17

The Spinning and Curving of Spacetime: TheElectromagnetic and Gravitational Fields InThe Evans Unified Field Theory

Summary. The unification of the gravitational and electromagnetic fields achievedgeometrically in the generally covariant unified field theory of Evans implies thatelectromagnetism is the spinning of spacetime, gravitation is the curving of space-time. The homogeneous unified field equation of Evans is a balance of spacetimespin and curvature and governs the influence of electromagnetism on gravitationusing the first Bianchi identity of differential geometry. The second Bianchi identityof differential geometry is shown to lead to the conservation law of the Evans unifiedfield, and also to a generalization of the Einstein field equation for the unified field.Rigorous mathematical proofs are given in appendices of the four equations of dif-ferential geometry which are the cornerstones of the Evans unified field theory: thefirst and second Maurer Cartan structure relations and the first and second Bianchiidentities. As an example of the theory the origin of wavenumber and frequency istraced to elements of the torsion tensor of spinning spacetime.

Key words: Evans unified field theory, spinning and curving spacetime, origin of thewavenumber and frequency.

17.1 Introduction

From 1925 to 1955 Einstein made various attempts to unify the gravitationaland electromagnetic fields within general relativity. These attempts are sum-marized in updated appendices of various editions of ref. [1] and are all basedon geometry. The gravitational sector of the unified field was developed byEinstein and others in terms of Riemann geometry with a symmetric Christof-fel connection, Γκ

µν , which implies the first Bianchi identity:

Rσµνρ +Rσνρµ +Rσρµν = 0 (17.1)

by symmetry [2]. In Eq (17.1) Rσµνρ is the Riemann or curvature tensor withlowered indices, defined by:

Rσµνρ = gσκRκµνρ (17.2)

302 17 Spinning and Curving of Spacetime

where gσκ is the symmetric metric tensor [1]. The Riemann curvature tensoris defined in terms of the gamma connection Γκ

µν by:

Rρσµν = ∂µΓ

ρνσ − ∂νΓ

ρµσ + Γ ρ

µλ Γλνσ − Γ ρ

νλ Γλµσ . (17.3)

Eq (17.3) is true for any kind of gamma connection, as is the second Bianchiidentity:

DλRρσµν +DρR

σλµν +DσR

λρµν := 0 (17.4)

where D∧ is the covariant derivative [2] defined with the general gamma con-nection of any symmetry. The symmetric Christoffel connection is the specialcase where the gamma connection is symmetric and defined by:

Γκµν = Γκ

νµ . (17.5)

Using the metric compatibility postulate [2]:

Dρgµν = 0 (17.6)

the symmetric Christoffel connection can be expressed in terms of the sym-metric metric:

Γ σµν =

12gσρ (∂µgνρ + ∂νgρµ − ∂ρgµν) . (17.7)

The use of Eq. (17.7) automatically implies that the torsion tensor Tκµν van-

ishes:Tκ

µν = Γκµν − Γκ

νµ (17.8)

So Einstein’s famous gravitational theory is one in which there is no space-time torsion or spinning. The first Bianchi identity (1) is also a special casetherefore, defined by Eq. (17.5). More generally the cyclic sum in Eq. (17.1)is NOT zero if the gamma connection is not symmetric, and this turns out tobe of fundamental importance for unified field theory: any mutual influence ofgravitation upon electromagnetism depends on the fact that Eq. (17.1) doesnot hold in general. In contrast, note carefully that the second Bianchi identity(4) is ALWAYS true for any type of connection, because it is fundamentallythe cyclic sum of commutators of covariant derivatives [2]:

[[Dλ, Dρ], Dσ] + [[Dρ, Dσ], Dλ] + [[Dσ, Dλ], Dρ] := 0. (17.9)

The above are the well known geometrical equations that are the cornerstonesof Einstein’s generally covariant theory of gravitation [1, 2].

The type of Riemann geometry almost always used by Einstein andothers [2] for generally covariant gravitational field theory is a special case ofthe more general Cartan differential geometry [2] in which the connection isno longer symmetric and which the metric is in general the outer or tensorproduct of two more fundamental tetradsqa

µ . Thus, in differential geometrythe metric tensor is in general an asymmetric matrix. Any asymmetric matrixis always the sum of a symmetric matrix and an antisymmetric matrix [3],

17.1 Introduction 303

so it is possible to construct an antisymmetric metric tensor. The symmetricmetric used by Einstein to describe gravitation is therefore the special casedefined by the inner or dot product of two tetrads [2]:

gµν = qaµq

bνηab (17.10)

where ηab is the diagonal, constant metric in the orthonormal or flat spacetimeof the tetrad, indexed a:

ηab = diag (−1, 1, 1, 1) . (17.11)

Any attempt to construct a generally covariant unified field theory of allradiated and matter fields must therefore be based on differential geometryand must be based on the tetrad rather than the metric. The fundamentalreason for this is that electromagnetism is known experimentally to be a spinphenomenon, and spin does not enter into Einstein’s theory of gravitationbecause the torsion tensor vanishes as we have argued. Therefore a unifiedfield theory must be based on geometry (such as differential geometry) thatconsiders a non-zero torsion tensor as well as a non-zero curvature tensor. Theconclusive advantage of a geometrical theory of fields over a gauge theory offields is that the tangent bundle spacetime indexed a in the former theory isgeometrical and therefore physical from the outset, whereas the fiber bundlespacetime of gauge theory is abstract and is a mathematical construct imposedfor convenience on the base manifold without any reference to geometry. Thusgauge theory can never be a valid theory of general relativity because the latteris fundamentally based on geometry and must always be developed logicallytherefrom. Proceeding on this fundamental geometrical hypothesis of generalrelativity, therefore, the Evans unified field theory follows straightforwardlyby tracing its origins to the fundamental equations that define differentialgeometry. These fundamental equations of differential geometry become thefundamental equations of the unified field theory through the Evans Ansatz[4]:

Aaµ = A(0)qa

µ . (17.12)

In Eq. (17.4) Aaµ is the potential field of the electromagnetic sector of the

unified field and the tetrad qaµ is the fundamental building block of the grav-

itational sector. Here A(0) denotes a C negative scalar originating in the mag-netic fluxon ~/e, a primordial and universal constant of physics. Here ~ is thereduced Planck constant h/(2π) and e the charge on the proton (the negativeof the charge on the electron):

~ = 1.05459× 10−34Js, (17.13)

e = 1.60219× 10−19C. (17.14)

In Section 17.2 we give the four fundamental equations of differential geome-try: the first and second Maurer Cartan structure relations and the first and

304 17 Spinning and Curving of Spacetime

second Bianchi identities and transform them into the equations of the Evansunified field [5]–[20]using Eq. (17.12). The rigorous mathematical proofs of allfour equations are given in Appendices A to D.

In Section 17.3 the Maxwell Heaviside and Einstein limits of the Evansunified field theory are derived and discussed and in Section 17.4 a discussionis given of the implications of the unified field theory in evolution and variousnew technologies based on the ability of the gravitational and electromagneticfields to be mutually influential.

17.2 The Fundamental Equations

The fundamental equations of the unified field theory are the fundamentalequations of differential geometry [2], namely the two Maurer Cartan struc-ture relations and the two Bianchi identities. There are two fundamental dif-ferential forms [2] that together describe any spacetime, the torsion or spinform and Riemann or curvature form. Any radiated or matter field in generalrelativity is therefore defined in terms of these forms. The structure relationsof differential geometry define the spin and curvature forms respectively as thecovariant exterior derivatives of the tetrad form and spin connection one-form:

T a = D ∧ qa = d ∧ qa + ωab ∧ qb, (17.15)

Rab = D ∧ ωa

b = d ∧ ωab + ωa

c ∧ ωcb (17.16)

It is shown rigorously in the Appendices that these definitions are equivalentto the definitions of the spin and curvature tensors in terms of the gammaconnection of any symmetry. Differential geometry is valid for any spacetimeand any type of connection, and this realization is a key step towards theevolution of the Evans unified field theory, the first successful unified theoryof the gravitational and electromagnetic fields.

The other two fundamental equations of differential geometry are thefirst and second Bianchi identities [2]:

D ∧ T a := Rab ∧ qb, (17.17)

D ∧Rab := 0. (17.18)

These are written out in tensor notation and rigorously proven in the Appen-dices. The first Bianchi identity (17) generalizes Eq. (17.1), and the secondBianchi identity, Eq. (17.18), is Eq. (17.4) defined for any type of connection.

Eqs. (17.15) to (17.18) are the four cornerstones of any unified fieldtheory based on geometry, i.e. of any generally covariant unified field theory.They are transformed into equations of the unified field using the Ansatz(17.12), and so Eqs. (17.15) and (17.17) become:

F a = D ∧Aa = d ∧Aa + ωab ∧Ab, (17.19)

D ∧ F a := Rab ∧Ab. (17.20)

17.3 Limiting Forms of the Evans Field 305

Eq. (17.19) defines the field in terms of the potential, Eq. (17.20) is the homo-geneous field equation of the electromagnetic sector. Eqs. (17.16) and (17.18)define the gravitational sector for any connection. In general both the spinand curvature forms are non-zero and so Eq. (17.20) demonstrates the wayin which the gravitational field may influence the electromagnetic field andvice-versa. The extent to which this occurs must be found experimentallybut Eq (17.20) shows that it is possible through a balance of spin and cur-vature in differential geometry. When both sides of Eq. (17.20) are non-zerothe electromagnetic field can be influenced by the gravitational field and thegravitational field can be influenced by the electromagnetic field. In the firstinstance it then becomes possible to build electric power stations from space-time curved by mass, and in the second instance it becomes possible to buildcounter gravitational devices built from electromagnetic technology. The pos-sibility of such technologies must be tested by high precision experiments [21].It seems likely that the chances of success are maximised by using high in-tensity femtosecond laser pulses incident on a high precision gravimeter in ahigh vacuum. The latter is used to remove ”ion wind” artifact, i.e. extraneouseffects due to atmospheric charging.

If it is found within experimental uncertainty that there is no effectof the gravitational field on the electromagnetic field and vice versa then theprimordial Evans field has split entirely during the course of billions of years ofevolution into what we term ”pure electromagnetism” and ”pure gravitation”.These independent fields are described by the unified field equation:

d ∧ F a = 0. (17.21)

This is evidently an equation of differential geometry in the limit:

d ∧ F a = 0, (17.22)

Rab ∧Ab = ωa

b ∧ F b, (17.23)

and so is a generally covariant unified field equation. In the following sectionwe discuss this equation further in order to define precisely the Einsteinianand Maxwell - Heaviside limits of the Evans field theory.

17.3 Limiting Forms of the Evans Field

The Einsteinian limit is defined by:

T a = 0, (17.24)

so the torsion or spin form vanishes and we recover the equations of theintroduction. In the language of differential geometry the Einstein field theoryis therefore:

306 17 Spinning and Curving of Spacetime

D ∧ qa = 0, (17.25)

Rab = D ∧ ωa

b , (17.26)

Rab ∧ qb = 0, (17.27)

D ∧Rab = 0. (17.28)

It is defined by the two Bianchi identities with a symmetric Christoffel symbol,and by the structure relations for zero torsion or spin form. The first structurerelation in the Einstein theory gives a differential equation for the tetrad interms of the spin connection:

d ∧ qa = −ωab ∧ qb (17.29)

which is equivalent to Eq. (17.7) of the introduction. The second Bianchiidentity of the Einstein field theory, Eq. (17.28), leads directly [2] to the wellknown Einstein field equation. In tensor notation this is:

Gµν = kTµν (17.30)

where:Gµν = Rµν −

12Rgµν (17.31)

is the Einstein field tensor. The Ricci tensor Rµν and the metric gµν aresymmetric in the Einstein field theory because the Christoffel connection issymmetric in its lower two indices (Eq. (17.5)). In the more general Evansunified field theory the Ricci tensor and the metric tensor are asymmetricmatrices with anti-symmetric components representing spin. In the Einsteintheory the tetrad postulate [2] of differential geometry:

Dνqaµ = 0 (17.32)

is specialized to the metric compatibility condition [2] for a symmetric metric:

Dνgµρ = Dνgµρ = 0. (17.33)

Finally the canonical energy-momentum tensor is symmetric in the Einsteinlimit and is used in the well known Noether Theorem [2]:

DµTµν = 0. (17.34)

More generally Tµν is asymmetric in the Evans theory and therefore hasan anti-symmetric component representing canonical angular energy/angularmomentum.

So the Einstein limit of the Evans unified field theory is a special casein which the spinning of spacetime is not considered.

The Maxwell Heaviside theory of the electromagnetic field is older thangeneral relativity and is conceptually a different theory in which the field is

17.3 Limiting Forms of the Evans Field 307

an abstract, mathematical entity superimposed on flat or Minkowski space-time. In general relativity on the other hand the field is always non-Euclideangeometry itself and so must be the frame of reference itself. General relativityis simpler (one concept, the frame, instead of two concepts, field and frame)and is therefore the preferred theory by Ockham’s Razor. The simplest way ofthinking about this conceptual jump is to think of a helix. In general relativitythe helix is the spinning and translating baseline, while in Maxwell Heavisidetheory the helix is the abstract field superimposed on a static frame. If we re-strict attention to three space dimensions the flat frame is the static Cartesianframe. In differential geometry the spinning and translating frame is the basemanifold for the electromagnetic field, labelled, µ, and the tangent bundle isdescribed by a Minkowski spacetime labelled a. The tetrad is defined for theelectromagnetic field by:

V a = qaµV

µ (17.35)

where V µ is a vector in the base manifold, and where V a is a vector in thetangent bundle. The tetrad is therefore the four by four invertible transfor-mation matrix [2] between base manifold and tangent bundle. The tetrad istherefore a geometrical construct as required for general relativity. Circularpolarization, discovered experimentally by Arago in 1811, is described geo-metrically by elements of Aa

µ from Eq. (17.12), i.e. by the following complexvalued tetrad elements:

A(1)

x =(A(0)/

√2)eiφ, (17.36)

A(1)

y = −i(A(0)/

√2)eiφ, (17.37)

where φ is the electromagnetic phase. The complex conjugates of these ele-ments are:

A(2)

x =(A(0)/

√2)e−iφ, (17.38)

A(2)

y = i(A(0)/

√2)e−iφ. (17.39)

Therefore these tetrad elements are individual components of the followingvectors:

A(1) =(A(0)/

√2)

(i− ij) eiφ, (17.40)

A(2) =(A(0)/

√2)

(i + ij) e−iφ, (17.41)

representing a spinning and forward moving frame. This frame is multipliedby A(0) to give the generally covariant electromagnetic potential field. In 1992it was inferred by Evans [22] that these vectors define the Evans spin field,B(3), of electromagnetism:

B(3)∗ = −igA(1) ×A(2), (17.42)

308 17 Spinning and Curving of Spacetime

where g is defined by the wavenumber:

g =κ

A(0). (17.43)

The Evans spin field is a fundamental spin invariant of general relativity andis observed through the fact [23] that circularly polarized electromagneticradiation of any frequency magnetizes any material. This reproducible andrepeatable phenomenon is known as the inverse Faraday effect.

In order to reach the Maxwell Heaviside limit from the Evans field the-ory we must make the above important conceptual adjustments and considerthe limit:

Rab ∧ qb = ωa

b ∧ T b (17.44)

i.e. the limit reached when there is no gravitation curvature, but when thereis spacetime spin defined by the following differential geometry:

T a = D ∧ qa, (17.45)d ∧ T a = 0, (17.46)

D ∧ ωab = 0. (17.47)

In this geometry the Riemann or curvature form is zero so the second Bianchiidentity becomes a differential equation for the gravitational spin connection:

d ∧ ωab = −ωa

c ∧ ωcb . (17.48)

This is therefore the underlying differential geometry that defines the pure,generally covariant, electromagnetic field if it has split away completely fromthe gravitational field during the course of billions of years of evolution. Ifthere is any residual influence of gravitation upon electromagnetism and viceversa to be found experimentally then BOTH the curvature and spin formsare experimentally non-zero, and the Evans unified field is described by thedifferential geometry of Section 17.2 - the geometry of the primordial Evansfield.

In general relativity, once we have found the geometry we understandthe physics.

For pure electromagnetism the geometry translates into its field equa-tions using the Ansatz (17.12) to give:

F a = D ∧Aa, (17.49)d ∧ F a = 0, (17.50)

and the gravitational equation

d ∧ ωab = −ωa

c ∧ ωcb . (17.51)

The electromagnetic field is thus always defined through the spin connectionby:

17.3 Limiting Forms of the Evans Field 309

F a = D ∧Aa = d ∧Aa + ωab ∧Ab. (17.52)

Generally covariant non-linear optics is therefore generated by expanding thespin connection in terms of the tetrad or potential as follows:

ωab ∧Ab = −gAb ∧

(Ac + gεcdeAdAe

+g2εcfghAfAgAh + · · ·).

(17.53)

All observable non-linear optical phenomena are therefore phenomena of spin-ning spacetime always describable by a well defined spin connection. TheMaxwell Heaviside theory is the weak field limit or linear limit described by:

F a = d ∧Aa. (17.54)

The Maxwell Heaviside theory is further restricted by the fact that it implicitlysuppresses the index a, meaning that only one unwritten scalar componentof the tangent bundle spacetime is considered, and then only implicitly. Inother words the tangent bundle is ill defined in the Maxwell Heaviside fieldtheory. More generally, in the Evans unified field theory, there are four physicalcomponents of the Minkowski spacetime of the tangent bundle: (ct,X, Y, Z),although a can represent any suitable set of basis elements such as unit vectorsor Pauli matrices [2] in any well defined mathematical representation spaceof the physical tangent bundle spacetime). Suppressing the index a gives thefamiliar equations:

F = d ∧A, (17.55)d ∧ F = 0. (17.56)

These are the Maxwell Heaviside equations in differential geometric notation.The second equation is a combination of the Gauss Law and the FaradayLaw of induction. Therefore in order to construct a unified field theory withdifferential geometry it is necessary to recognise that the Maxwell Heavisidestructure is incomplete.

The inference of the Evans spin field in 1992 [21] was the first steptowards this recognition of the incompleteness of the Maxwell Heaviside the-ory, and therefore towards the the unified field theory long sought after byEinstein[1] and others. The Evans spin field is the torsion or spin form com-ponent defined by the wedge product:

B(3)∗ = −igA(1) ∧A(2). (17.57)

The inference of B(3) led to the development [5]–[20] of O(3) electrodynamics,in which the field is defined by:

F (3)∗ = d ∧A(3)∗ − igA(1) ∧A(2)

et cyclicum(17.58)

310 17 Spinning and Curving of Spacetime

and so O(3) electrodynamics is a special case of Eqns. (17.49) to (17.53). TheGauss and Faraday Laws in O(3) electrodynamics are given by:

d ∧ F a = Rab ∧Ab − ωa

b ∧ F b = 0. (17.59)

Experimentally, it is found [5]–[20] that Eq. (17.59) must split into the par-ticular solution:

d ∧ F a = 0, (17.60)

ωab ∧ F b = Ra

b ∧Ab. (17.61)

These equations are obeyed by the circularly polarized electromagnetic po-tential field described in Eqs (17.40) and (17.41). Eqs. (17.60)and (17.61)alsoimply:

∇ ·B(3) = 0, (17.62)

∂B(3)

∂t= ∇×B(3) = 0. (17.63)

for the Evans spin field. Therefore the latter does not give rise to Faradayinduction, but gives rise to the magnetization observed in the inverse Fara-day effect. Once we recognise the existence of the index a the inverse Faradayeffect and all of non-linear optics follows logically. In the linear Maxwell Heav-iside theory these non-linear optical effects have to be described [5]–[20] withadditional ad hoc and non-linear constitutive relations which are obviouslyextraneous to the original linear Maxwell Heaviside structure. This originallinear structure was inferred in the nineteenth century, long before the adventof non-linear optics. The latter never became available to Einstein, who neverrealized its significance to unified field theory. All sectors of a generally covari-ant unified field theory must be non-linear, because geometry is non-linear.Self consistently therefore, the linear Maxwell Heaviside structure can be ob-tained only if the spin connection vanishes, in which case spacetime becomesthe Minkowski spacetime of special relativity, and Maxwell Heaviside theorywas the first theory of special relativity. The frame covariance of the latterwas first inferred (circa 1900 - 1904) by Poincare and Lorentz. Only later,in 1905, did Einstein finally extend the concept of special relativity to all ofphysics from electromagnetism.

The Evans unified field theory is therefore much more powerful thanthe earlier Maxwell Heaviside field theory, being a generally covariant theoryof all radiated and matter fields. One example out of many possible examplesis given to end this section, the description of the class of all Aharonov Bohmeffects [5]–[20] for all fields. This class of phenomena can be defined withinthe context of Evans’ field when F a is zero but the potential Aa is non-zero.For the electromagnetic sector this means that:

d ∧Aa = 0, (17.64)

F a = d ∧Aa + ωab ∧Ab 6= 0, (17.65)

17.4 Consequences for Evolutionary Theory and New Technology 311

and for the gravitational sector it means that:

d ∧ qa = −ωab ∧ qb, (17.66)

Rab = 0, (17.67)

The potential field Aaof electromagnetism for example can interact with mat-ter fields such as electrons when F a is zero. The first experimental evidencefor this inference was given by Chambers using a static magnetic field, butthe Evans unified field theory shows that there is also an optical or electro-magnetic Aharonov Bohm effect [5]–[20] and also a gravitational AharonovBohm effect. The latter has been observed precisely but the electromagneticAharonov Bohm effect has not been observed experimentally yet. Howeverthe theory of the latter effect has been given in considerable detail [5]–[20]and has major technological consequences if observed. The class of AharonovBohm effects is therefore explained straightforwardly as spacetime phenomenain experimental situations when F a or T a is zero but when qa is non-zero. It isalso possible to explain them when Ra

b is zero and when the spin connectionis non-zero. This will be the subject of a future paper. They are simply theconsequence of geometry in general relativity

17.4 Consequences for Evolutionary Theory and NewTechnology

Before proceeding to a discussion of the implications of the Evans unified fieldtheory we note the generally covariant wave equation:

D ∧ (D ∧ F a) = Rab ∧ F b (17.68)

which can be derived from the two Bianchi identities written as:

D ∧ (D ∧ qa) =(D ∧ ωa

b

)∧ qb, (17.69)

D ∧(D ∧ ωa

b

):= 0. (17.70)

Using the Ansatz (17.12) Eqs. (17.69) and (17.70) become:

D ∧ (D ∧Aa) :=(D ∧ ωa

b

)∧Ab, (17.71)

D ∧(D ∧ ωa

b

):= 0. (17.72)

Differentiating the right hand side of Eq (17.68) and using the Leibnitz The-orem [2]:

D ∧((D ∧ ωa

b

)∧ qb

):=(D ∧ ωa

b

)∧(D ∧ qb

)(17.73)

Using Eq (17.16), Eq (17.73) becomes:

D ∧((D ∧ ωa

b

)∧ qb

):= Ra

b ∧ T b. (17.74)

312 17 Spinning and Curving of Spacetime

Use Eq (17.17) in Eq (17.74) to give:

D ∧ (D ∧ T a) := Rab ∧ T b, (17.75)

which using the Ansatz (17.12) translates into Eq (17.68). Using the latterthe condition for independent fields (no mutual interaction of gravitation andelectromagnetism) becomes:

D ∧ (D ∧ F a) := Rab ∧ F b = 0 (17.76)

which means that the fields are independent when the wedge product of theRiemann tensor and electromagnetic field tensor vanishes. Conversely if thiswedge product is non-zero the fields can influence each other. This influence,if found to be non-zero experimentally with precise, well designed experi-ments, implies major new technology as discussed briefly already. This typeof technology is governed in general by Eq. (17.68) which is a wave equationor quantum equation with the field F a as eigenfunction or wave-function andthe field Ra

b as eigenvalues or quantum values. These new technologies wouldtherefore depend on the fact that the quantum values of F a are Ra

b within afactor A(0), in other words it would depend on a generally covariant quantummechanics of the unified Evans field.

Using the tetrad postulate [2]:

∂µqaν + ωa

µbqbν = Γλ

µν qaλ (17.77)

the torsion form becomes the equivalent torsion tensor (see Appendix One):

T aµν =

(Γλ

µν − Γλνµ

)qa

λ , (17.78)

Tλµν = qa

λTaµν = Γλ

µν − Γλνµ . (17.79)

Therefore, using the Ansatz (17.12) the magnetic field is defined by:

Baµν = Tλ

µν Aaλ (17.80)

and it becomes particularly clear from Eq (17.80) that spacetime spinninggives rise to electromagnetism.

The Maxwell Heaviside limit has the mathematical structure:

B(1) = ∇×A(1) (17.81)

where A(1) is given by Eq (17.40). So the magnetic field from Eqs (17.40) and(17.81) is:

B(1) =B(0)

√2

(ii + j) eiφ, (17.82)

and is the limit of Eq. (17.80) when the spin connection vanishes. Takingcomponents of Eqs. (17.39) and (17.82)

17.4 Consequences for Evolutionary Theory and New Technology 313

B(1)

x = −B(1)1 = −B(1)

23 = B(1)

32 = iB(0)

√2eiφ, (17.83)

B(1)

y = −B(1)2 = −B(1)

31 = B(1)

13 =B(0)

√2eiφ, (17.84)

A(1)

0 = A(1)

3 = B(1)

0 = B(1)

3 = 0. (17.85)

Therefore simple algebra gives:

iB(0) = A(0)(T 1

23 − iT 223

). (17.86)

As in any paradigm shift Eq (17.86) gives new insight into known things. Eq(17.86) shows that the Maxwell Heaviside field theory defines the magneticfield as a complex sum of torsion tensor components. From Eq (17.86):

B(0) = A(0)(T 2

32 + iT 132

). (17.87)

In the Maxwell Heaviside limit we know however that [5]–[20]:

B(0) = κA(0) (17.88)

whereκ =

ω

c. (17.89)

Here ω is the angular frequency and c the speed of light. From Eqs. (17.87)and (17.88) therefore

κ = T 232 + iT 1

32 , (17.90)

a result which traces the origin of wave-number in the Maxwell Heaviside limitto a complex sum of scalar valued torsion tensor components.

It is then possible to define the scalar curvature in the Maxwell Heav-iside limit:

R = κκ∗ =(T 2

32

)2 − (T 132

)2, (17.91)

a result which illustrates the fact that we are now thinking of the electromag-netic field as a spinning of spacetime, and not as an abstract mathematicalfield superimposed on a static frame of reference in Minkowski (flat) space-time. In flat spacetime the scalar curvature is zero, in spinning spacetime it isnon-zero. In flat spacetime initially parallel lines remain parallel, in spinningspacetime they become geodesics [2]of the electromagnetic field. The dielec-tric permittivity and the absorption coefficient [24] are defined in terms ofa complex wavenumber, so these fundamental spectroscopic properties aretraced to a geometrical origin. Photon mass is defined by the Evans Principleof Least Curvature [5]–[20] in which is subsumed the Hamilton Principle ofLeast Action in dynamics and the Fermat Principle of Least Time in optics.The Evans Principle asserts that in the limit of Minkowski spacetime:

κ→ 2πλ0

= 2πmc

~, (17.92)

314 17 Spinning and Curving of Spacetime

where λ0 is the Compton wavelength of any particle of mass m, including thephoton. Therefore mass is also identified has having a geometrical origin, aleast or minimized curvature in the Minkowski limit. Indeed, we see from Eqs.(17.90) and (17.92) that within a factor 2πc/~, mass is a torsion element forany matter field or radiated field:

T 232 → 2π

mc

~, (17.93)

Therefore the origin of all fundamental physical concepts in the Evans uni-fied field theory is differential geometry. Recently Pinter [25], in a remarkablemulti-disciplinary work, has extended this basic insight to infer in a rigorouslylogical sequence of arguments that the origin of life is also differential geome-try. Live organisms are extensions of general relativity itself and have evolvedto their present condition through a sequence of transitions brought about bythe theory of general relativity. The laws of physics, chemistry, biology, ge-ology and genetics for example, are the laws of differential geometry. Withinthe context of the Evans unified field theory it is now recognized that elec-tromagnetism as well as gravitation, originates in differential geometry, i.e. inthe primordial or unified Evans field. Electromagnetism and gravitation aretwo parts of the same thing, and both are essential for the evolution of life.This inference justifies the fundamental and closely argued hypothesis usedby Pinter [25], that gravitational effects evolve into effects driven by electro-dynamics, for example photosynthesis in the early planet Earth. In the olderMaxwell Heaviside field theory electrodynamics appears ”out of the blue” butin the Evans unified field theory its origin is the same as that of gravitation.Similarly the origin of the weak and strong fields is also differential geometryin the Evans unified field theory. There are no longer any abstract internalgauge spaces or strings in nature.

The technological implications of the Evans field theory depend on itsinference that one type of field may affect another, as briefly discussed already.If it were possible to obtain electromagnetic power from spacetime curved, forexample, by the Earth’s mass or by the mass of an electron in a circuit, theproblem of burning fossil fuel would be obviated. Conversely, if it were possiblefor electromagnetic devices to counter or enhance the gravitational force, greattechnological strides would be made in any future aerospace industry. TheEvans unified field theory shows that this is indeed possible but very careful,very precise, experiments are needed to measure the extent of the interaction(if any) between the sectors of the Evans field.

Acknowledgments

The Ted Annis Foundation, Craddock Inc., and Applied Science Associatesare thanked for funding, and the AIAS environment for many interesting dis-cussions. Paul Pinter is thanked for a copy of his important multi-disciplinarywork. Bob Gray, Jan Abas and Michael Anderson are thanked for the con-struction of websites.

A

The First Maurer-Cartan Structure Relation

The first structure relation [2] defines the torsion or spin form as the exteriorcovariant derivative of the tetrad form:

T aµν = (D ∧ qa)µν = (d ∧ qa)µν + ωa

µbqbν − ωa

νbqbµ (A.1)

where ωaµb is the spin connection. The torsion tensor is therefore:

Tλµν = qλ

aTaµν (A.2)

and using the tetrad postulate:

Tλµν = qλ

a

(∂µq

aν − ∂νq

aµ + ωa

µbqbν − ωa

νbqbµ

)(A.3)

we obtain:Tλ

µν = Γλµν − Γλ

νµ . (A.4)

This is an expression for the torsion tensor in terms of the gamma connec-tion of any symmetry. If the gamma connection is the symmetric Christoffelsymbol:

Γλµν = Γλ

νµ (A.5)

then the torsion tensor vanishes.

B

The Second Maurer-Cartan Structure Relation

The second structure relation defines the Riemann or curvature form as theexterior covariant derivative of the spin connection, regarded as a one-form:

Rab = D ∧ ωa

b (B.1)

i.eRa

bνµ = ∂νωaµb − ∂µω

aνb + ωa

νcωcµb − ωa

µcωcνb . (B.2)

It is proven in this appendix that the second structure relation is equivalent tothe definition of the Riemann tensor for a gamma connection of any symmetry.

The proof starts with the tetrad postulate expressed as:

ωaµb = qa

νqλbΓ

νµλ − qλ

b∂µqaλ . (B.3)

Multiplying both sides of Eq (B.3) by qbλ and using:

qbλq

λb = 1 (B.4)

the tetrad postulate can be expressed as:

∂µqaλ = qa

νΓνµλ − qb

λωaµb . (B.5)

Differentiating Eq (B.3) and using the Leibnitz Theorem:

∂νωaµb =∂ν

(qa

σqλbΓ

σµλ

)− ∂ν

(qλ

b∂µqaλ

)=∂ν

(qa

σqλb

)Γ σ

µλ + qaσq

λb∂νΓ

σµλ

−(∂νq

λb

) (∂µq

aλ

)− qλ

b

(∂ν∂µ

(qa

λ

)).

(B.6)

Now use the Leibnitz Theorem again:

∂ν

(qλ

bqaσ

)= qa

σ∂νqλb + qλ

b∂νqaσ (B.7)

to obtain:

318 B The Second Maurer-Cartan Structure Relation

∂νωaµb =

(qa

σΓσµλ − ∂µq

aλ

)∂νq

λb

+ qλbΓ

σµλ ∂νq

aσ + qa

σqλb∂νΓ

σµλ

− qλb

(∂ν∂µ

(qa

λ

)) (B.8)

Now use Eq. (B.5)in Eq. (B.8):

∂νωaµb =qb

λωaµb∂νq

λb + qλ

bΓσµλ ∂νq

aσ

+ qλbq

aσ∂νΓ

σµλ − qλ

b

(∂ν∂µ

(qa

λ

)).

(B.9)

Switching the µ and ν indices gives:

∂µωaνb =qb

λωaνb∂µq

λb + qλ

bΓσνλ ∂µq

aσ

+ qλbq

aσ∂µΓ

σνλ − qλ

b

(∂µ∂ν

(qa

λ

)) (B.10)

which implies:

∂νωaµb − ∂µω

aνb =qb

λ

(ωa

µb∂νqλb − ωa

νb∂µqλb

)+ qb

λ

(Γ σ

µλ ∂νqaσ − Γ σ

νλ ∂µqaσ

)+ qλ

bqaσ

(∂νΓ

σµλ − ∂µΓ

σνλ

) (B.11)

because(∂ν∂µ − ∂ν∂µ) qa

λ = 0. (B.12)

In order to evaluate the Riemann form:

Rabνµ = ∂νω

aµb − ∂µω

aνb + ωa

νcωcµb − ωa

µcωcνb (B.13)

we need:

ωaνc = qa

µqλcΓ

µνλ − qλ

c∂νqaλ (B.14)

ωaµb = qc

νqλbΓ

νµλ − qλ

b∂µqcλ (B.15)

ωaµc = qa

νqλcΓ

νµλ − qλ

c∂µqaλ (B.16)

ωcνb = qc

µqλbΓ

µνλ − qλ

b∂νqcλ . (B.17)

It is then possible to evaluate products such as:

ωaνcω

cµb =

(qa

µqλcΓ

µνλ − qλ

c∂νqaλ

)(qc

νqλbΓ

νµλ − qλ

b∂µqcλ

). (B.18)

The Riemann tensor can then be evaluated using:

Rσλνµ = qσ

aqbλR

abνµ. (B.19)

In order to evaluate Eq (B.19) first rearrange dummy indices in Eq. (B.18) asfollows:

B The Second Maurer-Cartan Structure Relation 319

qλcq

aµq

λbq

cνΓ

µνλ Γ

νµλ

↓ (µ→ σ)

qλcq

aσq

λbq

cνΓ

σνλ Γ

νµλ

↓ (λ→ ρ, ν → ρ)

qρcq

aσq

λbq

cρΓ

σνρ Γ

ρµλ = qa

σqλbΓ

σνρ Γ

ρµλ

(B.20)

Secondly cancel the term qλbΓ

σνρ ∂νq

aσ in Eq. (B.11) with the term−(qλ

c∂νqaλ)

(qλbq

cνΓ

νµλ ) In Eq. (B.18) by rearranging dummy indices as follows:

−qλcq

λbq

cνΓ

νµλ ∂νq

aλ

↓ (λ→ σ)

−qσcq

λbq

cνΓ

νµλ ∂νq

aσ

↓ (ν → σ)

−qσcq

λbq

cσΓ

σµλ ∂νq

aσ = −qλ

bΓσµλ ∂νq

aσ .

(B.21)

Finally cancel the term −qbλω

aνb∂µq

λb in Eq. (B.11) with the term qλ

cqλb

(∂νqaλ)(∂µq

cλ)−qa

µqλcq

λbΓ

µνλ ∂µq

cλ in Eq. (B.18). To do this rewrite the Eq

(B.18) term as qλcq

λb∂µq

cλ(∂νq

aλ − qa

µΓµνλ ) and use the tetrad postulate:

∂νqaλ = qa

µΓµνλ − qb

λωaνb (B.22)

to obtain:−qλ

cqλbq

bλω

aνb∂µq

cλ = −qc

λωaνb∂µq

cλ . (B.23)

We therefore obtain:

−qbλω

aνb∂µq

λb −

(−qλ

cωaνb∂µq

cλ

)= −ωa

νb

(qc

λ∂µqλc + qλ

c∂µqcλ

). (B.24)

In order to show that this is zero use:

qλcq

cλ = 1 (B.25)

and differentiate:∂µ(qλ

cqcλ) = 0. (B.26)

Finally use the Leibnitz Theorem to obtain:

qλc∂µq

cλ + qc

λ∂µqλc = 0. (B.27)

The remaining terms give the Riemann tensor for any gamma connection:

Rλσνµ = ∂νΓ

σµλ − ∂µΓ

σνλ + Γ σ

νρ Γρµλ − Γ σ

µρ Γρνλ (B.28)

quod erat demonstrandum.

C

The First Bianchi Identity

The first Bianchi identity of differential geometry is a balance of spin andcurvature

D ∧ T a := Rab ∧ qb (C.1)

and becomes the homogeneous field equation of the Evans unified field theory:

D ∧ F a := Rab ∧Ab (C.2)

using the Evans Ansatz:Aa = A(0)qa (C.3)

So it is important to thoroughly understand the structure and meaning of thefirst Bianchi identity as in this Appendix. In order to proceed we need thefollowing general definitions [2] of the exterior derivative and wedge productfor any differential form:

(d ∧A)µ1···µp+1= (p+ 1) ∂[µ1Aµ2···µp+1] (C.4)

(A ∧B)µ1···µp+q=

(p+ q)!p!q!

(p+ 1)A[µ1···µpBµp+1···µp+q ]. (C.5)

Eq (C.4) defines the exterior derivative of a p-form and Eq. (C.5) defines thewedge product of a p-form and a q-form. We also use the fact that the spinconnection is a one-form [2]. The exterior covariant derivative of a one-formXa

µ , for example, then follows as:

(D ∧X)aµν = (d ∧X)a

µν + (ω ∧X)aµν (C.6)

where:(d ∧X)a

µν = ∂µXaν − ∂νX

aµ (C.7)

(ω ∧X)aµν = ωa

µbXbν − ωa

νbXbµ . (C.8)

Eqs. (C.7) and (C.8) follow using:

322 C The First Bianchi Identity

p = 1, q = 1, µ1 = µ, µ2 = ν (C.9)

and(d ∧A)µ1µ2

= (d ∧A)µν = 2∂[µAν] = ∂µAν − ∂νAµ (C.10)

(A ∧B)µ1···µp+q= (A ∧B)µν =

2!1!1!

A[µBν] = AµBν −AνBµ (C.11)

Now extend this method to the exterior covariant derivative of a two-form,using:

(d ∧A)µ1µ2µ3= 3∂[µ1Aµ2µ3]

= ∂µAνρ + ∂νAρµ + ∂ρAµν

(C.12)

and

(A ∧B)µ1µ2µ3=

3!2!1!

A[µ1Bµ2µ3] = 3A[µBνρ]

= AµBνρ +AνBρµ +AρBµν

(C.13)

Therefore the exterior covariant derivative of the torsion or spin form used inthe first Bianchi identity is:

(D ∧ T )aµνρ = (d ∧ T )a

µνρ + (ω ∧ T )aµνρ (C.14)

where:(d ∧ T )a

µνρ = ∂µTaνρ + ∂νT

aρµ + ∂ρT

aµν (C.15)

(ω ∧ T )aµνρ = ωa

µbTbνρ + ωa

νbTbρµ + ωa

ρbTbµν (C.16)

and where:T a

µν =(Γλ

µν − Γλνµ

)qa

λ (C.17)

Similarly:

Rab ∧ qb = Ra

bµνqbρ +Ra

bνρqbµ +Ra

bρµqbν

= Raµνρ +Ra

νρµ +Raρµν

=(Rσ

µνρ +Rσνρµ +Rσ

ρµν

)qa

σ .

(C.18)

So the first Bianchi identity becomes:

∂µTaνρ + ωa

µbTbνρ + · · · = Rσ

µνρqaσ + · · · . (C.19)

Using Eq. (C.17), Eq. (C.19) becomes:

∂µ

((Γλ

νρ − Γλρν

)qa

λ

)+ωλ

µb

(Γλ

νρ − Γλρν

)qb

λ + · · ·

= Rλµνρq

aλ + · · ·

(C.20)

C The First Bianchi Identity 323

Using the Leibnitz Theorem Eq. (C.20) becomes:(∂µΓ

λνρ − ∂µΓ

λρν

)qa

λ +(∂µq

aλ + ωa

µbqbλ

)(Γλ

νρ − Γλρν

)+ · · · = Rλ

µνρqaλ + · · ·

(C.21)

Now use the tetrad postulate:

∂µqaρ + ωa

µbqbσ = Γλ

µσ qaλ (C.22)

in Eq. (C.21) to obtain:

∂µΓλνρ − ∂νΓ

λµρ + Γλ

µσ Γσνρ − Γλ

νσ Γσµρ

+ ∂νΓλρµ − ∂ρΓ

λνµ + Γλ

νσ Γσρµ − Γλ

ρσ Γσνµ

+ ∂ρΓλµν − ∂µΓ

λρν + Γλ

ρσ Γσµν − Γλ

µσ Γσρν

:= Rλρµν +Rλ

µνρ +Rλνρµ.

(C.23)

The Riemann tensor for any connection (Appendix two) is:

Rλρµν = ∂µΓ

λνρ − ∂νΓ

λµρ + Γλ

µσ Γσνρ − Γλ

νσ Γσµρ , (C.24)

and so Eq. (C.23) is an identity made up of the cyclic sum of three Riemanntensors on either side. The familiar Bianchi identity of the famous Einsteingravitational theory is the SPECIAL CASE when the cyclic sum vanishes:

Rλρµν +Rλ

µνρ +Rλνρµ = 0. (C.25)

Eq. (17.25) is true if and only if the gamma connection is the symmetricChristoffel symbol:

Γλµν = Γλ

νµ . (C.26)

It is not at all clear using tensor notation (Eq. (17.23)) that the first Bianchiidentity is a balance of spin and curvature. In order to see this we need thedifferential form notation of Eq. (C.1) and this is of key importance for thedevelopment of the Evans unified field theory.

D

The Second Bianchi Identity

The second Bianchi identity is:

D ∧Rab = d ∧Ra

b + ωac ∧Rc

b + ωcb ∧Ra

c

:= 0.(D.1)

using the results of Appendix Three we may write out Eq. (D.1) in tensornotation:

DρRabµν +DµR

abνρ +DνR

abρν := 0 (D.2)

where:

DρRabµν = ∂ρR

abµν + ωa

ρcRcbµν + ωc

ρbRacµν

et cyclicum.(D.3)

Now use:Ra

bµν = qσbR

aσµν . (D.4)

The Leibnitz Theorem and tetrad postulate give the result:

DρRabµν = Dρ

(qσ

bRaσµν

)= qσ

bDρRaσµν (D.5)

which implies:DρR

aσµν +DµR

aσνρ +DνR

aσρµ := 0 (D.6)

Now use:Ra

σµν = qaκR

κσµν (D.7)

The Leibnitz Theorem and tetrad postulate are used again to find:

DρRκσµν +DµR

κσνρ +DνR

κσρµ := 0 (D.8)

which is the second Bianchi identity in tensor notation for any gamma con-nection, quod erat demonstrandum.

The second Bianchi identity is true for ANY gamma connection becauseit is equivalent to:

326 D The Second Bianchi Identity

[Dρ, [Dµ, Dν ]] + [Dµ, [Dν , Dρ]] + [Dν , [Dρ, Dµ]] := 0. (D.9)

and Eq. (D.9) can be summarized symbolically as a round trip with covariantderivatives around a cube: The second Bianchi identity is the geometrical

-

6��

��*-����

?���*��

���

Fig. D.1. Covariant Derivatives Around a Cube

foundation for the conservation law of the Evans unified field theory:

D ∧ T ab := 0. (D.10)

Eq. (D.10) is the required generalization of the Noether Theorem for theunified field theory. The second Bianchi identity is also the foundation for thegeneralization of the Einstein field equation in the Evans unified field theory:

Rab = kT a

b . (D.11)

References

1. A. Einstein, The Meaning of Relativity (Princeton Univ. Press, 1921 and sub-sequent editions).

2. S. P. Carroll, Lecture Notes in General Relativity (Univ California Santa Bar-bara, graduate course) arXiv:gr-gq/9712019 v1 3 Dec 1997.

3. G. Stephenson, Mathematical Methods for Science Students (Longmans, Lon-don, 1968).

4. M. W. Evans, Generally Covariant Unified Field Theory: the Geometrization ofPhysics, (in press, 2005).

5. M. W. Evans, Found. Phys. Lett., 16, 367 (2003).6. M. W. Evans, Found. Phys. Lett., 16, 507 (2003).7. M. W. Evans, Found. Phys. Lett., 17, 25 (2004).8. M. W. Evans, Found. Phys. Lett., 17, 149 (2004).9. M. W. Evans, Found. Phys. Lett., 17, 267 (2004).

10. M. W. Evans, Found. Phys. Lett., 17, 301 (2004).11. M. W. Evans, Found. Phys. Lett., 17, 393 (2004).12. M. W. Evans, The Evans Lemma of Differential Geometry, Found. Phys. Lett.,

17, 433 (2004)13. M. W. Evans, Derivation of the Evans Wave Equation from the Lagrangian and

Action, Origin of the Planck Constant in General Relativity, Found. Phys. Lett.,17, 535 (2004)

14. M. W. Evans and the AIAS Author Group, Development of the Evans WaveEquation in the Weak Field Limit, Found. Phys. Lett., 17, 497 (2004)

15. M. W. Evans, New Concepts from the Evans Unified Field Theory: Part One,The Evolution of Curvature, Oscillatory Universe without Singularity, CausalQuantum Mechanics, and General Force and Field Equations, Found. Phys.Lett., in press (preprint on www.aias.us).

16. M. W. Evans, ibid., Part two: Derivation of the Heisenberg Equation and Re-placement of the Heisenberg Uncertainty Principle, Found. Phys. Lett., in press(preprint on www.aias.us).

17. M. W. Evans, Derivation of the Lorentz Boost from the Evans Wave Equation,Found. Phys. Lett., 17, 663 (2004)

18. M. W. Evans, The Electromagnetic Sector of the Evans Unified Field Theory,Found. Phys. Lett., in press.

328 References

19. M. W. Evans (ed.), Modern Non-Linear Optics, special topical issues of I. Pri-gogine an S. A. Rice, Advances in Chemical Physics (Wiley Interscience, NewYork, 1992 to 2001, first and second eds.), vol. 85 and 119; M. W. Evans, J.-P.Vigier et al., The Enigmatic Photon (Kluwer, Dordrecht, 1994 to 2002 hard-back and softback), in five volumes; M. W. Evans and L. B. Crowell, Classicaland Quantum Electrodynamics and the B(3) Field, (World Scientific, Singapore,2001); M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum fieldTheory, (World Scientific, Singapore, 1994).

20. L. Felker, The Evans Equations of Unified Field Theory preprint on www.aias.us.21. M. W. Evans, STAIF 2004 papers on www.aias.us.22. M. W. Evans, Physica B, 182, 227, 237 (1992).23. M. Tatarikis et al., Measurement of the Inverse Faraday Effect in High Inten-

sity Laser produced Plasmas, CFDL Annual report, 1998 / 1999, RutherfordAppleton Laboratory.

24. M. W. Evans, G. J. Evans, W. T. Coffey and P. Grigolini, Molecular Dynamicsand the Theory of Broad Band Spectroscopy (Wiley-Interscience, New York,1980).

25. P. Pinter, private communication.