Derivation of the Gauss Law of Magnetism, The Faraday Law of Induction, and O(3) Electrodynamics from The Evans Field Theory

Transcription

1 27 Derivtion of the Guss Lw of Mgnetism, The Frdy Lw of Induction, nd O(3) Electrodynmics from The Evns Field Theory Summry. The Guss Lws of mgnetism nd the Frdy lw of induction re derived from the Evns unified field theory. The geometricl constrints imposed on the generl field theory y these well known lws led self consistently to O(3) electrodynmics. Key words: Evns field theory, Guss lw pplied to mgnetism; Frdy lw of induction; O(3) electrodynmics Introduction In this pper the Guss lw pplied to mgnetism [1] nd the Frdy Lw of induction [1, 2] re derived from the Evns field theory [3] [33] y the imposition of well defined constrints in differentil geometry. Therefore the origin of these well known lws is trced to differentil geometry nd the properties of the generl four dimensionl mnifold known s Evns spcetime. Such inferences re not possile in the Mxwell-Heviside (MH) theory of the stndrd model [1, 2] ecuse MH is not n ojective theory of physics, rther it is theory of specil reltivity covrint only under the Lorentz trnsformtion. An ojective theory of physics must e covrint under ny coordinte trnsformtion[1] nd this is fundmentl philosophicl requirement for ll physics, s first relized y Einstein. This fundmentl requirement is known s generl reltivity nd the generl coordinte trnsformtion leds to generl covrince in contrst to the Lorentz covrince of specil reltivity. The fundmentl lck of ojectivity in the Lorentz covrint MH theory mens tht it is not le to descrie the importnt mutul effects of grvittion nd electromgnetism. In contrst the Evns unified field theory is generlly covrint nd is direct logicl consequence of Einstein s generl reltivity, which is essentilly the geometriztion of physics. The unified field theory is le y definition to nlyze the effects of grvittion on electromgnetism nd vice-vers.

2 Guss Lw, Frdy Lw, nd O(3) Electrodynmics In Section 2 fundmentl geometricl constrint on the generl field theory is derived from considertion of the first Binchi identity of differentil geometry. It is then shown tht this constrint leds to O(3) electrodynmics [3] [33] directly from the Binchi identity. These inferences trce the origin of the Guss lw pplied to mgnetism nd the Frdy lw of induction to differentil geometry nd generl reltivity, s required y Einsteinin nturl philosophy. Section 3 is discussion of the numericl methods needed to solve the generl nd restricted Evns field equtions Geometricl Condition Needed for the Guss Lw of Mgnetism nd the Frdy Lw of Induction nd Derivtion of O(3) Electrodynmics The geometricl origin of these lws in the Evns field theory is the first Binchi identity of differentil geometry [1]: which cn e rewritten s: D T = R q (27.1) d T = R q ω T. (27.2) Here T is the vector vlued torsion two-form, R is the tensor vlued curvture or Riemnn two-form; q is the vector vlued tetrd one-form;ω is the spin connection, which cn e regrded s one-form [1]. The symol D denotes the covrint exterior derivtive nd d denotes the ordinry exterior derivtive. The Binchi identity ecomes the homogeneous Evns field eqution (HE) using: A = A (0) q, (27.3) F = A (0) T. (27.4) Here A (0) is sclr vlued electromgnetic potentil mgnitude (whose S.I. unit is volt s / m). Thus A is the vector vlued electromgnetic potentil one-form nd F is the vector vlued electromgnetic field two-form. The HE is therefore: d F = R A ω F (27.5) = µ 0 j where: j = 1 µ 0 ( R A ω F ) (27.6) is the homogeneous current, vector vlued three-form. Here µ 0 is the S. I. vcuum permeility.

3 27.2 Geometricl Condition 447 The homogeneous current is theoreticlly non-zero. known experimentlly to gret precision tht: However it is d F 0. (27.7) Eqn. (27.7) encpsultes the two lws which re to e derived here from Evns field theory. These re usully written in vector nottion s follows. The Guss lw pplied to mgnetism is: B 0, (27.8) where B is mgnetic flux density. The Frdy lw of induction is: E + B t 0 (27.9) where E is electric field strength. The index ppering in Eqns. (27.8) nd (27.9) comes from eqution (27.5), i.e. from generl reltivity s required y Einstein. The physicl mening of is tht it indictes sis set of the tngent undle spcetime, Minkowski or flt spcetime. Any sis elements (e.g. unit vectors or Puli mtrices) cn e used [1]in the tngent spcetime of differentil geometry, nd the sis elements cn e used to descrie sttes of polriztion [3] [33], for exmple circulr polriztion first discovered experimentlly y Argo in Argo ws the first to oserve wht is now known s the two trnsverse sttes of circulr polriztion. It is convenient [3] [33] to descrie these sttes of circulr polriztion y the well known [34] complex circulr sis: = (1), (2) nd (3) (27.10) whose unit vectors re: e (1) = 1 2 (i ij) = e (2), (27.11) e (3) = k (27.12) where * denotes complex conjugtion. Ech stte of circulr polriztion cn e descried y two complex conjugtes. One sense of circulrly polrized rdition is descried y the complex conjugtes: A (1) 1 = A(0) 2 (i ij) e iφ, (27.13) A (2) 1 = A(0) 2 (i + ij) e iφ. (27.14) The other sense of circulrly polrized rdition is descried y the complex conjugtes:

4 Guss Lw, Frdy Lw, nd O(3) Electrodynmics A (1) 2 = A(0) 2 (i + ij) e iφ, (27.15) A (2) 2 = A(0) 2 (i ij) e iφ. (27.16) Here φ is the electromgnetic phse nd Eqns. (27.13) nd (27.16) re solutions of Eq. (27.7). The experimentlly oservle Evns spin field B (3) [3] [33] is defined y the vector cross product of one conjugte with the other. In non-liner optics [3] [33] this is known s the conjugte product, nd is oserved experimentlly in the inverse Frdy effect, (IFE), the mgnetiztion of ny mteril mtter y circulrly polrized electromgnetic rdition. In the sense of circulr polriztion defined y Eqns. (27.13) nd (27.14): B (3) 1 = iga (1) 1 A (2) 1 = B (0) k (27.17) where g = κ (27.18) A (0) nd where κ is the wvenumer. In the sense of circulr polriztion defined y Eqns. (27.15) nd (27.16) B (3) reverses sign: B (3) 2 = iga (1) 2 A (2) 2 = B (0) k (27.19) nd this is oserved experimentlly [3] [33] ecuse the oservle mgnetiztion chnges sign when the hndedness or sense of circulr polriztion is reversed. Liner polriztion is the sum of 50% left nd 50% right circulr polriztion nd in this stte the IFE is oserved to dispper. Thus B (3) in linerly polrized em vnishes ecuse hlf the em hs positive B (3) nd the other hlf negtive B (3). The B (3) field ws first inferred y Evns in 1992 [34] nd it ws recognized for the first time tht the phse free mgnetiztion of the IFE is due to third (spin) stte of polriztion now recognized s = (3) in the unified field theory. The Guss lw nd the Frdy lw of induction hold for = (3), ut it is oserved experimentlly tht A (3) = 0 [3] [33]. Therefore: B (3) 0 (27.20) B (3) t 0. (27.21) The fundmentl reson for this is tht the spin of the electromgnetic field produces n ngulr momentum which is oserved experimentlly in the Beth effect [3] [33]. The electromgnetic field is negtive under chrge conjugtion symmetry (C), so the Beth ngulr momentum produces B (3) directly, ngulr momentum nd mgnetic field eing oth xil vectors. The

5 27.2 Geometricl Condition 449 puttive rdited E (3) would e polr vector if it existed, nd would not e produced y spin. There is however no electric nlogue of the inverse Frdy effect, circulrly polrized electromgnetic field does not produce n electric polriztion experimentlly, only mgnetiztion. Similrly, in the originl Frdy effect, sttic mgnetic field rottes the plne of linerly polrized rdition, ut sttic electric field does not. The Frdy effect nd the IFE re explined using the sme hyperpolrizility tensor in the stndrd model, nd in the Evns field theory y term in the well defined Mclurin expnsion of the spin connection in terms of the tetrd, producing the IFE mgnetiztion: M (3) = i µ 0 g A (1) A (2) (27.22) where A (1) nd A (2) re complex conjugte tetrd elements comined into vectors [3] [33]. Similrly ll non-liner opticl effects in the Evns field theory re, self consistently, properties of the Evns spcetime or generl fourdimensionl se mnifold. The unified field theory therefore llows non-liner optics to e uilt up from spcetime, s required in generl reltivity. In the MH theory the spcetime is flt nd cnnot e chnged, so non-liner optics must e descried using constitutive reltions extrneous to the originl liner theory. In the unified field theory the Evns spin field nd the conjugte product re deduced self consistently from the experimentl oservtion: Eqs.(27.23) nd (27.6) imply: j 0. (27.23) R q = ω T (27.24) to high precision.in other words the Guss lw nd Frdy lw of induction pper to e true within contemporry experimentl precision.the reson for this in generl reltivity (i.e. ojective physics) is Eq.(27.24), constrint of differentil geometry. Using the Murer-Crtn structure equtions of differentil geometry [1]: T = D q (27.25) R = D ω. (27.26) Eq.(27.24) ecomes the following experimentlly implied constrint on the generl unified field theory: ( D ω ) q = ω ( D q ). (27.27) A prticulr solution of Eq.(27.27) is: ω = κɛ cq c (27.28)

6 Guss Lw, Frdy Lw, nd O(3) Electrodynmics where ɛ c is the Levi-Civit tensor in the flt tngent undle spcetime.being flt spcetime, Ltin indices cn e rised nd lowered in contrvrint covrint nottion nd so we my rewrite Eq.(27.28) s: ω = κɛ c q c. (27.29) Eqn.(27.29) sttes tht the spin connection is n ntisymmetric tensor dul to the xil vector within sclr vlued fctor with the dimensions of inverse metres. Thus Eq.(27.29) defines the wve-numer mgnitude, κ, in the unified field theory. It follows from Eqn.(27.29) tht the covrint derivtive defining the torsion form in the first Murer-Crtn structure eqution (27.25) cn e written s: T = d q + ω q (27.30) = d q + κq q c (27.31) from which it follows,using Eqs.(27.3) nd (27.4), tht: F = d A + ga A c (27.32) In the complex circulr sis,eq.(27.32) cn e expnded s the cycliclly symmetric set of three equtions: F (1) = d A (1) iga (2) A (3) (27.33) F (2) = d A (2) iga (3) A (1) (27.34) F (3) = d A (3) iga (1) A (2) (27.35) with O(3) symmetry [3] [33]. These re the defining reltions of O(3) electrodynmics, developed y Evns from 1992 to 2003 [3] [33] It hs een shown tht O(3) electrodynmics is direct result of the unified field theory given the experimentl constrints imposed y the Guss lw nd the Frdy lw of induction. It follows tht O(3) electrodynmics utomticlly produces these lws, i.e.: d F () = 0, = 1, 2, 3 (27.36) s oserved experimentlly.the existence of the conjugte product hs een DEDUCED in Eq.(27.32) from differentil geometry,nd it follows tht the spin field nd the inverse Frdy effect hve lso een deduced from differentil geometry nd the Evns unified field theory of generl reltivity or ojective physics. This is mjor dvnce from the stndrd model nd the MH theory of specil reltivity.

7 27.3 Numericl Methods of Solutions 27.3 Numericl Methods of Solutions 451 In generl the homogeneous nd inhomogeneous Evns field equtions must e solved simultneously for given initil nd oundry conditions. In this section the two equtions re written out in tensor nottion nd susidiry informtion summrized. The homogeneous field eqution in tensor nottion is: µ F νρ + ν F ρµ + ρ F µν = R µν A ρ + R νρ A µ + R ρµ A ν ω µ F νρ ω ν F ρµ ω ρ F µν (27.37) nd this is equivlent to the reones or minimlist nottion of differentil geometry: d F = R A ω F (27.38) where ll indices hve een suppressed. For the purposes of electricl engineering Eq.(27.37) is,to n excellent pproximtion: µ F νρ + ν F ρµ + ρ F µν = 0 (27.39) Eq.(27.39) is equivlent to the Guss Lw pplied to mgnetism: nd the Frdy Lw of induction: B = 0 (27.40) B + E t = 0 (27.41) for ll polriztion sttes. The exceedingly importnt influence of grvittion on electromgnetism nd vice vers must however e computed in generl when the right hnd side of Eq.(27.37) is non-zero. This tiny ut in generl non-zero influence leds to violtion of the well known lws (27.40) nd (27.41), nd this must e serched for with high precision instrumenttion. Eq.(27.39) my e rewritten s the Hodge dul eqution: µ F µν = 0 (27.42) nd for ech index this is homogeneous Mxwell-Heviside field eqution. In generl F µν is the Hodge dul of Fµν in Evns spcetime nd j ν is the Hodge dul of the chrge-current density three-form defined y the right hnd side of Eq.(27.37), i.e.y: µ F νρ + ν F ρµ + ρ F ( µν = µ 0 j µνρ + j νρµ + j ) ρµν (27.43) where:

8 Guss Lw, Frdy Lw, nd O(3) Electrodynmics j µνρ = 1 ) (R µν µ A ρ ω µ F νρ 0 etc. (27.44) Thus: j σ = 1 6 ɛµνρ σj µνρ = 1 6 ɛνρµ σj νρµ (27.45) = 1 6 ɛρµν σj ρµν nd F µσ = 1 2 ɛνρ µσf νρ, F νσ = 1 2 ɛρµ νσf ρµ, (27.46) F ρσ = 1 2 ɛµν ρσf µν. In computing these Hodge duls the correct generl definition mps from p-form of differentil geometry to n (n-p)-form of differentil geometry in the generl n dimensionl mnifold. The generl four dimensionl mnifold is the Evns spcetime, so nmed to distinguish it from the Riemnnin spcetime used in Einstein s field theory of grvittion. The Hodge dul is in generl [1]: X µ1...µ n p = 1 p! ɛν1...νp µ 1...µ n p X ν1...ν p. (27.47) The generl Levi-Civit symol is: 1 if µ 1 µ 2...µ n is n even permuttion ɛ µ 1µ 2...µ n = 1 if µ 1 µ 2...µ n is n odd permuttion 0 otherwise (27.48) nd the Levi-Civit tensor used in Eq.(27.47) is: ɛ µ1µ 2...µ n = ( g ) 1/2 ɛ µ 1µ 2...µ n (27.49) where g is the numericl vlue (i.e. numer) of the determinnt of the metric tensor g µν. The field tensor is defined y the torsion tensor: nd the potentil is defined y the tetrd: F µν = A (0) T µν (27.50)

9 where: 27.3 Numericl Methods of Solutions 453 A µ = A (0) q µ. (27.51) The metric is fctorized into dot product of tetrds: g µν = q µq ν η (27.52) η = (27.53) is the digonl metric tensor of the tngent undle spcetime, Minkowski or flt spcetime. The gmm nd spin connections re relted y the tetrd postulte: D µ q ν = µ q ν + ω µ q ν Γ λ µν q λ = 0. (27.54) The torsion nd curvture (or Riemnn) tensors re defined y the Murer-Crtn structure reltions of differentil geometry. The torsion tensor is the covrint derivtive of the tetrd nd is: T λ µν = qλ T µν = Γ λ µν Γ λ νµ. (27.55) It therefore vnishes for the Christoffel connection of Einstein s grvittionl theory: Γ λ µν = Γ λ νµ. (27.56) The curvture tensor is the covrint derivtive of the spin connection nd is: R σ λνµ = qσ q λr νµ. (27.57) It follows tht: nd R νµ = νω µ µω ν + ω νcω c µ ω µcω c ν (27.58) R σ λνµ = νγ σ µλ µγ σ νλ + Γ σ νρ Γ ρ µλ Γ σ µρ Γ ρ νλ. (27.59) The Evns spcetime is therefore completely defined y the structure reltions. The homogeneous field eqution is the first Binchi identity of differentil geometry within the C negtive fctor A (0). The second Binchi identity leds to the Noether Theorem of Evns field theory, nd sttes tht the covrint derivtive of the curvture tensor vnishes identiclly. Note tht Eq.(27.39) is equivlent to the use of Riemnn norml coordintes nd loclly flt spcetime, ecuse it is n eqution in ordinry derivtives, not covrint derivtives. In generl, g, the modulus of the determinnt of the metric, in Eq.(27.49) is function of x µ, ut t point p in

10 Guss Lw, Frdy Lw, nd O(3) Electrodynmics the Evns spcetime or mnifold M it is lwys possile to define the Riemnn norml coordinte system so the metric is in cnonicl form, nd: µ g = 0. (27.60) This defines the loclly flt spcetime. Note crefully tht the generl homogeneous eqution (27.37) cnnot e expressed s Hodge dul eqution of type (27.39), so the pproprite eqution for numericl solution must lwys e Eq.(27.37). Eq.(27.39) is mentioned only ecuse of the trditionl method of expressing the homogeneous Mxwell-Heviside eqution of Minkowski spcetime (HME) s the Hodge dul eqution. The correct form of the HME is the following Binchi identity of Minkowski spcetime: λ F µν + µ F νλ + ν F λµ = 0 (27.61) spcetime in which the Hodge dul is definle s: F µν = 1 2 ɛµνρσ F ρσ. (27.62) It my then e proven tht Eq.(27.61) is the sme eqution s: The proof is s follows. From Eq.(27.63): nd using Eq.(27.62): µ F µν = 0. (27.63) λ F λρ + µ F µρ + ν F νρ = 0 (27.64) 1 ( ( λ ɛ λρµν ) ( F 2 µν + µ ɛ µρνλ ) ( F νλ + ν ɛ νρλµ )) F λµ = 0. (27.65) Using the Leiniz Theorem nd the constncy of ɛ µνρσ in Minkowski spcetime, Eq.(27.65) ecomes: which is upon using: ɛ λρµν λ F µν + ɛ µρνλ µ F νλ + ɛ νρλµ ν F λµ = 0. (27.66) We my dd individul indices of Eq.(27.66), to give, for exmple: ɛ 1ρ23 1 F 23 + ɛ 2ρ31 2 F 31 + ɛ 3ρ12 3 F 12 + = 0 (27.67) 1 F F F 12 + = 0 (27.68) ɛ 1023 = ɛ 2031 = ɛ 3012 = 1. (27.69) Proceeding in this wy we see tht Eq.(27.61) is the sme s Eq.(27.63).Note crefully tht this proof is not true in generl for Evns spcetime, ecuse in tht spcetime we otin results such s:

11 27.3 Numericl Methods of Solutions 455 µ F µσ = 1 ( 2 µ ɛ νρ µσf ) 1 ( νρ = ɛ νρ 2 µσ µ F νρ + ( µ ɛ νρ ) µσ) F νρ nd since depends on, one otins: (27.70) µ ɛ νρ µσ 0 (27.71) nd there is n extr term which does not pper in Eq.(27.63) of Minkowski spcetime. Only in the specil cse of Eq.(27.62) do we otin: µ ɛ νρ µσ = 0. (27.72) Therefore in generl, the homogeneous Evns eqution (27.37) must e solved simultneously with the inhomogeneous Evns eqution.the ltter is n expression for the covrint exterior derivtive of the Hodge dul of Fµν. In the minimlist or reones nottion of differentil geometry this expression is: d F = R A ω F = µ 0 J (27.73) where R denotes the Hodge dul of the curvture form. Eq.(27.73) is the ojective or generlly covrint expression in unified field theory of the inhomogeneous Mxwell-Heviside eqution (IMH) of specil reltivity: d F = µ 0 J. (27.74) It is seen y comprison of Eqs.(27.73) nd (27.74) tht in the unified field theory d hs een replced y D s required. Since oth F nd R in Eq.(27.38) re two-forms it follows y symmetry tht if one tkes the Hodge dul of F on the left hnd side, one must tke the Hodge dul of R on the right hnd side. The reson is tht the Hodge dul of ny two-form in Evns spcetime is lwys nother two-form. It is convenient to rewrite Eq.(27.73) s: D F = R A (27.75) nd in nlogy with Eq.(27.37), the tensoril expression for Eq.(27.73) is: µ F νρ + ν F ρµ + ρ F µν = R µν A ρ + R νρ A µ + R ρµ A ν ω µ F νρ ω ν F ρµ ω ρ F µν. (27.76) Therefore the generl computtionl tsk is to solve Eqs.(27.37) nd (27.76) simultneously for given initil nd oundry conditions. In the lst nlysis this is prolem in simultneous prtil differentil equtions. We my now define the inhomogeneous current ( J µνρ = 1 µ R 0 µν A ρ ω F ) µ νρ (27.77).

12 Guss Lw, Frdy Lw, nd O(3) Electrodynmics nd we my note tht J µνρ is in generl much lrger thn the homogeneous current j µνρ. Therefore J µνρ is of gret prcticl importnce for the cquisition of electric power from Evns spcetime nd for counter grvittionl technology in the erospce industry. Finlly convenient form of the inhomogeneous field eqution my e otined from the following considertions.we first construct the following Hodge dul: F µν = 1 2 ɛµνρσ F ρσ (27.78) in the MH limit: nd note tht: d F = 0 (27.79) d F = µ 0 J (27.80) ( d F ) µν 1 2 ɛµνρσ (d F ) ρσ. (27.81) In convenient shorthnd nottion this result my e written s: result which ecomes: d F 1 2 ɛ d F (27.82) d F 1 2 g 1 2 ɛ d F (27.83) in Evns spcetime. This is key result ecuse it shows tht J is not zero if j is zero or lmost zero, nd it is J tht is the importnt current term for the cquisition of electric power from Evns spcetime. For the purposes of computtion systemtic method of constructing the inhomogeneous Evns field eqution (IE) is needed, where every term needed for coding is defined precisely. In order to do this egin with the fundmentl definitions of differentil geometry, the two Murer-Crtn structure equtions: nd the two Binchi identities: T = D q (27.84) R = D ω (27.85) D T = R q (27.86) D R = 0. (27.87) In order to correctly construct the Hodge duls of T nd R ppering in the IE the determinnt of the metric must e defined correctly in ech cse

13 27.3 Numericl Methods of Solutions 457 (see Eq.(27.49)). In order to proceed consider the limit of the Evns unified field theory tht gives Einstein s field theory of grvittion uninfluenced y electromgnetism. In the Einstein limit the metric tensor is symmetric nd is defined y the inner or dot product of two tetrds: (S) gµν = q (S) µ q (S) ν η. (27.88) The differentil geometry pproprite to the Einstein theory is then: T (S) = 0 (27.89) d q (S) = ω (S) q (S), (27.90) R (S) q (S) = 0. (27.91) The determinnt of the metric is defined in this limit y: (S) g (S) = gµν (27.92) nd so the Hodge dul of the Riemnn form is defined in Einstein s theory of grvittion y: R (S) = 1 2 g (S) µν 1 2 ɛ R (S). (27.93) Consider next the limit of the Evns field theory tht gives the free electromgnetic field when there is no field mtter interction, mtter eing defined y the presence of non-zero mss. The differentil geometry tht defines this limit is: T (A) = D q (A), (27.94) R (A) g c (A) µν = q (A) nd the determinnt of the metric is: = D ω (A), (27.95) µ q (A) ν, (27.96) g (A) = g c (A) µν. (27.97) The Hodge duls of the torsion nd Riemnn forms for the free electromgnetic field re therefore: T (A) = 1 2 g(a) 1 2 ɛ T (A), (27.98) R (A) = 1 2 g(a) 1 2 ɛ R (A). (27.99) Thirdly, when, the electromgnetic field intercts with mtter, s in the IE, the pproprite differentil geometry is:

14 Guss Lw, Frdy Lw, nd O(3) Electrodynmics nd the determinnt of the metric tensor is now: T = D q, (27.100) R = D ω, (27.101) g µν = q µq ν, (27.102) g = g µν. (27.103) The metric tensor itself is the sum of symmetric nd nti-symmetric component metric tensors: q µq ν = 1 ( (q 2 µ q ) (S) ( ν + q µ q ) (A) ) ν. (27.104) The Hodge dul of the Riemnn form in the IE is therefore: R = 1 2 g 1 2 ɛ R, (27.105) ecuse in generl oth the symmetric nd ntisymmetric metrics contriute to the Riemnn tensor or Riemnn form when there is field mtter interction However, the Hodge dul of the torsion form in the IE is: T = 1 2 g(a) 1 2 ɛ T, (27.106) ecuse only the ntisymmetric metric contriutes to the torsion tensor or torsion form from Eq.(27.94). Therefore the computtionl lgorithm is fully defined, nd the computtionl tsk in generl is to solve the HE nd IE SIMULTANEOUSLY for given initil nd oundry conditions. The HE nd IE contin more informtion thn the equivlents in Mxwell-Heviside field theory, nd the engineering tsk is to CAD/CAM circuit tking electric power from Evns spcetime, defined y the generl four dimensionl mnifold in which the Riemnn nd torsion tensors re oth non-zero. In the Minkowski spcetime of Mxwell-Heviside field theory oth tensors re zero. We must use the computer to define this extr source of power nd to optimize circuits which utilize this extr source of power in prcticl devices. Acknowledgments The Ted Annis Foundtion, Crddock Inc., John B.Hrt nd other scholrs re thnked for funding this work, nd the AIAS group for mny interesting discussions.

15 References 1. S.P.Crroll., Lecture Notes in Generl Reltivity ( grdute course t Hrvrd, UC Snt Brr nd U.Chicgo, pulic domin rxic:gr-gq v1 Dec 1997, pulic domin). 2. L.H.Ryder, Quntum Field Theory, (Cmridge Univ Press,1996,2nd ed.). 3. M.W.Evns, A Unified Field Theory for Grvittion nd Electromgnetism, Found.Phys.Lett., 16, 367 (2003). 4. M.W.Evns, A Generlly Covrint Wve Eqution for Grnd Unified Field Theory, Found.Phys.Lett., 16, 507 (2003). 5. M.W.Evns, The Equtions of Grnd Unified Field Theory in Terms of the Murer-Crtn Structure reltions of Differentil Geometry, Found.Phys.Lett.,17,25 (2004). 6. M.W.Evns, Derivtion of Dirc s Eqution from the Evns Wve Eqution, Found.Phys.Lett.,17,149 (2004). 7. M.W.Evns, Unifiction of the Grvittionl nd Strong Nucler Fields, Found.Phys.Lett., 17,267 (2004). 8. M.W.Evns, The Evns Lemm of Differentil Geometry, Found.Phys.Lett., 17,433 (2004). 9. M.W.Evns, Derivtion of the Evns Wve Eqution form the Lgrngin nd Action,Origin of the Plnck Constnt in Generl Reltivity, Found.Phys.Lett.,17,535 (2004). 10. M.W.Evns et. l. (AIAS Author Group), Development of the Evns Wve Eqution in the Wek Field Limit: the Electrogrvitic Eqution, Found.Phys.Lett., 17,497 (2004). 11. M.W.Evns, Physicl Optics, the Sgnc Effect nd the Ahronov Bohm Effect in the Evns Unified Field Theory, Found.Phys.Lett., 17, 301 (2004). 12. M.W.Evns, Derivtion of the Geometricl Phse from the Evns Phse Lw of Generlly Covrint Unified Field Theory, Found.Phys.Lett.,17,393 (2004). 13. M.W.Evns, Derivtion of the Lorentz Boost from the Evns Wve Eqution, Found.Phys.Lett.,17,663 (2004). 14. M.W.Evns,The Electromgnetic Sector of the Evns Field Theory, Found.Phys.Lett., in press (2005),preprint on M.W.Evns, New Concepts from the Evns Field Theory,Prt One:The Evolution of Curvture, Oscilltory Universe without Singulrity nd generl Force nd Field Equtions, Found.Phys.Lett., in press (2005),preprint on

16 460 References 16. M.W.Evns, New Concepts from the Evns Field Theory,Prt Two: Derivtion of the Heisenerg Eqution nd Reinterprettion of the Heisenerg Uncertinty Principle, Found.Phys.Lett., in press (2005), preprint on M.W.Evns, The Spinning nd Curving of Spcetime, the Electromgnetic nd Grvittionl Fields in the Evns Unified Field Theory, Found.Phys.Lett., in press,(2005), preprint on M.W.Evns, Derivtion of O(3) Electrodynmics from Generlly Covrint Unified Field Theory, Found.Phys.Lett., in press, preprint on M.W.Evns, Derivtion of O(3) Electrodynmics form the Evns Unified Field Theory, Found.Phys.Lett., in press, preprint on M.W.Evns, Clcultion of the Anomlous Mgnetic Moment of the Electron, Found.Phys.Lett., in press, preprint on M.W.Evns, Generlly Covrint Electro-wek Theory, Found.Phys.Lett., in press, preprint on M.W.Evns, Evns Field Theory of Neutrino Oscilltions, Found.Phys.Lett., in press, preprint on M.W.Evns, The Interction of Grvittion nd Electromgnetism, Found.Phys.Lett., preprint on M.W.Evns, Electromgnetic Energy from Grvittion, Found.Phys.Lett., in press, preprint on M.W.Evns, The Fundmentl Invrints of the Evns Field Theory, Found.Phys.Lett., in press, preprint on M.W.Evns, The Homogeneous nd Inhomogeneous Evns Field Equtions, Found.Phys.Lett., in press, preprint on M.W.Evns, Generlly Covrint Unified Field Theory: the Geometriztion of Physics, (in press, 2005), preprint on L.Felker, The Evns Equtions, (freshmn text in prep.). 29. M.Anderson, Artspeed Wesite for the Collected Works of Myron Evns, M.W.Evns nd L.B.Crowell, Clssicl nd Quntum Electrodynmics nd the B (3) Field (World Scientific,Singpore,2001). 31. M.W.Evns (ed.), Modern Nonliner Optics, specil topicl issue in three prts of I.Prigogine nd S.A.Rice (series eds.), Advnces in Chemicl Physics, (Wile Interscience, New York, 2001, 2nd ed., nd 1992,1993,1997,1st ed.), vols 119 nd M.W.Evns, J.-P.Vigier et l, The Enigmtic Photon, (Kluwer,Dordrecht,1994 to 2002,hrdck nd softck), in five volumes. 33. M.W.Evns nd A.A.Hsnein, The Photomgneton in Quntum Field Theory (World Scientific,Singpore,1994) 34. M.W.Evns, Physic B, 182,227,237 (1992).