PROOF OF THE ANTISYMMETRY OF THE CHRISTOFFEL CONNECTION USING THE BASIC THEOREMS OF DIFFERENTIAL GEOMETRY.

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1 PROOF OF THE ANTSYMMETRY OF THE CHRSTOFFEL CONNECTON USNG THE BASC THEOREMS OF DFFERENTAL GEOMETRY. by M.W. Evans, H. Eckardt and D. W. Lindstrom, Civil List, AlAS and UPTEC, ( www. upitec.org) ABSTRACT The Christoffel connection is proven rigorously to be antisymmetric by consideration of the complete set of equations of differential geometry, combined with consideration of the general coordinate transformation. The complete set of equations available include the two structure equations, the Cartan identity and the tetrad postulate. A further rigorous proof of antisymmetry is given by consideration of the inhomogeneous term in its general coordinate transformation. t is proven that this term is zero, so the connection can have no symmetric component in any frame of reference. These proofs all refute the Einsteinian general relativity, which is incorrectly based on a symmetric connection. Keywords: ECE theory, antisymmetry of the Christoffel com1ection, refutation of Einsteinian general relativity.

2 1. lntroducton t has been known for almost a hundred years { 1-10} that the Einsteinian general relativity is incorrect. t was first refuted by Schwarzschild on December 22nct 1915, a month after the publication of Einstein's first paper on the perihelion precession. A translation and analysis by Vankov { 11} of the Schwarzschild paper is available on the net. n ref. ( 1) several clear and simple refutations of the theory are given. n Section 2, it is proven straightforwardly that the Christoffel connection must be antisymmetric. ts antisymmetry refutes the Einsteinian general relativity (EGR), whch used a symmetric connection because at the time of development ofegr ( ) the concept oftorsion was unknown. Torsion was inferred by Cartan { 12} in the early twenties, and its inference effectively refuted EGR at that time. n Section 2 this is made clear by combined consideration of the basic theorems available in differential geometry, the definitions oftorsion and curvature, the Cartan identity, the tetrad postulate, and the general coordinate transformation of the connection. The Einstein Cartan Evans (ECE) theory is the only one at present that correctly accounts for the connection antisymmetry. 2. PROOF OF CONNECTON ANTSYMMETRY Assume that the Clu istoffel connection is in general asymmetric: ~ (r;~ t '~r) t ~ (r;~ ~~) -(') with symmetric and antisymmetric components. Eq. (1) is a well known theorem of matrices. Now note that the Cartan identity is true for eq. (1). The Cartan identity is an exact identity consisting of a cyclical sum of terms on both sides. n UFT209 it was given in Riemannian format:

3 -(~ t The sum of curvature tensors appears on the left hand side, and the sum on the right hand side is made up of derivatives of torsion and products of connection and torsion in cyclical permutation. The left and right hand sides of eq. ( :l. ) are identically the same. The rollov.ring is a mathematical solution ofthe identity: ~ A. cs _ ~ --r ~r k c -: kv..~~-,. t rr, 1..,r Now note that the curvature and torsion may be defined by the commutator of covariant derivatives acting on a vector V ( : where and n general, the connection in Eqs. ( 4- ) to ( (, ) is defined by eq. (1 ). This is also true for cq. ( :;). ). This set of equations is rigorously self-consistent as is well known {12}.

4 Therefore the curvature tensor defined by eqs. ( 1 ) and ( 5 ) is the same curvature tensor. t follows that: 'X R!"'"( _ ' ~ ~ ~T: 1:- r r r >-- >--,.. - )... ~y \- 'r cr f the connection is symmetric the torsion vanishes ±,& m Eq. ( b ), and ~rm Eq. ( { ) the curvature vanishes also, reductio ad absurdum. Therefore the connection is antisymmetric: For non zero torsion and curvature the connection is antisymmetric, Q. E. D. This proof i.s confirmed by considerations of Eq. ( ~ ). f in that equation the connection is assumed to be symmetric, then: in which case: () and, from Eq. ( 4- ): )._ f<rr reductio ad absurdum. J D The cmvature ten sor may also be obtained from the Cartan identity as follows: \ }. \ )._ cr r).. (( - J,_, -\-- ),_~ \~'"" + \f T., -\-... ~ }i r.j r 1 >- r >- 11 }.. T ~ d' + ~ ~~ - J., ~~ t f).'"~" r,.,./i

5 -- fr-, in which the connection may be asymmetric in general. f the connection is assumed to be symmetric then Eq. ( \ J ) reduces to: because the torsion terms vanish. For a symmetric connection the only possible definition of curvature is, from Eq. ( \1 ): )... '>.. Rf"f - )r r,jr A - J~~r This result implies that the curvature cannot be defined by Eq. (.3 absurdity because Eq. ( j connection is in general asymmetric. \ ($' \ tr t ~'r~ r... r - r ~ ~ ~r _. -- (\l~.) ). This is a reduction to ) is a mathematical solution ofeq. ( ~ ), in which the so: or in equivalent differential form notation:. ~ \, + w ~" ~, T,r. This is a new result of differential geometry which again demonstrates antisymmetry under exchange of r and t...:j. n ECE theory Eq. ( \r ) is an equation both of gravitation

6 -- \ fc. ancl of electrodynamics. T he sol uti on: is the Evan s identity, first deri ved using Hodge duals { 1-10}. This result is shown as foll ows. First raise indices: \ "t or\ so that: and consider the case: --l')) r~~ then: ~ ~T~( '/" - >- )Ar Rr ~ {J)) which is the Evan s id entity, Q. E. D. Eq, ( ).) ) is true if repeated indices are summed. The above is the inhomogeneous term in the.transformation of the Christoffel connection, as is well known { 1-10, 12}. Consider the connection defined by: ts transformation contains th e following inhomogeneous term:

7 J~A ))(_ ~.,_ 1~- ) J-,c" - 0 "''...---~ J)(_.M )~ d)( d)c ~. - (JS) To prove this result it is proven that: 1 j)(.<v 0 - ()l) j)(._ 0.. To prove Eq. ( )..L ) note that: The tangent spacetime of Cm tan's differential geometry is a Minkowski spacetime as is well known { - 1 0, 12}. Therefore by orthogonality: JX- b 0 ~ J.,eand Eq. ( ).S) follows, Q.E.D. Therefore the inhomogeneous term in the transformation of,~ r~ vanishes. The inhomogeneous term is however symmetric in ja and because partial derivatives with these indices commute. t follows that: - Jx fa JK <A J:x....J - 9 ),c._" ~ ~ a.., because it can have no symmetric component in any frame of reference, Q.E.D. From Eq. ( } ~) it follows that: r )..' r"' These proofs each show that the Clu istoffel connection is antisymmetric and. th8t the EGR 'is refuted at its most fundamental level.

8 ACKNOWLEDGMENTS The British Government is thanked for the award of a Civil List Pension and rank of Armiger. The staff of AlAS and others are thanked for many interesting discussions. lj8vc Burleigh is thanked for vol untary posting, A lex Hill, Robert Cheshire and Simon CJ i Tord are thanked for translation and broadcasting. The AlAS is established under the aegis of the Newlands Family Trust. REFERENCES {1} M.W. Evans, Ed., "Definitive Refutations ofthe Einsteinian General Relativity" (C8mbridge nternational Science Publishing, CSP, March 2012), spcci81 issue six of rererenee two. {2) M. W. Evans, Ed., Journal of Foundations of Physics and Chemistry, (CSP from June 2011, six issues a year). : J) M.W. Evans, S. Crothers.. H. Eckardt and. Pendergast, "Criticisms of the Einsteinian General Relativity" (CSP, Spring 2011). { 4} M. W. Evans, H. Eckardt and D. W. Lindstrom, "Generally Covariant Unified Field Theory" (Abramis Academ ic to 2011) in seven volumes. { 5) L.Felker, "The Evans Equations of Unified Field Theory'' (Abramis, 2007). { 6) K. Pendergast, "The Life of Myron Evans" (CSP, 2011 ). f7} M.W. Evans and S. Kielieh (Eds.), "Modern Nonlinear Optics" (Wiley, 1992, 1993, l), in six volumes and two editions. { 8} M. W. Evans and L. B. Crowell, "Classical and Quantum Electrodynamics and the 8(3) Field" (World Scientific, 2001). (9) rvl. W. Evans and.j.-p. Vigier, 'The Enigmatic Photon" (Kiuwer, 1994 to 2002), in ten

9 volumes. { 10} M. W. Evans and A. A. Hasanein, "The Photomagneton in Quantum Field Theory" (World Scientific, 1994). { 11} A. A. Vankov, translation and analysis on the web. { 12} S. P. Carroll, "Spacetime and Geometry: an ntroduction to General Relativity" (Ad dison Wesley, New York, 2004).