-- I f<-. SIMPLE PROOFS OF THE ANTISYMMETRY OF THE CHRISTOFFEL CONNECTION.

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1 I f<. SIMPLE PROOFS OF THE ANTISYMMETRY OF THE CHRISTOFFEL CONNECTION. by M.W. Evans, H. Eckadt and D. W. Lindstom Civil List and A. I. A. S ( ABSTRACT It is poven staightfowadly in two ways that the Chistoffel connection is antisymmetic in its lowe two indices, thus diectly efuting the Einsteinian geneal elativity. The coect deivation is given of the Newton equation fom the geodesic equation, using a coectly antisymmetic Chistoffel connection. Keywods: ECE theoy, antisymmety of the Chistoffel connection, deivation ofnewton's equation fom the geodesic equation. === ~

2 INTRODUCTION The Chistoffel connection was intoduced into geomety in 1869 in such a way that it was defined to be symmetic in its lowe two indices. Thee ae symmetic Chistoffel symbols of the fist and second kind. The Chistoffel connection augmented the ealie wok of Riemann, who intoduced the idea of the symmetic metic tenso. About thity yeas late tenso analysis was developed and the concept of cuvatue intoduced by LeviCivita, Ricci, Bianchi and co wokes. At that time, the symmetic Chistoffel connection was the only definition available, and was used by Einstein in his development of geneal elativity as is well known. The Einstein field equation is coect if and only if the Chistoffel connection is symmetic. In about 1923 howeve Cm tan and his co wokes intoduced the concept of tosion, which exists if a~only if the Chistoffel connection is antisymmetic. So it became clea that the oiginal 1869 definition by Chistoffel was too esticted. Catan communicated with Einstein as is well known but the antisymmetic pat of the connection continued to be ignoed. In the ECE seies of papes { 11} it has been poven in seveal ways that the Chistoffel connection has no symmetic pat, it is wholly antisymmetic. The entie ea of Einsteinian geneal elativity (EGR) has been efuted. In Section 2 a completely simple poof of the antisymmety of the connection is given using the well known commutato method { 11} of geneating tosion and cuvatue simultaneously. The EGR theoy incoectly assets a symmetic connection as axiomatic, so the tosion is incoectly zeo by axiom. The commutato method on the othe hand isolates the connection, showing that its lowe index symmety is that of the commutato, i.e. anti symmetic. The null commutato is a commutato and is zeo because it is symmetic. Theefoe the symmetic connection is zeo, Q.E.D. This finding is enough to efute EGR completely, EGR is a meaningless theoy and claims to have veified it ae misplaced

3 entiely. In Section 3 a futhe simple poof of the anti symmety of the c1mection is given using the geneal coodinate tansfomation. Full details ae given in the notes accompanying this pape on vvvvw.aias.us. Finally, in Section 4, it is shown vey simply that the Einsteinian deivation of the Newton equation fom the geodesic equation fails immediately because of its use of a symmetic Chistoffel connection. An example of a coect deivation is given using an antisymmetic Chistoffel connection. 2. COMMUTATOR PROOF OF CONNECTION ANTISYMMETRY The commutato of covaiant deivatives { 11} in geomety is an opeato that <lets on any tenso in a space of' any dimensions to poduce the tosion and cuvatue tensos. f' Jts action on a vecto V poduces the following esult: _ T).. {) v _(t) )'~ ~ whee the tosion tenso is defined by: and whee (Z f '?<""is the cuvstue tenso. It is well known that both the tosion and cuvatue ae antisymmetic: because the commutato is antisymmetic:

4 "'" I I' Let: ~ ", A 'na ~ ' N ::.. ) then L \)/" 1\)~Vf L o,., ~~ vf c~ A (At 1\:nw let: anc! Eq. ( l ) gives:. ~A " hile Eq. ( ~ ) gives: So it follows that: ~A~ A\\ d. A ~ J.\1:.. The symmetic connection is zeo Q. E. D. and the Chistoffel connection is always <t nt i ~ ' mmeti c: because it is symmetic: This poof uses the fact that the null commutato is a commutato which is zeo

5 [O) D.. l ""! f = 6', ;v.::...~ The EGR theoy uses a symmetic connection and is a meaningless theoy. 3. TRANSFORMATlON OF THE CHRISTOFFEL CONNECTION. It is well known [I 1 } that the Chistoffel connection tansfoms as: ;,,.., unde geneal coodinate tansfomation. In this notation: \ )~~ ~I '\,('' Jx, ~ ::. \j~ ) Conside: /"' ~I ~ >' dx }..' ),t. ~ J,(..). ).,.>I ),:' J~t..JI J)L...> :. ~~ dx t:. (n) ~.., ( t") In the abitay manifold howeve (Eq. (2.15) of ef ( 11 ), 1997 online notes): ( l"') \\hee the Konecke delta!'unction is defined by: \ ' T heefoe: ll l11c ss:

6 ll'[q. ( ~~ ) is tue then: Jx...v I Jx_~ unless: So: ~. ( J\,) J \in> I (JS) unless: ~ tv \. (:l This is inconsistent with the fact that: A + "' f \. ( :n) in geneal. Theefoe: ( )1i) in geneal and the connection tansfoms as Cl tenso: The accompanying notes show that this esult can be poven in seveal othe '' ~ws. and the eade is efeed to these notes in the UFT section ofwww.aias.us.

7 I 1~ The EGR used the ideo of a local Loentz fame { 1 1} in which spacetime is locally flat and the connections vanish. Fom Eq. ( \ b ) the tansfomed connection vanishes if: i. e. if: ~ \{~>then: \!N~';~ )., ),(...)I )xfo' I (36),_:) Jx~ ) _(3),... J, IV' )..' ()\vcve the coect esult is obtain ed fom Eqs. ( :l~ ) and ( ~ \ ), i.e. if (3J) I the connection vanishes in one fame it vanishes in all fames. The idea of Riemann nomal coodinates cannot be usee!. ln tun the E in stein equivalence pinciple cannot stand as defined in EGR, because it depends on the locally flat Loentz fame. The oiginal 1869 definition of the connection by Chistoffel is simila to Eq. (3 \) Fom Eq. ( )~ ) it is seen howeve that

8 i.e. the symmetic connection is zeo Q. E.D. The antisymmetic connection is the only non zeo connection and tansfoms as a tenso: I,,.> I v \.v. )..' \("" I ~~I \(~ A The tosion tansfoms as a tenso: I" I,.J T, )..' ~ \j and if it is zeo in one fame it is zeo in all fames. A \_ > I,.. N I \i \/ \i,.) '> (n) (1 4. ;\CORRECT DERIVAT1N OF THE NEWTON EQUATION FROM THE GEODESIC I~QUAT IO N. The geodesic equation { l ll} is: _o whee the affine (o manifestl y covaiant) paamete has been chosen to be the pope time (. Jn standad EGR the eduction ofthis equation to the Newton equation consisted of assuming th:n in the SJXICC JXlt ol'the cqucllion with index i: \\ h~tc tis the time in the obsen c fame. This assumption educes Eq. c.>~) to: ~J) ~ I lowcve:

9 b ~, antisymmety. so the EGR theoy fails immediately. The!a ll owing is one coect way ~feducing Eq. ( 3~) to the fomat of Newtonian dynamics. Conside the spacelike pat ofeq. (.3 ~ ):. ' JvJ :;..' j~ ~ tl"t using the method developed in UFT212 on \. whee is the magnitude of th e adial vecto. Multiply both sides ofeq. ( lt3) by )(_ to give: a 1,c: ~) '. ' ~ 9 A)(. )'Y2 ~ Nc,vtonian dynamics ae gi ven by : at d_,l l a._ a.,\.;.l ( so:. l X (. ' 't ' l"lll'c!o c Ne wt onian dynami cs ac defined by:

10 I J. t 'J.?t ".} l ( with summation ove epeated indices. The anti symmetic commutato gives Newtonian d: nnmics, a commutato defined by:. Moe geneally: l j~ ck_"j~' ( \ ' t )~ ( S1) ' A.)._~ J.x. ~ (s)j :::. \ )~ clt) J.t cjj and most geneally: llc cl) M.? ~ ~"A).. (s) )(_ \ ). cl)\.,.._) A"C ") c,l"'\.. give non Newtonian dynamics. The EGR deivation uses a symmetic connection and is 1!1CO JTect and meaningless. ACKNOWLEDGMENTS The Bitish Govenment is thanked fo the awad of a Civil List Pension and ank of Amige to MWE. The AlAS and othe colleagues ae thanked fo many inteesting discussions. David Buleigh, CEO of Annexa Inc., is thanked fo voluntay posting, Alex Hill. Robet Cheshie and Simon Cl iffod fo tanslation and boadcasting. The AlAS is established unde the aegis o the Newlancls Family Tust (Est. 2 12).

11 1\ FFI:::RENCES { l} M.W. Evans, Ed., "Definitive Refutations ofthe Einsteinian Geneal Relativity" (CISP, Cambidge lntenational Science Publishing, WW\v.cisppublishing.com. 212). : ~: 1\1!. W!::: va ns. Ed...1. Found8tions Physics and Chemisty (CISP, six issues a yea, fom 211 ). {3] M.\V. Evans, S. Cothes. H. Eckadt and K. Pendegast, "Citicisms ofthe Einstein Fiel d Eqw1ti on. ClSP 211). (4] 1\1!. W. Evans, H. Eckadt and D. W. Lindstom, "Geneally Covaiant Unified Field Theoy' (Abamis Academic, Suffolk, 25 to 211) in seven volumes. ( ~ ] L.Felke. The Evans Equations of Unified Field Theoy" (Abamis 27). (6] K.Pendegast, The Lil'e of" Myon Evans" (CJSP 211). {7} M.W. Evans and S. Kielich, Eels., "Moden Nonlinea Optics" (Wiley, 1992, 1993, 1997, 2 1 ). in six volumes and two editions. l ~; ivi. W.[vans and L. 13. C owell. ''Classical and Quantum Electodynamics and the B(3) Field" (Wold Scientific 2 I). (9} M. W. Evans and J.P. Vigie, "The Enigmatic Photon" (Kluwe, Dodecht, 1994 to =2). in ten volumes hadback md softback. {1} M.W. Evans and A. A. llasanein, "The Photomagneton in Quantum Field Theoy" (Wold Scientific 1994). : l I: S. P. Caoll. Spacetime and Geomety: an Intoduction to Geneal Relativity" (Addison Wesley, New Yok 24. and online notes 1997).