GENERALLY COVARIANT UNIFIED FIELD THEORY

Transcription

1 GENERALLY COVARIANT UNIFIED FIELD THEORY THE GEOMETRIZATION OF PHYSICS VOLUME VII Myron W. Evans, Horst Eckardt, Douglas Lindstrom February 18, 2010

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3 i This book is dedicated to true progress in physics

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5 Preface In this volume fteen further papers of Einstein Cartan Evans (ECE) eld theory are given dealing with aspects of torsion based ECE cosmology and the ECE antisymmetry law and applications. As usual the papers are arranged into chapters. Chapter 1 gives an explanation of the whirlpool galaxy from constant spacetime torsion, eliminating the need for dark matter, a gment of imagination of the now obsolete ("standard") physics. Chapter 2 develops the link between spacetime torsion in ECE theory and the theory of angular momentum. Chapter 3 oers a plausible explanation of the resonant initial event in ECE cosmology in order to replace the incorrect and obsolete big bang theory. Chapter 4 develops orbital dynamics in terms of spacetime angular momentum, eliminating the need for the Einstein eld equation. Chapter 5 is on the fundamental origin of angular momentum in ECE theory and chapter 6 on metric compatibility from Cartan geometry. Chapter 7 is a entirely novel ECE theory of the rest fermion, which is extended to the fermion with nite momentum in chapter 8. Chapter 9 is on the potential antisymmetry equations of electromagnetic and gravitational theory, and the antisymmetry law is extended to the natural sciences and engineering in chapter 10 and applied to the ECE engineering model in chapter 11. The nal antsymmetry law of Cartan geometry is developed in chapter 12. In chapter 13 the ECE theory of SU(2) quantized electrodynamics is developed and in chapter 14 the unied eld is developed in SU(2) and SU(3) representation spaces. Chpater 15 gives a rigorous sef-checking proof of the Cartan Evans identity based on the antisymmetry of the connection of Riemann geometry. For ease of reference a general overview of ECE theory and the collapse of the standard model is given as follows as part of this preface. The overview gives full details of the unprecedented interest in ECE theory recorded daily for ve and a half years. This is clear evidence for the fact that ECE theory has been accepted as the leading unied eld theory. Craigcefnparc, Wales November 2009 Myron W. Evans The British and Commonwealth Civil List Scientist iii

6 PREFACE iv

7 Contents 1 Explanation of the Whirlpool Galaxy from Constant Space- Time Torsion: The Case Against Dark Matter Introduction Calculation of the Logarithmic Spiral Stellar Orbit due to Constant Space-Time Torsion Approximations for Angular Momentum and Velocity Simulation of Dynamical Galaxy Behaviour The Link between Space-Time Torsion in ECE Theory and the Theory of Angular Momentum Introduction The Link Between Torsion and Angular Momentum Resonant Initial Event in ECE Cosmology Introduction Spin Connection Resonance as an Initial Condition Spacetime Dynamics Underlying a Logarithmic Spiral Trajectory 44 4 Orbital Dynamics in Terms of Spacetime Angular Momentum Introduction Conservation of the Angular Momentum of Spacetime The Origin of Kinetic and Potential Energy and Force, and Free Fall Limit On The Fundamental Origin of Angular Momentum in Cartan Geometry Introduction Relation between Angular Momentum and Spacetime Torsion The Plane wave Tetrads On Metric Compatibility from Cartan's Geometry Introduction Development of the Cartan Tetrad in the Base Manifold The Existence of Spin Connection Resonance

8 CONTENTS 7 Einstein Cartan Evans (ECE) Theory of the Rest Fermion Introduction ECE Rest Fermion Equation With 2 x 2 Matrices The Anti-Fermion ECE Equation of the Fermion with Finite Momentum Introduction Derivation of the ECE Fermion Equation Comparison of the ECE and Dirac Equations Potential Anti-Symmetry Equations of Electromagnetic and Gravitational Theory Introduction Commutator Method and Antisymmetry Equations Some Field Potential Equations on the ECE Level ECE Antisymmetry Laws in the Natural Sciences and Engineering Introduction The Incompatibility of U(1) Gauge Symmetry Electrodynamics and Fundamental Potential Antisymmetry Novel Antisymmetries of the Riemann Curvature Discussion Antisymmetry Constraints in the ECE Engineering Model Introduction The Commutator Antisymmetry Law Antisymmetry in the ECE Engineering Model The Antisymmetry Law of Cartan Geometry: Applications to Electromagnetism and Gravitation Introduction Geometrical Antisymmetry Laws and Application to Physics Hodge Duality, Inhomogeneous Field Eqaution and Electromagnetic Potential Electromagnetic Equations Suitable for Numerical Analysis Appendix A - Dependence of the Coulomb and Gauss Law on the Ampère-Maxwell Law Appendix B - Derivation of Standard Electromagetic Theory from Specialized Antisymmetry Constraints Theory of SU(2) Quantum Electrodynamics Introduction Equations of the Fermion and Photon Interaction of the Fermion and Photon

9 CONTENTS 14 Development of the Unied Field in SU(2) and SU(3) Representation Spaces Introduction Factorization of the d'alembertian Operator Development of the Unied Field in SU(3) Proof of the Cartan Evans Identity Introduction Detailed Proof Solution of the ECE Vacuum Equations Introduction Antisymmetry conditions and equations of state Direct setup of an equation set Pre-evaluation of magnetic constraints Solutions of the vacuum equations Energy and momentum density Standard theory ECE theory Discussion Lindstrom constraint Relations of time dependent constants Character of solutions Resonant Coulomb law Topological charge density Reduction of the ECE Theory of Electromagnetism to the Maxwell- Heaviside Theory Introduction Reduction of ECE EM Theory to that of Maxwell-Heaviside Conclusions Appendix: Limitations Imposed When Using the Lindstrom Constraint Overview of ECE Theory: The Collapse of the Standard Model The Source Papers Articles by Colleagues Summary Analysis of Feedback Patterns Monthly Feedback statistics for to May2009) Appendix: Selected Feedback Interest in ECE, April 30th 2004 to Present 253 3

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11 Chapter 1 Explanation of the Whirlpool Galaxy from Constant Space-Time Torsion: The Case Against Dark Matter by Myron W. Evans 1 and H. Eckardt 2 Alpha Institute for Advanced Study (AIAS) ( Abstract Constant torsion of space-time in general relativity produces a constant angular momentum of space-time by volume integration. It is shown that this constant angular momentum gives rise straightforwardly to a potential energy proportional to inverse square distance - a negative valued centrifugal potential energy that does work on a star. The potential energy attracts the star into a logarithmic spiral orbit through a negative valued force law that is inversely proportional to the cube of distance. The orbit gradually becomes a circle with constant orbital velocity. This theory explains the main features of a whirlpool galaxy without any "dark matter". The angular momentum is a constant of motion and is conserved, i.e. does not change with time. Total energy is conserved, and consists of the kinetic energy of a star with velocity v moving in a 1 emyrone@aol.com 2 horsteck@aol.com 5

12 1.1. INTRODUCTION plane, added to the potential energy of the spinning space-time. A whirlpool galaxy is a direct demonstration of the potential energy of spinning space-time. Keywords: Einstein Cartan Evans eld theory, dark matter, cosmology, galaxies. 1.1 Introduction It is well known that the now obsolete Einsteinian general relativity omits consideration of space-time torsion, and in so doing uses an incorrect symmetric connection [1] - [10] with multiple sequential mathematical errors that render the theory meaningless in physics. By correctly considering the space-time torsion [1] - [10] a new cosmology has been constructed in Einstein Cartan Evans (ECE) theory, a cosmology that is based directly on Cartan geometry. In so doing the space-time torsion plays a central role. In Section 2 it is shown that a constant space-time torsion is sucient to produce the main features of a whirlpool galaxy, in which stars move on a logarithmic spiral orbit contrary to Newtonian dynamics. The constant torsion is integrated over a volume to produce a constant angular momentum of space-time and a negative valued potential energy that does work on the star, attracting it into a logarithmic spiral orbit. The potential energy is inversely proportional to r 2, and produces a negative valued force of attraction that is inversely proportional to r 3. A Lagrangian analysis of the problem shows that the orbit due to such a force law is a logarithmic spiral as observed experimentally. The orbit gradually becomes a circle in which the orbital linear velocity is constant as observed experimentally. The angular momentum is a constant of the motion and does not change with time in this simplest theory. The angular momentum is therefore conserved. The total energy is also conserved, and is the sum of the kinetic energy due to the linear velocity of the star in a plane, and the potential energy caused by the constant torsion of space-time. In Section 3, a graphical analysis of the evolution of the whirlpool galaxy is given. The orbital equations may also be animated for direct visualization. Some discussion is given of this theory and of the main experimental features of a whirlpool galaxy. This simplest theory is soluble analytically, and is designed to produce only the main features of the galaxy. More realistic models would include a varying torsion and the dynamical equations of ECE theory solved numerically. Severe criticism of the obsolete "dark matter" speculation is summarized. 1.2 Calculation of the Logarithmic Spiral Stellar Orbit due to Constant Space-Time Torsion The space-time torsion tensor is dened [1] - [10] in ECE theory by the Cartan - Evans dual identity: D µ T κµν = R κ µν µ (1.1) 6

13 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY... where its covariant derivative is the curvature tensor appearing on the right hand side of Eq. (1.1). In general the torsion tensor may be integrated over a hyper-surface to give a rank two anti-symmetric tensor: T µν = T νµ = T κµν dσ κ. (1.2) σ This tensor denes the angular momentum tensor of space-time through the following proportionality: J µν = c k T µν (1.3) where k is Einstein's constant and c the vacuum speed of light. Eq. (1.3) is a hypothesis that asserts that the integrated torsion tensor T µν is proportional to the angular momentum tensor. It is well known [11] that the angular momentum is dened by the following volume integration of the angular momentum/angular energy density tensor in eld theory: J µν = J 0µν dv (1.4) and similarly: T µν = T 0µν dv. (1.5) Therefore the hypothesis Eq. (1.3) is one way of correcting the Einstein eld equation for the presence of torsion [1] - [10]. Consider now the space-time angular momentum in the Z axis dened by: J Z = J 12 = J 012 dv. (1.6) This is a Z axis angular momentum generated by space-time itself. It does not exist in Einsteinian theory, and does not exist in Newtonian theory. It is a concept of the ECE unied eld theory. The space-time angular momentum produces a negative valued potential energy: U = J 2 2mr 2 (1.7) where m is the mass of a star moving in the spinning space-time and r is the radial distance of the star from the force centre (the centre of the spacetime "whirlpool"). Work is done on the star by the spinning space-time and changes the star's potential energy from U 1 to U 2 while keeping the star's kinetic energy constant. In a whirlpool galaxy the stars move in a plane to a good approximation, so a star's kinetic energy is dened by: T = 1 2 mv2 (1.8) 7

14 1.2. CALCULATION OF THE LOGARITHMIC SPIRAL STELLAR... where the linear velocity in the plane is expressed [11] in terms of plane polar coordinates: v = ṙ e r + r θ e θ. (1.9) Therefore the kinetic energy of a star that moves in a plane with any velocity (v) is: T = 1 2 m (ṙ 2 + r 2 θ2 ). (1.10) The force on the star due to the constant angular momentum Eq. space-time is: (1.7) of 2 1 F dr = U 1 U 2 (1.11) and changes the star from state 1 to 2 while keeping the kinetic energy constant. This is the denition of potential energy [11]. If: U 2 > U 1 (1.12) the force is attractive and negative valued. The initial state is chosen such that: U 1 = 0. (1.13) The force on the star due to the spinning space-time is negative valued: F = U = J 2 mr 3 e r (1.14) and attracts the star into an orbit. It is shown as follows that this is a logarithmic spiral orbit. The total energy of the system is: E = T + U (1.15) and consists of the kinetic energy of the star moving at v in a plane: T = 1 2 m (ṙ 2 + r 2 θ2 ) (1.16) and potential energy due to the spinning space-time: U = J 2 2mr 2. (1.17) The Lagrangian of the system is: L = E U (1.18) 8

15 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY... and the Euler Lagrange equations of motion are: and L r = d L dt ṙ L θ = d dt (1.19) L = 0. (1.20) θ Eq. (1.19) can be rewritten [11] as: d 2 dθ 2 ( ) 1 + r 1r = mr2 F (r) (1.21) J 2 by using a change of variable. If the potential energy Eq. (1.17) is expressed as: U(r) = J 2 2mr 2 ( 1 + α 2 ) (1.22) the force is: F (r) = J 2 and Eq. (1.21) shows that: mr 3 ( 1 + α 2 ) (1.23) r = r 0 exp(αθ). (1.24) This is a logarithmic spiral orbit as observed experimentally. The star evolves with time as follows: θ(t) = 1 ( ) 2αJ 2α log t + C (1.25) and r(t) = mr 2 0 ( ) 1/2 2αJ m t + r2 0C (1.26) where C is an integration constant. Both quantities (with all constants set to unity) are depicted in Figs. 1.1 and 1.2, showing their sublinear time dependence. The angular velocity is dened [11] as: ω = θ = dθ dt = J (1.27) mr 2 and the radial velocity is dened as: v r = ṙ = dr dt = αj mr. (1.28) 9

16 1.2. CALCULATION OF THE LOGARITHMIC SPIRAL STELLAR θ(t) dθ/dt θ(t), dθ/dt t Figure 1.1: Time dependence of θ coordinate for a spiralling star r(t) dr/dt r(t), dr/dt t Figure 1.2: Time dependence of r coordinate for a spiralling star. 10

17 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY v(t) t Figure 1.3: Time dependence of velocity v for a spiralling star. The total velocity of the star is therefore dened by: v 2 = ṙ 2 + r 2 θ2 = ( 1 + α 2) ( ) 2 J (1.29) mr (see Fig. 1.3) and so the angular momentum magnitude is: mvr J =. (1.30) (1 + α 2 1/2 ) It is observed in a whirlpool galaxy that v in the arms of the galaxy is a constant, and that the arms are logarithmic spirals of stars, i.e. stars moving on a logarithmic spiral. The angular momentum J is constant and is dened by the lagrangian: J = L = U θ θ = mr2 θ (1.31) and so the angular momentum of the spinning space-time is related to the potential energy by: J = U θ = ( θ 1 ) 2 mr2 θ2 (1.32) If v and J are constants then from Eq. (1.30), r is also constant, meaning that the orbit evolves to a circle. The Newtonian attraction of the star to the 11

18 1.3. APPROXIMATIONS FOR ANGULAR MOMENTUM AND... heavy mass at the centre of the galaxy is balanced by the centrifugal force of the spinning space-time which attracts the stars outwards. The angular momentum is constant and given by: J mv = r = constant. (1.33) (1 + α 2 1/2 ) For each spiral of the galaxy, the parameter α is characteristic of that spiral, and the observed rotation curve is such that the velocity v is constant over large distances from the centre, meaning that the velocity v is much greater than that expected from Kepler's equation: v 2 = k m ( 2 r 1 a ), F = k r 2. (1.34) This fact is explained in this paper by an additional v due to Eq. (1.30), i.e. due to spinning space-time. The simplest model of this paper may be elaborated in many dierent ways. 1.3 Approximations for Angular Momentum and Velocity The angular momentum J of spinning space-time is given by Eq. (1.30). J is assumed to be constant throughout the spiral arms of a galaxy. Experimentally it is found that the velocity v of stars in the galaxy arms is constant too. According to Eq. (1.30) then the spiral parameter α has to be variable with r. Assuming this, the potential U Eq. (1.22) takes the form U 1 (r) = J 2 ( 1 + α 2 (r) ) 2mr 2. (1.35) Correspondingly the force can be written with F 1 (r) = U 1 r = J 2 ( 1 + α 2 (r) ) mr 3 + J 2 mr 2 α(r)α (r) (1.36) α (r) = dα(r) dr. (1.37) In order to obtain the original force law which gives the logarithmic spiral orbits we dene a potential U 2 (r) = J ( α 2 (r) ) J 2 2mr 2 + mr 2 α(r) α (r) dr. (1.38) Then we get the original force F 2 (r) = U ( α 2 r = J (r) ) (1.39) mr 3 12

19 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY... of spiral orbits. Now we make the ansatz α(r) := r r 0 (1.40) with a characteristic length r 0. From Eq. (1.38) we then obtain ( U 2 (r) = J 2 ( ) ) 2 r 2mr J 2 log(r) r 0 mr0 2. (1.41) From the Lagrange Function L = T U 2 (1.42) and the Lagrange Equation L r d L dt ṙ = 0 (1.43) the radial part of the equation of motion becomes m r = mr θ J 2 (1 + α 2 (r)) mr 3 or with Eq. (1.40): + J 2 α(r)α (r) mr 2 (1.44) ( ) ) 2 J 2 r (1 + m r = mr θ r 0 mr 3 + J 2 mr0 2r (1.45) There is a strong centrifugal term proportional to 1/r now. From Eq. (1.30) the angular momentum for a constant v = v 0 becomes J = mv 0 r (1 + ( rr0 ) 2 ) 1/2 (1.46) (see Fig. 1.4) which in the limit r goes towards J = mv 0 r 0. (1.47) The orbits can be derived from the Euler Lagrange equation. We make the ansatz r = r 0 exp(α(r) θ) (1.48) and will show that this fullls the force law Eq. (1.39). From Eq. (1.48) follows 1 r = 1 r 0 exp( α(r) θ) (1.49) 13

20 1.3. APPROXIMATIONS FOR ANGULAR MOMENTUM AND J(r) r Figure 1.4: Angular momentum J(r) for radius-dependent α v(r) r Figure 1.5: Velocity v(r) for radius-dependent α. 14

21 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY U Spiral U Torsion 2 1 U(r) r Figure 1.6: Potential for radius-dependent α, model of Eq. (1.40). d dθ d 2 dθ 2 ( ) 1 = α(r) exp( α(r) θ) (1.50) r r 0 ( ) 1 = α2 (r) (1.51) r r The Euler Lagrange equation is d 2 ( ) 1 1r dθ 2 + = mr2 F (r), (1.52) r J 2 so F (r) = J 2 mr 3 ( 1 + α 2 (r) ) (1.53) q.e.d. The orbital velocity Eq. (1.29) is v 2 = ( ) 2 J ( 1 + α 2 (r) ). (1.54) mr Inserting the above approach Eq. (1.40) for α leads to ( ) ( 2 ( ) ) 2 J r v 2 = 1 + mr r 0 (1.55) 15

22 1.3. APPROXIMATIONS FOR ANGULAR MOMENTUM AND... which again has a constant limit for large r: v 0 = J mr 0 (1.56) and is consistent with Eq. (1.47). It is shown in Fig The logarithmic potential U 2 which is induced by constant angular momentum is compared in Fig. 1.6 with the pure spiral potential being proportional to 1/r 2. As an alternative approach, let's start directly with the condition that Eq. (1.30) is exactly constant: mv 0 r J = (1 + α 2 (r)) = J 1/2 0 = const. (1.57) From this condition we obtain α(r) directly: r α(r) = 2 1 (1.58) with r 2 0 r 0 := J 0 mv 0. (1.59) Computeralgebra then delivers quite simple expressions for the potential and force law: ( U 2 = m v0 2 log (r) 1 ), (1.60) 2 F 2 = m v2 0. (1.61) r This is the potential and force law for spiralling orbits where angular momentum and orbital velocity are strictly constant. The velocity condition changes the spiral 1/r 3 force law to a 1/r force law. The potential is a logarithmic function with a constant shift (see Fig. 1.7, in comparison to pure spiral potential). It is known that a 1/r force is longer reaching than a Newtonian 1/r 2 force. This explains in a natural way why galaxies are developing spiral structures outside the central bulge region. Work done on the star implies a negative valued potential energy by convention, and an attractive force by convention. This attracts the star outwards from the centre of the galaxy. To obtain a constant velocity as observed, the second positive valued term of Eq. (1.38) is needed. This means that the galaxy develops spirals of stars which reach a constant velocity - the graph of velocity against r is a plateau. In a third approach we start with the expression for the asymptotically contant velocity which according to Eq. (1.54) is v 2 = ( ) 2 J0 ( 1 + α 2 (r) ). (1.62) mr 16

23 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY U Spiral U Torsion 2 1 U(r) r Figure 1.7: Potential for radius-dependent α, model of Eq. (1.58). It is observed that in the limit r we have v v 0, so 1 + α 2 (r) r 2 1 r 2 0 (1.63) with r 0 = J 0 mv 0 (1.64) as in Eq. (1.59). The force Eq. (1.53) therefore becomes F 2 = m v2 0 r (1.65) which is self-consistently the same result as Eq. (1.61). 1.4 Simulation of Dynamical Galaxy Behaviour Finally we describe a method for numerical solution of the galaxy problem. According to [12] the equations of motion can be derived from the kinetic energy Eq. (1.16) and potential energy Eq. (1.60) via the Lagrange function Eq. (1.42). Besides the radial and angular coordinate, we introduce additional variables to 17

24 1.4. SIMULATION OF DYNAMICAL GALAXY BEHAVIOUR Radius r Figure 1.8: Orbit for 1/r 3 potential. obtain dierential equations of rst order which can be solved by the Runge- Kutta method: ṙ = v r, (1.66) θ = ω, (1.67) v r = rω F (r), m (1.68) v θ = 2 v r ω. r (1.69) The rst approach we analyse is the 1/r 3 force law which should give spiralling orbits. The problem is that such orbits are only obtained for a negative force F (r) = 1 r 3. (1.70) Then the orbits spiral inwards, not outwards as in galaxies. Taking the positive value of F (r) does not give spiral orbits since the equations of motion do not exhibit mirror symmetry in space. Therefore we restrict to the statement that a time reversal t t together with a sign reversal of the force F (r) F (r) gives at least conceptually the desired behaviour. Fig. 1.8 shows the orbits of Eq. (1.70), Fig. 1.9 the velocity, angular momentum and total energy. The velocity is not constant because a force component in diretction of θ would be 18

25 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY v J E* Figure 1.9: Velocity, angular momentum and total energy for 1/r 3 potential. required to achieve this. Angular momentum and total energy are conserverd, with exception near to the end of the calculation where numerical errors become signicant. This is because the radius of curvature of the orbit becomes very small near to the center. The next example is more realistic. We added a Neutonian potential with a repulsive term γ/r F (r) = γ r 1 r 2 (1.71) where γ has been adopted to a suitable value of In this combination we obtain outward-spiralling orbits with a limit of a straight line as is predicted by logarithmic spirals (Fig. 1.10). This is an indication that our result Eq. (1.61) is able to predict the correct orbitals in connection with Newtonian attraction in the inner region of a galaxy. From Fig the velocity components v r and ω are shown. One sees that for t the angular component goes to zero and the radial component dominates. It has to be noted nally that in none of the simulations the velocity is constant. The velocity is a dependent variable and is completely determined by the Eqs. ( ). To enforce constancy of velocity we have to dene ω = J 0 mr 2 (1.72) from constancy of angular momentum J 0. Then ω is no more an independent 19

26 1.4. SIMULATION OF DYNAMICAL GALAXY BEHAVIOUR Radius r Figure 1.10: Orbit for potential of type γ/r 1/r 3. variable but dened by r. Therefore the equation system Eqs. ( would have to be modied. ACKNOWLEDGMENTS The sta of A.I.A.S. / T.G.A. is thanked for many interesting discussions. 20

27 CHAPTER 1. EXPLANATION OF THE WHIRLPOOL GALAXY v r ω Figure 1.11: Velocity components for potential of type γ/r 1/r 3. 21

28 1.4. SIMULATION OF DYNAMICAL GALAXY BEHAVIOUR 22

29 Bibliography [1] M. W. Evans, "Generally Covariant Unied Field Theory" (Abramis, 2005 onwards), vols. 1-5, vol. 6 in prep., (see [2] L. Felker, "The Evans Equations of Unied Field Theory" (Abramis 2007). [3] K. Pendergast, "The Life of Myron Evans" (Abramis 2009, preprint on [4] M. W. Evans, "Modern Non-Linear Optics" (Wiley 2001, second edition); M. W. Evans and S. Kielich (eds., ibid., rst edition, 1992, 1993, 1997). [5] M. W. Evans and L. B. Crowell, "Classical and Quantum Electrodynamics and the B(3) Field" (World Scientic 2001). [6] M. W. Evans and J.-P. Vigier, "The Enigmatic Photon" (Kluwer, 1994 to 2002, hardback and softback), in ve volumes. [7] ECE Papers and Articles on [8] M. W. Evans et al., Omnia Opera section of notably from 1992 to present on the ECE theory and its precursor gauge theories homomorphic with those of Barrett, Harmuth and Lehnert. [9] M. W. Evans, Acta Phys. Polonica, 400, 175 (2007); M. W. Evans, Physica B, 403, 517 (2008) [10] M. W. Evans and H. Eckardt, invited papers to journal special issue, [11] L. H. Ryder, "Quantum Field Theory" (Cambridge, 2nd ed., 1996). [12] J. B. Marion and S. T. Thornton, "Classical Dynamics" (HBC, New York, 1988, 3rd.ed.). 23

30 BIBLIOGRAPHY 24

31 Chapter 2 The Link between Space-Time Torsion in ECE Theory and the Theory of Angular Momentum by Myron W. Evans 1 Alpha Institute for Advanced Study (AIAS) ( Abstract The link between space-time torsion and angular momentum is established by integration over a hyper-surface of the rank three angular energy momentum density tensor. The latter is assumed to be proportional by hypothesis to the rank three torsion tensor. Angular momentum theory is highly developed and appears for example in the theory of central orbits, so this link implies that ECE theory can be developed in terms of angular momentum theory, introducing several new techniques from several areas of physics. Keywords: ECE theory, angular momentum theory, integration of a hypersurface. 1 emyrone@aol.com 25

32 2.1. INTRODUCTION 2.1 Introduction In the recently developed [1] - [10] Einstein Cartan Evans (ECE) eld theory the role of space-time torsion is of key importance. It appears for example in the Cartan-Bianchi identity [11] and the Cartan-Evans dual identity [1] - [10]. A valid theory of relativity must obey both identities, and on these grounds the Einstein eld equation has been rejected as incorrect. The torsion denes the acceleration due to gravity, g, and the static electric eld strength E. More generally the torsion denes the electromagnetic eld and the complete eld of gravitation, encompassing the Newtonian g and other elds such as the gravitomagnetic eld. Such phenomena as the equinoctial precession have been shown to be due to the gravitomagnetic eld. The ECE equations of electrical engineering and cosmology are based on the Cartan Bianchi identity and the Cartan Evans dual identity, together with the Cartan Maurer structure equation [1] - [11] that denes torsion as the covariant exterior derivative of the Cartan tetrad. In cosmology, the torsion denes central orbits, and more generally, orbits of all kinds. It is therefore important to develop the theory of spacetime torsion. In Section 2, the rank three spacetime torsion tensor T κµν [1] - [10] is assumed by hypothesis to be proportional to the rank three angular energy/momentum tensor density, J κµν, which is dened [12] from the canonical energy momentum density tensor as is well known in eld theory. The tensor J κµν can be integrated over a hypersurface [12] to give a rank two tensor J µν. Integration over volume [12] gives the well known rank two, antisymmetric, angular momentum tensor: J µν = J νµ = J 0µν dv. (2.1) The theory of angular momentum is well developed [1] - [10], [12] and the procedure given in this paper therefore links ECE theory with angular momentum theory, simplifying the derivation of the ECE engineering model [1] - [10] and introducing concepts about angular momentum from several other areas of physics, both classical and quantum. 2.2 The Link Between Torsion and Angular Momentum The ECE theory and engineering model can be simplied and strengthened by developing it in terms of rank two tensors derived from the general rank three tensor of spacetime torsion. The complete information is still contained in the rank three tensor, but or practical purposes the two index tensors are sucient in general. A three index tensor density may always be integrated over a hypersurface to give a rank two tensor [12]. For example, the conserved charge in the usual theory of the Noether Theorem [12] is dened by: Q ν = J ν µ dσ µ (2.2) σ 26

33 CHAPTER 2. THE LINK BETWEEN SPACE-TIME TORSION IN... where the integral is over a spacelike hypersurface σ µ. This integration gives a rank one tensor form a rank two tensor. In the same way as a two-dimensional surface can be dened in 3-D Euclidean space, a three dimensional hypersurface can be dened in a 4 - D spacetime. Thus: Q ν = Jν 0 dv (2.3) in which: and µ = 0 (2.4) dv = dσ 0 (2.5) where V denotes volume in cubic metres. The rank three angular momentum/energy density tensor is dened [12] as: J µρσ = 1 2 (T µρ x σ T µσ x ρ ) (2.6) where T µρ (not to be confused with torsion) denotes the symmetric canonical energy momentum density tensor and x µ is the coordinate vector: x µ = (ct, X, Y, Z). (2.7) As argued in previous work [1] - [10]: T κµν = k c J κµν (2.8) where T κµν is the spacetime torsion tensor, k is Einstein's constant: k = 8πG c 2 = m kgm 1 (2.9) and c is the vacuum speed of light. The angular momentum tensor is dened in eld theory [12] by: J µν = J κµν dσ κ. (2.10) σ Similarly dene: T µν = T κµν dσ κ (2.11) σ which is an anti-symmetric tensor with the units of torsion (m 1 ) multiplied by cubic metres. Thus: J µν = c k T µν. (2.12) 27

34 2.2. THE LINK BETWEEN TORSION AND ANGULAR MOMENTUM The electromagnetic eld tensor in ECE theory is in general a rank three tensor, but may be expressed as a rank two tensor: and related directly to angular momentum: F µν = A0 V T µν = A (0) T 0µν dv. (2.13) The theory of angular momentum is highly developed, and angular momentum operators [1] - [10], [12] are innitesimal generators within, the reduced Planck constant. The electromagnetic eld tensor is the following volume integral over the electromagnetic tensor density, the latter being the most general expression of the electromagnetic eld in relativity: F µν = F 0µν dv. (2.14) Similarly the Hodge dual eld tensor is: F µν = F 0µν dv. (2.15) The homogeneous and inhomogeneous eld equations of ECE theory are based on the Cartan Bianchi identity: D µ T κµν = R κ µ µν (2.16) and the Cartan Evans dual identity: D µ T κµν = R κ µ µν. (2.17) The two identities may be integrated over volume on both sides as follows: D µ T 0µν dv = R 0 µ µν dv (2.18) and D µ T 0µν dv = R 0 µ µν dv. (2.19) The covariant derivatives are dened by [1] - [10]: D µ T κµν = µ T κµν + ω κ T µλ λµν (2.20) and D µ T κµν = µ T κµν + ω κ µλt λµν. (2.21) Therefore: µ T κµν = j κν (2.22) 28

35 CHAPTER 2. THE LINK BETWEEN SPACE-TIME TORSION IN... µ T κµν = j κν (2.23) where the currents are dened by: j κν = R κ µ µν ω κ µλ T λµν (2.24) j κν = R κ µ µν ω κ µλt λµν. (2.25) Now use: T µν = and j ν = σ σ T κµν dσ κ (2.26) j κν dσ κ (2.27) and similarly for the homogeneous equation The hypersurface is dened by: σ κ = (σ 0, σ) = (V, σ) (2.28) where the volume is dened by its zero'th element: V = σ 0. (2.29) Therefore: T µν = T 0µν dv (2.30) and j ν = j 0ν dv. (2.31) Thus: T 0µν = T µν V, (2.32) j 0ν = jν V. (2.33) With these denitions, Eqs. (2.22) and (2.23) reduce to: µ T µν = j ν, (2.34) µ T µν = j ν, (2.35) where: j ν = V(R 0 µ µν ω 0 µλt λµν ), (2.36) 29

36 2.2. THE LINK BETWEEN TORSION AND ANGULAR MOMENTUM j ν = V( R 0 µ µν ω 0 µλ T λµν ). (2.37) In electrodynamics: µ F µν = A (0) j ν, (2.38) µ F µν = A (0) j ν, (2.39) and if there is no magnetic monopole: µ F µν = 0, (2.40) µ F µν = A (0) j ν. (2.41) In vector notation, Eq. (2.40) becomes the homogeneous eld equations: B = 0 (2.42) E + B t = 0 (2.43) and Eq. (2.41) becomes the inhomogeneous eld equations: E = ρ ɛ 0 (2.44) B 1 c 2 E t = µ 0J (2.45) These are the generally covariant equations of classical electrodynamics. The charge density in coulombs per cubic metre is: ρ = cɛ 0 A (0) (R 0 µ µ0 ω 0 µλt λµ0 ) (2.46) where the units of ca (0) are volts (joules per coulomb) and where the S.I. units of vacuum permittivity, ɛ 0, are J 1 C 2 m 1. The current density is dened as: where: J = J X i + J Y j + J Z k (2.47) J x = ɛ 0 A (0) (R 0 µ µ1 ω 0 µλt λµ1 ) (2.48) J y = ɛ 0 A (0) (R 0 µ µ2 ω 0 µλt λµ2 ) (2.49) J z = ɛ 0 A (0) (R 0 µ µ3 ω 0 µλt λµ3 ) (2.50) The index λ is restricted to 0 in Eq. (2.48) to 2.50 because of Eqs. (2.34) and (2.35) so: ρ = cɛ 0 A (0) (R 0 µ µ0 ω 0 µ0t 0µ0 ) (2.51) J X = ɛ 0 A (0) (R 0 µ µ1 ω 0 µ0t 0µ1 ) (2.52) 30

37 CHAPTER 2. THE LINK BETWEEN SPACE-TIME TORSION IN... J Y = ɛ 0 A (0) (R 0 µ µ2 ω 0 µ0t 0µ2 ) (2.53) J Z = ɛ 0 A (0) (R 0 µ µ3 ω 0 µ0t 0µ3 ) (2.54) It is seen that this method of deducing the vector structure of the ECE engineering model uses: κ = 0 (2.55) throughout and so simplies previous work, in which κ was varied. Charge and current densities are dened by combinations of curvature, spin connection and torsion. The eld potential equations of the ECE engineering model are of key importance because they introduce the spin connection of general relativity [1] - [10], allowing spin connection resonance to occur. These eld potential equations are based on the Cartan Maurer structure equation: T a µν = µ q a ν ν q a µ + ω a µbq b ν ω a νbq b µ. (2.56) Dene the tensor: T µν = T a µν dσ a (2.57) where the hypersurface is: σ σ a = (σ 0, σ) (2.58) in the Minkowski spacetime labelled by a. When: a = 0 (2.59) then: T µν = T 0 µν dv (2.60) and similarly: q µ = qµ 0 dv. (2.61) Thus: T 0 µν = T µν V, q0 µ = q µ V. (2.62) Therefore Eq. (2.56) simplies to: T µν = µ q ν ν q µ + ω µb q b ν ω νb q b µ. (2.63) Finally dene: ω µ q ν = ω µb q b ν. (2.64) 31

38 2.2. THE LINK BETWEEN TORSION AND ANGULAR MOMENTUM This means that the b index is restricted to 0 because: and Therefore: and: where: q ν = Vq 0 ν (2.65) ω µ = ω µ0 V. (2.66) T µν = µ q ν ν q µ + ω µ q ν ω ν q µ (2.67) = ( µ + ω µ )A ν ( ν + ω ν )A µ (2.68) F µν = ( µ + ω µ )A ν ( ν + ω ν )A µ (2.69) µ = ( 1 c, ), (2.70) t A ν = ( φ c, A), ω µ = ( ω 0, ω). (2.71) c Therefore we obtain the electric and magnetic elds of the ECE engineering model in terms of potentials and spin connections, Q.E.D.: E = Φ A t + ωφ ω 0A, (2.72) B = A ω A. (2.73) ACKNOWLEDGMENTS The British Government is thanked for the award of a Civil List pension in 2005 and armorial bearings in 2008 for services to Britain in science, and the sta of A.I.A.S. / T.G.A. for many interesting discussions. 32

39 Bibliography [1] M. W. Evans, "Generally Covariant Unied Field Theory" (Abramis, 2005 onwards), vols. 1-5, vol. 6 in prep., (see [2] L. Felker, "The Evans Equations of Unied Field Theory" (Abramis 2007). [3] K. Pendergast, "The Life of Myron Evans" (Abramis 2009, preprint on [4] M. W. Evans, "Modern Non-Linear Optics" (Wiley 2001, second edition); M. W. Evans and S. Kielich (eds., ibid., rst edition, 1992, 1993, 1997). [5] M. W. Evans and L. B. Crowell, "Classical and Quantum Electrodynamics and the B(3) Field" (World Scientic 2001). [6] M. W. Evans and J.-P. Vigier, "The Enigmatic Photon" (Kluwer, 1994 to 2002, hardback and softback), in ve volumes. [7] ECE Papers and Articles on ECE source papers, ECE papers by other authors and ECE educational articles. [8] M. W. Evans et al., Omnia Opera section of notably from 1992 to present on the ECE theory and its precursor gauge theories homomorphic with those of Barrett, Harmuth and Lehnert. [9] M. W. Evans, Acta Phys. Polonica, 400, 175 (2007); M. W. Evans, Physica B, 403, 517 (2008) [10] M. W. Evans and H. Eckardt, invited papers to journal special issue, [11] L. H. Ryder, "Quantum Field Theory" (Cambridge, 2nd ed., 1996). [12] J. B. Marion and S. T. Thornton, "Classical Dynamics" (HBC, New York, 1988, 3rd.ed.). 33

40 BIBLIOGRAPHY 34

41 Chapter 3 Resonant Initial Event in ECE Cosmology by Myron W. Evans 1 and H. Eckardt 2 Alpha Institute for Advanced Study (AIAS) ( Abstract The resonant initial event in ECE cosmology is dened by spin connection resonance in a Bernoulli Euler type dierential equation. A very small amount of oscillatory mass density in a driving force may cause resonance in potential energy. The equations governing the initial event are generally covariant and based on the Cartan Bianchi identity and Cartan Evans dual identity. This mechanism is the simplest possible mechanism that can result in a resonant initial event and so is developed on the basis of relativity theory and Ockham's Razor. In a galaxy for example such a resonant initial event may dissipate itself and form the observed logarithmic spirals of stars on the basis of generally covariant space-time dynamics. Keywords: ECE theory, angular momentum theory, integration of a hypersurface.einstein Cartan Evans (ECE) eld theory, spin connection resonance, resonant initial event, Bernoulli Euler resonance. 1 emyrone@aol.com 2 horsteck@aol.com 35

42 3.1. INTRODUCTION 3.1 Introduction It is well known that the metrics of the big bang view of a single initial cosmological event are incorrect because of the neglect of spacetime torsion [1] - [10]. Big bang theory has eectively been discarded and replaced by the Einstein Cartan Evans (ECE) unied eld theory, currently the leading unied eld theory. Cosmological initial events appear to be a plausible idea, but in the course of evolution there is no reason to assume as in big bang theory that only one such event occurred. In Section 2 a plausible mechanism is proposed for an initial event that is a resonant initial condition of a Bernoulli Euler type equation derived from ECE theory's concept of spin connection resonance (SCR). The latter concept has been shown experimentally to be the key to industrial descaling technology, and so the concept has been proven on an industrial scale [11]. It has several other applications in the search for new energy sources [1] - [10] and in the development of counter gravitational technologies [12]. The cosmological resonant initial event can be caused theoretically by a very small amount of mass density that plays the role of an oscillatory driving force in the simplest type of Bernoulli Euler resonant equation [13], a well known aspect of classical mechanics that conserves energy/momentum. It is shown in section 2 that at resonance, a large amount of potential energy may be created by the small amount of oscillatory mass density. This process is a Bernoulli Euler resonant process of classical dynamics, known since the eighteenth century. In Section 3 the general theory is applied to the evolution of a whirlpool galaxy [14], in which the resonant initial event is a resonant initial condition of a dierential equation. The potential energy and angular momentum of such a system may dissipate until equilibrium is reached. The point of equilibrium may be exemplied on the simplest conceptual level by constant spacetime angular momentum, which may be shown to produce logarithmic spirals of stars as observed in a whirlpool galaxy. It seems plausible that there were N such initial events during cosmological evolution, N going to innity. The universe appears to have no beginning and no end, (these are anthropomorphic concepts) and the obsolete big bang model was hopelessly simplistic and mathematically self inconsistent. 3.2 Spin Connection Resonance as an Initial Condition As an introduction to the concept of spin connection resonance in cosmology consider the orbital problem. In the received opinion the orbit is the result of balance between a negative valued, central and attractive force law, and positive valued, repulsive centripetal force. The potential energy of attraction depends only on the distance of an object from the force centre, dened by the reduced mass: µ = m 1m 2 m 1 + m 2 (3.1) 36

43 CHAPTER 3. RESONANT INITIAL EVENT IN ECE COSMOLOGY In a planar orbit, angular momentum is conserved [13] and rotation of the system about any xed axis through the centre of force cannot aect the equation of motion. Conservation of angular momentum L means that: L t = 0 (3.2) where the angular momentum is dened by: L = r p. (3.3) However, in general: L 0. (3.4) In recent work [1] - [10] the conserved angular momentum has been considered to be the angular momentum of spacetime itself, derived from spacetime torsion [1] - [10]. Time independent or conserved torsion produces a potential energy that is a function only of r. It has been shown [15] that this type of torsion produces a positive valued potential energy that is 1/r dependent. When used in combination with the Newtonian inverse square law of attraction, the result is a logarithmic spiral of stars as observed in a whirlpool galaxy. Orbital theory of any kind is due to spacetime angular momentum, a generally covariant view based on ECE theory. The orbital force is dened as: F = µ r and the potential energy may be in general: (3.5) so U(r) = ( k 1 r + k 2 r 2 + +k n r n ) (3.6) F (r) = ( k 1 r 2 + 2k 2 r n k n ) (3.7) rn+1 The terms in this series expansion indicate force laws which are all consistent with conservation of spacetime angular momentum 3.2. The Newtonian force law and orbit are respectively: and F = k 1 r 2, U = k 1 r (3.8) 1 r = 1 (1 + ɛ cos θ). (3.9) α In this case conserved spacetime angular momentum produces universal gravitation. A force of type: F = L2 (1 + α) (3.10) µr3 37

44 3.2. SPIN CONNECTION RESONANCE AS AN INITIAL CONDITION gives the potential energy: U = L2 (1 + α) (3.11) 2µr2 and a logarithmic spiral orbit inwards towards the force centre: r = r 0 exp (αθ). (3.12) Therefore spacetime angular momentum plays a central role in classical dynamics an electrodynamics in the ECE generally covariant unied eld theory [1] - [10]. The ECE equations of motion of dynamics and electrodynamics are angular momentum equations. The basic ECE hypothesis is: F κµν = A (0) T κµν (3.13) where F κµν is the electromagnetic eld, T κµν is the spacetime torsion tensor, and ca (0) is a primordial or vacuum voltage observed in the radiative corrections [1] - [10]. To dene S.I. units consider the Einstein eld equation: R µν 1 2 Rg µν = κt µν (3.14) where R µν is the Ricci tensor, R is the Ricci scalar, g µν is the symmetric metric, k is the Einstein constant, and T µν is the symmetric canonical energy momentum density tensor. It is seen that k has the units of m kgm 1, and the left hand side of [14] has the units of m 2. So T µν has the units of mass density, kgm m 3. So in eld theory [16] the three index canonical angular momentum / angular energy density tensor dened by: J κµν = 1 2 (T κµ x ν T κν x µ ) (3.15) has the units of kgm 2. These are the units of angular momentum kgm m 2 s 1 divided by cv, where V is a volume in cubic metres. Unfortunately these are non SI units inherited from standard eld theory. In these non-si units: T κµν = k J κµν. (3.16) Now dene: and T µν = J µν = V 0 V 0 T 0µν dv = V T 0µν (3.17) J 0µν dv = V J 0µν. (3.18) The volume V is taken to be a xed volume which does not uctuate with time. This is the basis in eld theory of the derivation of conservation of the total 38

45 CHAPTER 3. RESONANT INITIAL EVENT IN ECE COSMOLOGY angular momentum of a dynamically conservative system contained within the volume V. Since V is a constant: D µ (V T 0µν ) = D µ T µν = V D µ T 0µν. (3.19) Similarly dene the curvature tensor: R µ µ ν = V 0 R 0 µν µ dv = V R 0 µ µν. (3.20) It is found from the Cartan Bianchi identity and the Cartan Evans dual identity that: µ T µν = j ν (3.21) whose Hodge dual is: µ T µν = j ν. (3.22) Here we dene: T µν = k c J µν. (3.23) Eqs. (3.21) and (3.22) are equations in angular momentum, QED. It follows that the electromagnetic eld tensor is: F µν = A(0) V T µν (3.24) and the electromagnetic eld is proportional to angular momentum as follows: F µν = A(0) k cv J µν. (3.25) In units of tesla the magnetic ux density is: B µν = F µν (3.26) and in units of volts per metre the electric eld strength is: E µν = cf µν. (3.27) If it is assumed that there is no magnetic monopole, Eqs. (3.21) and (3.22) give four vector equations. The pair of homogeneous equations are: and B = 0 (3.28) E + B t = 0, (3.29) 39

46 3.2. SPIN CONNECTION RESONANCE AS AN INITIAL CONDITION and the pair of inhomogeneous equations are: E = ρ ɛ 0 (3.30) and B 1 c 2 E t = µ 0J. (3.31) Here ρ is the electric charge density in Cm 3, J is the electric current density,ɛ 0 is the vacuum permittivity, and µ 0 is the vacuum permeability. These are the generally covariant formulation of the well known Maxwell Heaviside structure of special relativity, a Lorentz covariant structure. It is seen from Eqs. (3.26) and (3.27) that these are equations in angular momentum of spacetime: and µ J µν = 0 (3.32) µ J µν = k c jν. (3.33) In ECE theory Eqs. (3.32) and (3.33) are also the equations of dynamics. Most generally, their structure in both dynamics and electrodynamics are: and µ J µν = j ν = k c j ν (3.34) µ J µν = j ν = k c jν (3.35) i.e. ECE allows a non-zero â œmagnetic monpoleâ which is actually a spacetime structure [1] - [10]. It is well known [1] - [10] that the electromagnetic eld tensor is: 0 E X c E Y c E Z c F µν E Xc = 0 B Z B Y E Yc B Z 0 B X. (3.36) E Z c B Y B X 0 Therefore this tensor is based on the following angular momentum structure: 0 L X L Y L Z J µν = L X 0 S Z S Y L Y S Z 0 S X (3.37) L Z S Y S X 0 and consists of orbital L and spin S angular momentum components. It follows that Eq. (3.35) becomes two inhomogeneous vector equations of eld theory: L = j 0 (3.38) 40

47 CHAPTER 3. RESONANT INITIAL EVENT IN ECE COSMOLOGY and Here: and S 1 c L t = j. (3.39) L X = J 10 = J 01, S Z = J 21 = J 12 L y = J 20 = J 02, S Y = J 13 = J 31 (3.40) L Z = J 30 = J 03, S X = J 32 = J 23 L = L X i + L Y j + L Z k, (3.41) S = S X i + S Y j + S Z k. (3.42) The acceleration due to gravity may be dened as: g = ck V L (3.43) and the electric eld strength may be dened as: E = A (0) k L. (3.44) V In previous work Eq. (3.38) was reduced to the form: where: g = 4πGρ m = c 2 (R ωt ) (3.45) k = 8πG c 2, (3.46) G being Newton's gravitational constant. Here ρ m denotes mass density in kgm per cubic metre, R is a well dened curvature, ω a well dened spin connection, and T a well dened torsion. Therefore: L = 1 2 c V ρ m. (3.47) If it is assumed that the mass density is: ρ m = m V (3.48) then: Similarly: L = 1 mc. (3.49) 2 S 1 c L t = 1 2 V j m (3.50) 41

48 3.2. SPIN CONNECTION RESONANCE AS AN INITIAL CONDITION where the mass four-current is: j µ = (ρ m, j m ). (3.51) c The dual Eq. (3.34) gives: and S = 1 2 c V ρ m (3.52) L + 1 S c t = 1 2 V j m. (3.53) Eq. (3.47) is the dynamical equivalent of the Coulomb law, Eq. (3.50) is the dynamical equivalent of the Ampere Maxwell law, Eq. (3.52) is the dynamical equivalent of the Gauss law of magnetism, and Eq. (3.53) is the dynamical equivalent of the Faraday law of induction. Usually in classical dynamics, only the Newtonian structure 3.45 is used in the non-relativisic limit, whereas all four ECE equations are generally covariant. In the Newtonian limit: g = 1 U m r = mg r 2 = ckl V so the force due to gravity is: (3.54) F = 8πGm L. (3.55) cv This result shows that spacetime is inherently chiral, or handed, because the force, a polar vector, is proportional to orbital angular momentum, an axial vector. In the Newtonian interpretation the potential energy produces an attractive, negative valued, force between masses m and M and there is no angular momentum of spacetime. In ECE theory the dynamics are controlled by the angular momentum of spacetime and by spacetime geometry. For example, the conserved angular momentum of orbits in a plane is the conserved angular momentum of spacetime itself. The latter produces the r and p of orbits. In Newtonian theory the converse is thought to be the case, the angular momentum is produced by the r and p of already existing orbits in a uniform space distinct from time. Spin connection resonance enters into EEC dynamics by considering the rst Cartan structure equation [1] - [10] in addition to the Cartan Bianchi identity and the Cartan Evans dual identity. The rst structure equation may be written in vector notation as: where: g = 1 c ( U 1 c U t + Uω ω 0U) (3.56) U µ = (U, U) (3.57) 42