−≡+ (4.10)

and the set of homogeneous equations, )3,2,1,( ,0])([ ==++ bawctfgsrg bababab δ . (4.11)

This involves g and f and their “Lax pair” fg given by

,

00

0000

31

31

32

−

−=abg ,030300

000

−=abf

−=

00

00000

)(

31

31

abfg ; (4.9)

so that Eq.(4.11) has an explicit form analogous to Eq.(4.7):

0

0

0

00

3

2

1

33

33

32

=

+−

−+−

+

www

ct

ct

ct

rs

sr

r

(4.10)

The vanishing of the secular determinant gives ( ) 0)( 3

233

2 2=

++−+ srr ctct (4.11)

which reduces (for s=0) to an analog of Eq.(2.14): 33

2722 )()(

31 ctrctr += , with 222 zyxr ++= . (4.12)

We regard this as a realization of Eq.(1.6) for the 2=n hierarchy. By virtue of the geometric principle of duality, Eq.(4.12) can be rewritten as done in catastrophe theory[13] in the form:

0)()( 3 =++ ctctYX . (4.13) This means that for fixed r, Eq.(4.13) represents a line in (X,Y)-space which is normal to a parabola parametrized by ct in such a way that any point on the parabola has coordinates ( )2)(, ctct (p.78 of ref.[13]) and the envelope of the normal (as ct varies) is a semi-cubical parabola (called canonical cusp catastrophe) defined by the discriminant equation:

027 32 =+ YX (4.14) This represents the involute of the parbola (shown in Fig. 3a) . The corresponding involute of an ellipse is an hypocycloid shown in Fig. 3b and can be produced as “light caustics” formed by refraction of a laser beam in a periodically grated piece of glass (p. 271 of ref.[13]).

15

For completeness, let us again rehearse the construction[5b & c] of a realization of the n=3 hierarchy:

.)( 433

432

431

4303 zgygxgctg +++=η (4.15)

The fact that this is a curve of fourth order or quartic curve suggests that it is associated with the interaction of two quadratic forms, two of which are of physical interest, namely:

{ } ,)( 222 rcts −+= . (4.16a) characterizing “Einstein states” of special relativity theory and its dual (as defined in the classical theory of tachyons by Recami[14]),

{ } ,)( 222 rcts −−= (4.16b)

where 222 zyxr ++= . The novel (“Oyibo”) states also of interest are obtained from the four linear homogeneous equations:

0])([ =++ νµνµνµν δ wsfgctrg (4.17a) which involves the usual Minkowski space-time metric tensor µνg and its “Lax pair”

µν)( fg with an antisymmetric tensor )( µνf where

,

1000010000100001

−−

−=µνg ,

0010000110000100

−−

=µνf

−−

−

=

001000011000

0100

)( µνfg ; (4.17b)

so that Eq.(4.17a) has the explicit form:

Fig. 3a Fig. 3b

16

0

0000

0000

4

3

2

1

=

+−+−−

−+−−+

wwww

srctsrct

ctsrctsr

. (4.18)

The vanishing of the secular determinant leads to the quartic equation 0])()][()([ 22222 =+−−− ctrsctrs , (4.19)

from which one obtains the two quadratic relations: 222 )(ctrs −− =0, i.e., 222 )(ctrs += ; (4.20a)

ictrsctrs ±==+− i.e., ,0)()( 22 or 22 )(ctrs += (4.20b) By putting τcs = , we may rewrite (4.20a & b) in the respective forms:

22222 )( zyxcts +++= , (4.21a) 0 )( 22222 =−−−± zyxitc τ (4.21b)

and observe that they are identical when 0== τcs . However, by putting )/(tan,)()( 1222222 τθττ θ tetcitc i −± =+=±

and re-defining θiecc 222ˆ ±= , Eq.(4.21b) can also be re-written so as to have the same form as (3.16b) i.e.,

} )ˆ{()ˆ( 22222 zyxtcc −−−−=τ . (4.21c) We recognize Eq.(4.21b) as a light cone with complex time, as proposed heuristically by Jannussis and co-workers[14] while Eq.(4.21c) is the dual of the Minkowski space defined in “extended relativity” by Recami[14], (or isodual space in Santilli’s terminology [14]) with complex speed of light (c ). And we note that if ),,, ˆ,ˆ( zyxtccτ are treated as homogeneous coordinates of a projective space, then the invariance group of Eq.(4.20c) would be the O(3,2)-de Sitter group and its projection on the plane 0=z , has the form

0 )ˆ()ˆ( 2222 =−−+ yxtccτ , (4.21d) which is a ruled quadric in ),, ˆ,ˆ( yxtccτ -space having real generating lines (see, Fig. 4)

Fig. 4

17

In the corresponding momentum space with five homogeneous momentum coordinates, ),,,,(),,,,( 3210 mcppppppppp mzyxt ≡ , the analogs of Eqs.(4.21a) and (4.21b) are

0)( 23

22

21

20

2 =−−−− ppppmc or 220 ppmc +±= , (4.22a)

0 )( 23

22

21

20 =−−−± pppipmc , (4.22b)

where 23

22

21 pppp ++= . Observe that the difference between Eq.(4.22a) and the

usual Einstein’s mass-energy-momentum relation, 222

0 )(mcpp =− or 220 ppmc −±= .

is a mere interchange of mass (mc) and energy ( 0p ), which is possible because the two are treated on equal footing in 5-momentum space. Moreover, we can re- interpret

cipmcE )(ˆ0±≡ in Eq.(4.22b) as the complex energy of a massive particle traveling

at the speed of light(c ) in a conformal invariant space-time. We are thus lead again, from this point of view, to Oyibo’s conclusion that mass conservation equation is a conformal invariant of energy conservation equation. Alternatively, by using

20

2220

220 ˆ)ˆ(])[()( pcmepmcipmc i +≡+=± ± ϕ

)/(tan where 0-1 mcp=ϕ , to rewrite Eq.(4.22b) in the form ,

}ˆ{)ˆ( 23

22

21

20

2 ppppcm −−−−= , (4.22c) we obtain the mass-energy-momentum relation of “extended relativity” theory[14], in which cEpep i ˆ/ˆ 00 ≡= ± ϕ , where ϕicec ±≡ˆ is a complex velocity of light. Such a complex velocity of light implies that the underlying medium is dispersive.

4.4 Gauge Transformation in ( µµ px , ) Phase Space. With regards to the relevance of homographic transformations for gauge theories of the basic (strong, electromagnetic, and weak) forces in nature, we now proceed to establish the connection of conformal transformation with gauge theory. We begin by observing that the conformal current µK may, by virtue of Eqs.(3.5a) and (3.6a) be rewritten in the form

µµνµνν KDgMx =+ )( , (4.23)

where µννµµν pxpxM −= is the antisymmetric angular momentum tensor. This may be compared with the Maxwell equation in the form

µµνµνν φ JgF =+∂ )( (4.24)

where ρρφ A∂≡ is the divergence of the electromagnetic vector potential which

vanishes if the Lorentz condition )0( =∂ ρρ A is obeyed. Thus in terms of the vector

potential ( µA ), the electromagnetic vector current λ

µλλµµ AgJV )2( 2∂−∂∂= (4.25) is seen to be the momentum space analog of the conformal current. This analogy makes it necessary to unify conformal transformations in space-time and in momentum space geometries, by defining a relativistic −),( µµ px “phase space” analogous to the usual (qp)-“phase space” of classical (Hamiltonian) mechanics. Such

18

a unification occurs in an algebraically symmetric manner at short distance, ms /1= , (in units such that )1=c , i.e. at distances on the order of the Compton wavelength of an elementary particle, as follows. We rewrite equation Eq. (3.3) and a corresponding transformation for the momentum in the form (treating µx and µp as c-numbers)

)0( /)(' 2222 ≠≡=−=→ λλ

µµµµ σ xxxspsxxx (4.26a) )0( /)(' 2222 ≠≡=−=→ λ

λµµµµ σ pppmxmppp (4.26b)

where )21( 222 pxpx ++= λλσ . But, because )/( dsdxmp µµ = , we have from λ

λ xxs =2 , that smpxD == µ

µ and hence 22222 Dmspx == , so that 22 )1( D−=σ . Now (4.26a) and (4.26b) define an SU(2) group of transformations if the determinant of the transformation is unity, i.e., if

1)1( 24 +=−− Dσ i.e., 322 )1()1( DD −=− Accordingly, this gauge symmetry is broken when 1=D , i.e., when 1=sm , corresponding to short distances. In this circumstance,

0 ' 2 =−= µµµ xmpp is the analog of the gauge-invariant substitution in the electromagnetic case and the singularity at

µµ xmp 2= (4.27a) is the analog of µµ eAp = in the electromagnetic case. However, the relation (4.27a) has a simple geometrical interpretation, namely that since λ

λ pppm ≡= 22 , it may be rewritten in the form

µµ xpp =2/ (4.27b) which, geometrically speaking, means that µx is the inverse point of µp with respect to the quadric surface 2mpp =λ

λ . This is a new type of inversion symmetry, unlike parity (space- inversion) and time-reversal (time- inversion). Its significance is that it transforms a time-like vector into a space-like vector, or slower-than- light particles (tardyons) into faster-than-light particles (tachyons), just as charge conjugation and parity transform particles into their antiparticles. It is this novel feature of conformal transformations that has enriched the geometrical and physical content of Oyibo’s GUT.

5. DISCUSSION AND CONCLUSION

The use of symmetry principles or groups and their associated conservation laws are anchored on the paradigm for development of applied mathematics which, since the time of the French mathematician, B. Fourier (1768-1830), is to achieve a description of physical phenomena in the general framework of a pure mathematical theory so that, if such an applied mathematical description is mathematically consistent and in agreement with physical observations, then the underlying axioms and theorems of the pure mathematical theory would be presumed to hold for the physical phenomena. Such a paradigm has, indeed, been a source of theoretical predictions and revolutionary ideas in physics. For example, the symmetry of Maxwell’s equations with respect to interchange of electric and magnetic quantities led the prediction of the existence of the magnetic charge (or monopole) reciprocal to isolated electric charges in nature.

19

We have, in this paper, addressed the problem of unifying various mathematical approaches to conformal invariance in physical phenomena at high energies in the general framework of the axioms of modern projective geometry. Unlike Maduemezia[6b], we have not targeted specific results of general relativity, like the precession of the perihelia of the planets Mercury, Venus, Earth and Icarus, but we have brought Maduemezia’s de-Sitter group approach within the purview of projective space-time geometry and exhibited its conceptual limitations. We have related Oyibo’s new conformal group to the geometric principle of duality, which is a universal concept. For example, there is an object in P3(C), known as Segre cubic primal [p. 387 of ref.7a]) – a “prime” is the dual of a “point” in P4(C) – which is a three-fold locus characterized by a quadratic transformation law,

)('),('),(');(),(),(xtyZztxYytzXytxZxtzYztyX

−=−=−=−=−=−=

(5.1)

obeying the principle of duality (in the set-theoretic sense):

''''''''''''

ZYXZYXZYXZYXZYXZYXXYZZYX

∪∪≡++=++≡∪∪∩∩≡=≡∩∩ . (5.2)

Such a relationship is presumably a valuable geometrically dual way of looking at the transition from “fission” to “fusion” of subparticles (such as a quarks triplet) within hadrons, due to the mutual penetration/overlap of their extended wavepackets, as envisaged in Santilli’s Hadronic Mechanics[14]. Perhaps by far the most important aspect of the geometric principle of duality is the realization of Oyibo’s new group in terms of the Lax pair method [5a] for a non-trivial unification of symmetric and antisymmetric tensor fields. As a result, projective geometry is to Oyibo’s GUT what Minkowskian (psudo-Euclidean) geometry has been to Einstein’s special relativity theory; and one is led to the conclusion from the mathematical studies in this paper, that the new group theory envisaged in Oyibo’s Grand Unified Theorem(GUT) is a viable mathematical (geometrical) framework for a “theory of everything”..

REFERENCES 1. (a) R. Jackiw, Introducing Scale Symmetry in Physics Today, Jan. 1970; (b)

A.O.E. Animalu, Scale Symmetry, Physics Today (June, 1972). 76. 2. A.O.Barut, Reformulation of the Dirac Theory of the Electron, Phys. Rev

Letters, 20, 893 (1968). 3. See for example, Yuval Ne’eman, Algebraic theory of Particle Physics

(Chapter 3) (W.A. Benjamin, Inc. 1967); H. Bacry, Projective Geometry and Dynamical Groups, in Second International Colloquium on Group Theoretical Methods in Physics, Nijmegen (June 1973).

4. G.A. Oyibo, (a) Generalized Proof of Einstein’s Theory Using A New Group Theory, published in the Russian Journal, “Problems of Nonlinear Analysis and Engineering Systems, International Journal Vol. 2, (1995); reviewed in 1998 by American Mathematical Society in its Mathematical Reviews journal (MR 98c83007); press-released in Our Time Press (February, 1999) (b) Grand Unified Theorem, with Subtitle, Representation of the Unified Theory or the Theory of Everything (Nova Science Publishers, New York, 2001)., (c) New Group Theory for Mathematical Physics, Gas Dynamics and Turbulence, a monograph.

20

5. A.O.E. Animalu, (a) , Lie-Santilli Iso-Approach to the Unification of Gravity and Electromagnetism, Hadronic J. 19, 255 (1996); (b) Dirac Equation in Five Dimensions and Consequences , an invited paper presented at the Conference on Ordinary Differential Equations, organized by the National Mathematical Centre, Abuja, July 28-29, 2000; (c) A Review of Oyibo’s Grand Unified Theorem With Realizations of a Hierarchy of Oyibo-Einstein Relativities (2005 Preprint)

6. Awele Maduemezia (a) Private Communication (2005), (b) De-Sitter Group Approach to the Theory of Gravitation, J. Nig. Ass. Math. Phys. 3, 1 (1999); (c) L.P. Eisenhart, Riemannian Geometry, (Princeton Univ. Press, Princeton, N.J. (1926).

7. (a) a Semple & Kneebone, Algebraic Projective Geometry (Oxford at the Clarendon Press, 1952). (b) J.A. Todd, Projective and Analytical Geometry (Sir Isaac Pitman 7 Sons, Ltd, London 1958).

8. H.T.H. Piaggio An Elementary Treatise on Differential Equations and Their Applications (G. Bell and Sons, London 1958)

9. C.N. Animalu, Proof of the Existence and Smoothness of Incompressible Spatially Periodic Solutions of the Navier-Stokes Equations by an Exact Solution Technique, Hadronic Journal Supplement, 15, 393 (2000).

10 A.O.E. Animalu and C.N. Animalu, Birkhoffian Mechanics of Velocity-Dependent Forces in the Exterended Relativity of Deformable Bodies in Proc. of the 5th Workshop on Hadronic Mechanics, Nova Science N.Y. 1991.

11. A. Einstein, Relativity, The Special and General Theory, University Paperbacks U.P. 10 (Methuen & Co, Ltd, London 1962)

12. T. Frankel, Gravitational Curvature, An Introduction to Einstein’s Theory (W.H. Freeman & Co, San Fransisco 1979).

13. T. Poston and I. Stewart, Catastrophe Theory and its Applications (Pitman London, 1978).

14. E. Recami, Classical tachyons and possible applications, Riv. Nuov. Cim. 9, N.6, 1986. A. Jannussis, Lie-Admissible Complex Time and its Applications to Quantum Gravity, (Dept. of Physics, University of Patras, preprint July, 1989). R.M. Santilli, Isodual Spaces and Antiparticles, a contribution for Courants, Amers, Ecuils en Microphysique, de Broglie Commemorative Volume, de Broglie Foundation, Paris (France). See also J.V. Kadeisvili, N. Kamiya and R.M. Santilli, Hadronic J. 16, 169 (1993). R.M. Santilli, Elements of Hadronic Mechanics, Vol. II (Naukova Dumka Publishers, Kiev, 1994).

21

APPENDIX A: THE NAVIER-STOKES EQUATIONS The Navier-Stokes Equation are comprised of five equations, namely:

1. The Continuity Equation:

0).( =∇+∂∂

Vt

rrρ

ρ (A.1)

2. 3. & 4. Momentum equation:

( )[ ] ( )[ ]VVBpVVtV

DtVD rrrrrrrrrrr

rr.)().(

34 ∇+∇+×∇×∇−+∇−=∇+

∂∂≡ µςµρρρ (A.2)

which, for constant density and viscosity, become (C.N. Animalu[2]):

VBpDtVD rrrr

2∇−+∇−= µρ (A.3)

5. Energy equation for temperature (a thermodynamic variable):

'".. 2 qqTVpDtDTc rv ++∇−∇+∇−= φκρ

rrr (A.4)

In these equations, Vr

is the velocity 3-vector, ρ is the density, p the pressure, T the temperature, µ the viscosity, …as explained in ref.[10]. Accordingly, in one dimension (x) plus time (t), the momentum equation is:

2

2)(

xVB

xp

tVV

tV x

xx

xx

∂∂++

∂∂−=

∂∂+

∂∂ µρρ

r (A.5)

ACKNOWLEGEMENTS I wish to thank Professor Awele Maduemezia for his incisive comments on my earlier review (ref.[5c]) of Oyibo’s Grand Unified Theorem and Professor R.M. Santilli for his independent review of ref.[5b]. These reviews have stimulated this research paper.