A response to the papers by Hehl and Hehl and Obukhov

Transcription

1 Response to the Ppers y Hehl nd Hehl nd Oukhov 573 Journl of Foundtions of Physics nd Chemistry, 011, vol. 1 (5) A response to the ppers y Hehl nd Hehl nd Oukhov M.W. Evns 1 *Alph Institute for Advnced Studies (AIAS) ( "If electrodynmics hd nothing to do with geometry of spce-time (s Hehl sserts), the WHOLE of well-tested nd very precise rel tivity theory would collpse ecuse reltivity is sed on geometry nd must pply to the whole of physics, not just grvittion" MWE, 007. A detiled scientific reuttl is given to the criticisms y Hehl nd Oukhov of Einstein Crtn Evns (ECE) unified field theory s reported in two ppers in the current edition of this journl. It is shown tht oth ppers contin sic errors nd misunderstndings. In ddition, it is pointed out tht cittions of other criticisms of ECE theory contined in the two ppers re mde without citing existing reuttls. Some of these reuttls hve een ville for over decde in the literture nd re either referenced in this pper or reproduced in ppendices for the record, This new reuttl is structured in the sme wy s the ppers of Hehl nd Ojkhov for ese of comprison. More thn 80 detiled points of error, misunderstnding or confused thought re discussed -this representing smll smple of the errors tht exist. They re chosen to illustrte tht these criticl ppers demonstrte cler misunderstnding of ECE theory often t very sic level. Keywords: Einstein Crtn Evns unified field theory. 1 Introduction Einstein Crtn Evns (ECE) unified field theory hs recently een completed nd is eing widely red nd studied s evidenced y the lrge numer of visits to the AIAS wesites where the ppers re ville, from cittions in ooks nd ppers, nd from enquiries tht re regulrly received from le scientists, technologists nd more recently usinessmen nd government deprtments worldwide. The numer of criticisms tht hve een pulished or reported re reltively few (restricted to hndful of scientists worldwide) -ll hve een responded to nd misunderstndings nd errors corrected in the ccepted scientific 1 e-mil: EMyrone@ol.com

2 574 M.W. Evns mnner. It seems, therefore, tht the theory is lrgely ccepted y the scientific community [1-1] nd y the stndrd nonymous peer review process. The level of cceptnce cn e mesured y intensive dy to dy usge of the ppers on the wesite. The numer of visits compres fvorly with most scientific wesites - nd often fr exceeds the interest tht is shown in most university reserch deprtment wesites. ECE theory is fundmentlly sed on long ccepted nd widely used Crtn geometry [13]. Estlished in 19 Crtn Geometry is tught in courses in university deprtments round the world. In 78 ppers nd ooks on the www. is.us wesite, ECE theory hs een rigorously tested for self consistency nd ginst experimentl dt in systemtic wy in the vrious fields of physics. It reduces in pproprite limits to ll the well known lws of physics (see the tles nd flow chrts on the AIAS wesite). It unifies clssicl nd quntum mechnics consistently nd coherently for the first time mking concepts such s uncertinty osolete. It reduces to ll the well known equtions of quntum mechnics ginst which it hs so fr een tested, It hs numerous experimentl dvntges over the stndrd model reproducing the results of mny experiments tht the stndrd model cnnot explin. It predicts new effects nd phenomen some of which re lredy strting to e exploited commercilly. It mkes notions such s Drk Mtter unnecessry -there is no missing energy in the universe on the sis of ECE theory. Despite this, some ttempts hve een mde to criticize nd question spects of the theory - s to e expected of new theory tht unifies ll of physics for the first time. It will e shown in this pper tht the cur rent criticisms demonstrte fundmentl misunderstnding of the theory nd s consequence contin mny errors nd misconceptions. Most of these errors nd misunderstndings hve lredy een corrected rigorously in pre vious reuttls ( ut Hehl nd Oukhov fil to cite these reut tls. This pper therefore ssesses gin in detil the remrks y Hehl nd Oukhov, nd highlights the min points of their scientific inccurcies. It reviews the misunderstndings nd errors tht hve een mde y Hehl nd Oukhov (nd other uthors they cite) nd shows the necessry corrections tht need to e mde to cquire correct understnding of ECE theory. The pper contins numerous reuttl sttements, references to lredy existing reuttls, nd technicl ppendices outlining some of the key reuttls in more mthemticl detil s nd when considered pproprite. The vlidity, reltive simplicity, nd gret strength of ECE theory is in fct self-evident sed directly s it is on Crtn geometry. Some misleding remrks y Hehl nd Oukhov [14] concerning the sic ECE hypothesis re corrected. 1.1 Astrct nd Introduction In the strct of the first pper y Hehl it is climed tht the ECE theory is strictly clssicl wheres it hs een shown to e fully quntized nd hs een

3 Response to the Ppers y Hehl nd Hehl nd Oukhov 575 pplied to quntum field theory (unifying consistently for the first time clssicl nd quntum mechnics). Tht is, ECE succeeds in unifying generl reltivity nd quntum mechnics where the stndrd model is well known to fil. It is climed tht there is not single pper in which ECE is fully set out, wheres the theory is descried in ooks, numerous individul ppers, tles nd flow chrts, vrious reuttls nd ppendices in ppers etc in very detiled wy. Hehl repets elementry errors y Bruhn in series of cittions of unrefereed wesite documents, while the existing reuttls ( re not cited. For this reson, some lredy pulished reuttls re summrized gin in Appendices ttched to this pper. Hehl clims for exmple tht there is no nti-symmetric metric, wheres this oject is clerly defined in ECE theory s the wedge product of tetrds. This is definition, so Hehl effectively sttes tht wedge product of tetrds cnnot e mde, this is reductio d surdum. The symmetric metric is well known to e the sclr product of tetrds. Hehl clims further tht the wedge product of two tetrds is somehow inconsistent with Crtn geometry [13]. A wedge product is well defined. The eqution: D T = 0 (1) follows from the second Binchi identity: D R = 0 () where R nd T re well defined [1-1]. Hehl does not pper to e wre of this fct. Both Hehl nd Rodrigues clim tht the tetrd postulte is somehow not pplicle, wheres it is well known nd long ccepted tht the tetrd pos tulte is the fundmentl requirement tht vector field e independent of its components nd sis elements. The tetrd postulte lwys holds for ny ppliction in ny re of science nd nturl philosophy. Hehl mkes some oscure remrks out the constnt of proportionlity A (0) in the fundmen tl ECE hypothesis: ( 0) = A A q (3) where A is the electromgnetic potentil form nd q is the Crtn tetrd form. If c is the velocity of light then ca (0) hs S.I. units of volts: ( 0) 1 1 ( 0) 1 A = JsC m, ca = JC = Volt. (4) Hehl lso mkes some confused remrks out the trnsformtion properties of

4 576 M.W. Evns A. Its trnsformtion properties re those of the tetrd [1-13], vector vlued one-form, nd re discussed in comprehensive detil on They re: A x = Λ x A u (5) x where A is the Lorentz trnsform nd where x denotes generl coor dinte trnsformtion. This is entirely stndrd nd well known, eing sic prt of Crtn geometry. Hehl grees tht ECE theory is sed on Cr tn geometry, nd refers to it s "Einstein Crtn" geometry (even though Einstein did not ply role in developing this geometry - it ws developed entirely y Crtn in 19). Hehl therefore AGREES tht ECE theory is techniclly correct, ut indulges in sujective ofusction. As n exmple of this ofusction, Hehl ccepts tht the ECE wve eqution is techniclly cor rect ut proceeds to descrie it s "redundnt". This is nonsense ecuse it would imply tht the tetrd postulte is "redundnt". It is climed tht the pper y Hehl nd Oukhov constructs lgrngin in n originl mnner. Lgrngins for ECE theory re lredy ville [1-1] nd ccepted in the usul mnner. Hehl points out tht the ECE theory is entirely originl to the present uthor, nd clims tht Crtn did not suggest the theory to Einstein. This clim concerning Crtn my or my not e correct. Hehl mkes some remrks out others who hve pprently used Crtn geometry, ut ll this is lrgely irrelevnt to the intended discourse on ECE theory. Geometry: Riemnn-Crtn Geometry of Spcetime In section of the first pper in ref. (14) stndrd textook type mte ril is introduced tht is irrelevnt to ECE theory [1-1]. The ltter is sed on the two stndrd Crtn structure equtions nd the two stndrd Binchi identities. These hve een rigorously tested [1-1] in mny wys for self consistency, nd hve een reduced [1-1] to their equivlents in Riemnn nottion in order to demonstrte self consistency. ECE theory hs een rigor ously tested ginst experimentl dt in mny wys ( success fully reproducing the results of mny experiments tht the stndrd model fils to explin for resons tht ecome cler within the ECE theory frme work. Mny new experiments nd phenomen re predicted, some of these re lredy eing developed into new technologies - prcticl testimony to the technologiclly importnt new insights provided y ECE theory. Just efore section. Hehl gin dmits tht ECE theory is stndrd Crtn geometry, which he gin clls Einstein Crtn geometry. Inconsis tently, he then cites Bruhn who clims tht the tetrd postulte is somehow "incorrect", so ccording to Bruhn, Crtn is incorrect. This inconsistency is compounded y

5 Response to the Ppers y Hehl nd Hehl nd Oukhov 577 his cittion of Rodrigues, who lso clims tht the tetrd postulte is "incorrect". He lso cites Lkhtki, who clims on the other hnd tht Crtn geometry is correct. Hehl cites none of the relevnt reut tls tht re ville ( us) even though some of them hve een in print for decde or more. We need no knowledge of mthemtics to see tht there is internl inconsistency in these cittions of unrefereed wesites nd the confused trin of thought tht is scttered throughout these ppers. Hehl mkes the clim tht the Crtn torsion is unrelted to spin. This is untrue. The first Crtn structure eqution sttes in the nottion [13] of Crtn geometry tht: T = d q +ω q (6) where T is the Crtn torsion form, d is the exterior derivtive, relized y wedge product, nd ω is the spin connection form. In tensor nottion Eq. (6) ecomes: T = q q +ω q ω q (7) v v v v v nd clerly hs the nti-symmetry needed for ngulr momentum or torque. These quntities involve spin nd this is meticulously defined in ECE the ory [1-1] in mny plces. Hehl oviously misunderstnds the originl points in ECE theory, (see Appendices to this pper) ecuse spin torsion nd oritl torsion hve een clerly defined in ECE theory. Hehl cites other work con cerning ngulr momentum ut this is not ECE theory. Hehl sttes in footnote tht "we re not told wht sort of spin we hve to think of". Spin torsion nd oritl torsion re clerly defined in ECE theory [1-1], so re ll dynmicl quntities, in n originl wy. In footnote to section.3 Hehl gin grees tht Crtn geometry s used y Crroll [13] nd the present uthor [1-1] is correct. Yet he then cites Bruhn nd Rodrigues (gin), who clim tht it is incorrect. So Hehl et l. re self contrdictory. In Section.3 there is (gin) long nd irrelevnt textook type reitertion. This is not relevnt to ECE theory. In Section.4. Hehl confuses the ECE Lemm [1-1] with wht he clls the "Ricci Identity". In this confused stte he then sserts tht there is n error in the ECE Lemm. The ECE Lemm is otined from the tetrd postulte in simple nd self-consistent mnner (for exmple Appendix J of ref. [1]). Hehl sttes tht Crtn is correct, Bruhn nd Rodrigues stte tht Crtn is incorrect - so centrl prt of Hehl's thesis is ctully sed on the vlidity of Crtn geometry itself. Needless to sy, the entire community of twenty first century mthemticins [13] ccept its vlidity nd regulrly tech Crtn geometry, the sis of ECE theory, in university deprtments.

6 578 M.W. Evns 3 Electromgnetism: Evns' Anstz for Extended Electromgnetism In Section 3.1, Hehl sttes once more tht ECE theory is techniclly correct, i.e. "Up to now everything is conventionl". He sttes therefter tht "Evns hs highly unconventionl d hoc nstz" (sic.). This oscure grmmr pre sumly mens tht the ECE hypothesis is originl (ut in n unconventionl wy?). Since every hypothesis must e originl, this sttement is tutology. In footnote to Section 3.1 Hehl contrdicts his previous sttements tht Crtn geometry is correct, now clims tht it is incorrect, nd tht Crtn hs "committed mistke". The mthemticl truth (Appendices) is tht the two Crtn structure reltions nd two Binchi identities re rigorously self consistent within their well defined terms of definition, s Hehl himself dmits in other prts of the sme pper. Wht re we to mke of this porridge of self contrdictions? Hehl compounds his confusion y further sujective sttements - "idols of the cve" tht hve no plce in nturl philosophy. He clims tht A (0) must e universl constnt on the sis of units nlysis. If so, ca (0), eing voltge, must e universl. The truth is tht A (0) is the sclr mgnitude of vector potentil nd s shown in the Appendices, the minimum ca (0) universl constnt. The four potentil is wht Feynmn descried s universl influence in guge theory, ut this does not men tht its mgnitude is universl constnt. This is sic confusion of concepts y Hehl. The ltter sttes tht "this does not smell prticulrly mc universl". In the Appendices it is shown tht the minimum ca (0) is, e where mc is the rest energy of the photon nd e is the chrge on the electron. This is universl constnt. Hehl mkes some pprently erroneous ttempts t defining the units of ca (0), they re volts. Hehl's confusion is further compounded y sttements out the trns formtion properties of the A form. It is well known tht vector vlued one-form trnsforms ccording to Eq. (5). Agin this is detiled in ECE theory [1-1], detil which Hehl never cites. Hehl sttes ritrrily tht A must e identified with Mxwellin potentil. This is n oscure clim. If it is sed on Lorentz covrince it is untrue, ecuse ECE is generlly covri-nt, not Lorentz covrint s in wht Hehl wrongly descries s Mxwell's theory. Presumly he is referring to the Mxwell Heviside theory. The cor rect wy to reduce A to Mxwell Heviside potentil is to use complex circulr indices for nd Crtesin indices for μ. Hehl does not seem to under stnd this, s his sttements revel. The nture of A hs een detiled in ECE theory. Hehl sttes tht he "would like to kill the index". The index is tht of circulr polriztion, nd indictes the fundmentl complex circulr sis. This is just s fundmentl s the Crtesin sis or sphericl polr sis for exmple. Hehl oscurely refers to "O(3) covrince". Presumly he is referring to the well known nd ccepted B Cyclic Theorem of the mid nineties:

7 Response to the Ppers y Hehl nd Hehl nd Oukhov 579 ( 1) ( ) ( 0) ( 3* ) B B =ib B et cyclicum (8) which hs O(3) symmetry. Bruhn's incorrect ssertions concerning this theo rem re corrected in the ppendices. The electromgnetic field form in ECE theory is proportionl to the Crtn torsion form, vector vlued two -form whose trnsformtion properties re determined y well known nd well ccepted geometry [1-13]. Hehl gin cites Bruhn who flsely clims tht the equtions (8) re somehow "not covrint". The truth is tht the equ tions (8) re the equtions of the frme itself, nd the frme itself is well known to e covrint (see Appendices). These flse clims y Bruhn hve een reutted in gret detil on nd the reuttls hve not een further criticized so presumly re ccepted. In his Eq, (31) Hehl mnges to see the correctness of: F ( 0) v = A T v (9) where F v is the electromgnetic field form of ECE theory. This is sym metry conserving eqution s follows: F = F v v, (10) T = T v v. (11) Hehl reitertes well known textook mteril concerning the form nottion of the Mxwell Heviside field theory, ut exhiits lck of understnding of the textook mteril in his clim tht the homogeneous nd inhomoge-neous equtions re independent. It is well known tht they re relted in the vcuum y the trnsform: E = icb. (1) The Mxwell Heviside equtions tht he cites (nd confuses with the Mxwell quternion equtions) re otined in the limit when the spin connection goes to zero. For exmple the homogeneous field eqution of ECE theory is: d F = 0 j (13)

8 580 M.W. Evns where μ 0 is the S.I. permeility in vcuo nd where j is the ECE homogeneous current. When there is no interction etween grvittion nd electromgnetism: j = 0 (14) nd d F = 0 (15) which for ech polriztion index is the homogeneous eqution of Mxwell Heviside field theory, Q.E.D. The only thing left for Hehl to question is circulr polriztion itself. In his section 3. Hehl flsely clims tht the Lorentz force eqution hs not een otined in ECE theory. In truth the Lorentz force eqution hs een otined [1-1] from the trnsformtion properties of the field form F (see Appendices). Hehl correctly cites the homogeneous ECE eqution in his Eq. (37) (using his own oscure nottion) ut not stisfied with this, he clims tht the homogeneous ECE eqution is n "nlog" of wht he clls Mxwell theory, presumly Mxwell Heviside theory. He misses the fct tht ECE is much more generl thn Mxwell Heviside theory. In the pr grph following his Eq. (39) Hehl tries to find wys of stting tht correct eqution is incorrect, ut on the other hnd correct. The present uthor prefers ojectivity nd Crtn geometry. He then sttes tht his Eq. (9) is the "rel" ECE homogeneous field eqution, the truth eing tht Eq. (13) of this reuttl is the ECE homogeneous field eqution [1-1]. Hehl then sttes tht the homogeneous current of ECE is "strnge". This is unscien tific sujectivity. It is simply quntity tht hs no counterprt in ny other theory. The ojective nd geometricl truth is tht the homogeneous current of ECE theory is defined y: ( 0) A : = ω 0 ( ) j R q T (16) where R the Crtn curvture. The current (16) is otined strightfor wrdly from the first Binchi identity: d T +ω T : = R q (17) y rerrnging terms s follows:

9 Response to the Ppers y Hehl nd Hehl nd Oukhov 581 d T : = R q ω T. (18) Now multiply oth sides y ca (0) (the primordil voltge) to otin eq. (13). Wht is strnge out this simple lger? Contrry to Hehl's clims, the trnsformtion properties of oth sides of Eq. (13) re the sme, those of vector vlued threeform [1-13]. Hehl then criticizes the ECE Lemm nd wve eqution. The ECE wve eqution nd Lemm re otined strightforwrdly from the tetrd postulte s in ref. [1], Appendix J. Hehl clims tht the homogeneous field eqution (13) somehow rep resents n "dditionl ssumption" re nonsensicl. Additionl to wht? It is otined simply from the Binchi identity. Then we re told tht the homogeneous field eqution is not Lorentz covrint. We ll know tht the homogeneous eqution is not Lorentz covrint, it is generlly covrint. In his Eq. (46) Hehl ppers to revert to greement once more, nd cites the present uthor's inhomogeneous eqution correctly. He correctly cites the inhomogeneous ECE eqution in his Eq. (48), leit in different nottion. He then sserts, incorrectly, tht this hs een derived fiom Hodge dul trnsformtion of d F, wheres he himself hs just greed tht it is derived from Hodge dul trnsform of F itself, not of d F. He then mkes the tutologicl deduction tht the inhomogeneous current cnnot e derived from Hodge dul of d F. In fct he himself hs just shown tht the inhomogeneous current hs een derived from Hodge dul of F. The truth is s Hehl himself sttes in his own eqution (46), the sme Hodge trnsformtion is correctly pplied oth sides of the eqution to the nti-symmetric tensors present on oth sides of the eqution. He descries this s "recipe", ut it is stright forwrd lger. Following his eqution (48) we re given whole series of consequen tilly erroneous comments. In Section 3.5 it is gin flsely stted tht the Lorentz force eqution hs not een derived in ECE theory. The truth is [1-1] tht it hs een derived from the trnsformtion properties of the F form (see Appendices). At this stge in this immensely long document y Hehl [14] sequentilly erroneous comments re compounded, comments tht merely compound n initilly flse clim. We re then told tht the homogeneous nd inhomogeneous cur rents of ECE theory re mysteriously "non-covrint". The fct is tht oth re vector vlued three-forms which trnsform s such under the usul rules of Crtn geometry [1-13]. The correct wy to look t the generl covrince is to trnsform oth sides of the originl eqution (in indexless nottion) D F = R A (19) which within A (0) is the first Binchi identity:

10 58 M.W. Evns D T = R q (0) This trnsforms to D T = R q (1) then rerrnge terms: d T = R q ω T () nd d F = R A ω F. (3) For ll quntities use the rules for generl coordinte trnsformtion given y Crroll. It is in ny cse ovious tht the first Binchi identity is generlly covrint, no mtter how it is rerrnged, provided the coordinte trnsform is crried out correctly s ove. 4 Grvittion: Evns Adopted Einstein-Crtn Theory of Grvity In section 4 Hehl ppers to revert to greement once more ut decides to reinterpret the uthor's originl ECE theory in terms of his own theory, climing incorrectly tht ECE is fter ll somehow "not originl". If it were not originl Hehl would hve hd no motive to write this long document. Severl clims re mde nd flsely ttriuted to the present uthor. The truth is tht the dynmics of the ECE theory re exemplified s follows: d T = j, d T = J, (4) where j nd J re the dynmicl equivlents of the electromgnetic current defined lredy. It is simple nd ovious to see tht the Crtn torsion form T is nti-symmetric, it is defined s two-form, nd two-form is nti symmetric, hving the required nti-symmetry properties of well defined spin [1-1]. A spinning vector is illustrted in the Appendices. The electrodynmic tetrd A represents one frme spinning nd propgting with respect to nother. Hehl does not seem to understnd this, even though he himself frequently dmits tht ECE theory is techniclly correct. He ppers still to e thinking of tetrd s n oject in grvittionl theory only, wheres in ECE it is more thn tht. Hehl speks of "the trce of the first field eqution". Nowhere in ECE theory is such trce mentioned, nowhere is it needed, nowhere is it used.

11 Response to the Ppers y Hehl nd Hehl nd Oukhov 583 In Section 4.4 Hehl ppers to disgree once gin nd re-sserts incor rectly tht there is n "error" in Crtn geometry, one tht the present uthor hs repeted. The truth is tht the ECE theory is sed on the rigorously cor rect structure equtions nd Binchi identities of Crtn, tught in ll good universities. These re summrized in the Appendices. Hehl ppers to some times gree with this, nd then to oscurely disgree. The reltion etween index reduced cnonicl energy momentum density T nd sclr curvture R in ECE theory is generliztion of Einstein's well known: R = kt (5) where k is the Einstein constnt. Here R is defined in ppendix J of ref. [1]. Hehl now directs ttention to construction of lgrngin. We re told tht the present uthor does not use covrint derivtives in lgrngin. Hehl quotes eq, (116) of pper 57 of This ws used to derive the ECE Lemm, eq. (46) of the sme pper, Lemm which hd lredy een derived from: ( ) : = 0 D Dq v (6) i.e. from covrint derivtives. So the lgrngin hs no need for covrint derivtives to derive n eqution (46) in which there re ordinry deriv tives lredy derived from covrint derivtives. 5 Assessment Hehl finlly dmits in section 5.1 tht "the equtions of Evns nd ssocites (sic) re not very trnsprent to us". Quite so, it is cler from the discourse tht Hehl does not understnd ECE theory (so why write critique)? Hehl criticizes the convention on the normliztion of tetrds. Due to orthogonlity of the vieleins we hve qq = δ v v (7) nd qq = δ (8) From this follows for the doule sum: qq = 4 (9)

12 584 M.W. Evns The convention used in ECE theory is to normlize this to unity: qq = 1 (30) Thus hving to crry round fctors of 4 nd qurter in complicted clcultions is voided. The Einstein convention used in ECE theory is g v g = v 4 (31) for the metric. It is then sserted tht equtions (19), discovered y the present uthor in 006 [1-1] were ctully discovered in ECE theory hs nothing to do with the equtions of Kile or those of Scim. These re equtions of generl reltivity including mechnicl spin while ECE includes electromgnetism nd ll forces of nture. If ECE were Kile or Scim, Hehl would not e criticizing ECE ecuse, to this uthor's knowledge, he does not criticize Kile or Scim. It is this kind of ofusction tht renders this long Hehl document of little vlue to discussion of ECE theory. Hehl then sks wht the present uthor mens y spin. Wht Evns mens y spin is given in references (1) to (1). Hehl cites unrefereed, long reutted, mteril y Bruhn tht clims tht the wedge product of two tetrds cnnot e mde. The wedge product of two oneforms is two-form [1-13] with the properties of nti-symmetry. This is definition of ECE theory. Hehl chooses to deny definition, nothing could e more sujective. Towrds the end of Section 5. [14] there is some compounded ofusc-tion concerning the ECE wve eqution. The ltter is correctly derived in Appendix J of ref, [1]. We re told in section (5.3.1), tht electrodynmics hs nothing to do with the geometry of spce-time. This is sserted without ny reference t ll to experimentl dt. The truth is tht in the "News" section of long list is given of experimentlly tested dvn tges of ECE theory over the stndrd model (see lso the Appendices). ECE theory reduces, in pproprite limits, to ll thec well known lws of physics (therey explining ll the well known experiments of physics consistently, within one frmework for the first time). It is to e noted tht Frncis Bcon dvocted the testing of theory with experimentl dt, not nother theory or construct of the humn mind. - this is the long ccepted wy of chemistry nd physics. If electrodynmics hd nothing to do with geometry of spce-time, the WHOLE of well-tested nd very precise reltivig theory would collpse ecuse reltivity is sed on geometry nd must pply to the whole of physics, not just grvittion. This is the sme reltivity theory tht Hehl cites s "Einstein Crtn" theory, i.e. Crtn geometry. Sometimes Hehl sttes tht Crtn is correct, sometimes he sttes tht he is

13 Response to the Ppers y Hehl nd Hehl nd Oukhov 585 incorrect. Wht ECE mens y Crtn geometry is stndrd nd well defined, nd set out in the mthemticl Appendices to this pper. The equiv lence principle is well defined in ECE theory [1-1] ecuse ll prticles hve mss, including the photon. If Hehl mens the equivlence of inertil mss nd grvittionl mss, then pply it to the photon nd its mss to find tht the equivlence principle exists in electrodynmics too. A pper on neutrino oscilltions is ctully ville in ref. [1]. Hehl seems to hve overlooked this neutrino mss s well? The ECE hypothesis: ( 0) = A A q (3) defines the electromgnetic potentil s vector vlued one-form. The electromgnetic potentil is Feynmn's "universl influence", nd the minimum is universl constnt (see Appendices). The universlity of ECE theory enters through the usul Einstein constnt, i.e. in the universl proportionl ity of R to T for ALL rdited nd mtter fields. The ca (0) is primordil voltge [1-1] mc with minimum vlue where mc e is the photon rest energy nd e the chrge on the proton. A mtter field with spin, such s the electron or neutrino, is descried in ECE y wve-function which is C positive tetrd. Similrly ny mtter field without spin is descried y curvture tetrd. Thus, the spin of the electron is descried in the specil reltivistic limit of ECE theory [1-1] y C positive spinor with four components, the Dirc four-spinor. The ltter does not contin e. In order to descrie the interction of the Dirc electron with the clssicl electromgnetic field, the miniml prescription is used within the Dirc eqution. The chrge e enters into the description in this wy, ut the spin of the electron is descried y C positive spinor. In prticle such s the neutrino with spin nd no chrge, there is no e present. The eigenfunction of the ECE wve eqution is there fore the pproprite q nd not A (0) q. The eigenfunction of the spin hlf electron is lso the pproprite q, ut the eigenfunction of the spin one elec tromgnetic field is A (0) q. The photon with mss is descried y the Proc eqution, with this eigenfunction. All elementry prticles nd ll four fundmentl fields cn thus e descried y one eqution the ECE wve eqution. This is the sis of ECE theory ll of known physics is now descried within the sme mthemticl frmework. Physics is geometry ojective nd deterministic (s mny scien tists like Einstein lwys elieved it hd to e). In generl therefore ll the mtter fields re descried y: ( + ) = 0 kt q (33)

14 586 M.W. Evns nd the vrious rdited fields y equtions similr to: ( + ) = 0. (34) kt A In this respect, Hehl ecomes confused nd sttes incorrectly tht the sic ECE hypothesis cnnot e true. This is done in sujective zel to try to destroy theory, s he himself dmits, nd this is neither the scientific method nor the scientific spirit. We re told tht there is somehow no "chrge - current conservtion" in ECE theory. The rgument y Hehl is so oscure tht the following is the present uthor's interprettion of wht he my men. Hehl gin rgues in the context of wht he incorrectly descries s Mxwell's theory. In fct he descries Heviside's theory, in which chrge current conservtion is descried y: J x = 0 (35) where J is the inhomogeneous current. In ECE theory the equivlent equ tion is: J x = 0 (36) nd is true for ech polriztion descried y (trnsverse or longitudi nl nd lso time-like). Chrge current conservtion is developed in severl plces in the three volumes [1-3], e.g. pp. 484, 508 nd 515 of volume one. Eq. (E.13) on pge 484 is n exmple. This hs the sme structure s the Mxwell Heviside eqution for ech index. Here = (1) nd () indicte complex conjugte trnsverse polriztions, nd = (3) indictes longitudi nl polriztion. A plne wve, for exmple, hs polriztions (1) nd (), nd it is well known tht plne wve conserves chrge 1current density. So it is esy to show tht ECE reduces to Heviside's theory for ech nd lso conserves chrge/current density for ech, i.e. for two senses of trnsverse circulr polriztion ( = (1) nd ()) nd one sense of longitudinl polriz tion ( = (3)). The time-like sense is = (0). The ECE theory thus reduces in well defined wy [1-1] to type of Heviside theory in which the senses of polriztion re well defined y the upper index = (1), (), (3) of the potentil A. The lower index / denotes X, Y ; nd Z in three dimensionl spce. Hehl then cites Bruhn's erroneous comment on the B Cyclic Theo rem without citing the cler nd wtertight reuttl on www. is.us (given gin in the Appendices to this pper). There re mny reuttls ville ( of oth Bruhn nd Lkhtki, none of which re cited

15 Response to the Ppers y Hehl nd Hehl nd Oukhov 587 y Hehl. So Hehl seems to e ised, s well s confused, nd is not evluting ECE theory in n ojective, imprtil, nd scientific wy. For exmple he cites decde old pper y Lkhtki, ut gin does not cite the decde old reuttl [1-1] of Lkhtki's nonsensicl clim tht the ECE spin field is somehow not oservle. It is oservle in the inverse Frdy effect [1-1] nd in severl other wys to e found in the ooks nd ppers pulished y this uthor. The energy momentum in ECE theory is defined for ll prticles nd fields y: R = -kt (37) nd this is universl proportionlity ecuse the constnt k is universl nd the sme for ll prticles nd fields. All prticles nd fields in ECE theory re unified y Crtn geometry. Hehl's comments on energy momentum re wholly irrelevnt to this simple nd cler definition. A sujective lgrngin construction is mentioned y Hehl towrds the end of his first document [14]. This then develops into second document with nother uthor of his own group. 6 Second Document of Hehl nd Oukhov In the pproch to document two, it is mentioned tht lgrngin multiplier hs een found tht elimintes torsion. This oscurity presumly mens tht the lgrngin tht gives zero torsion must e lgrngin of the Einstein Hilert theory, which hs no torsion. In wht wy cn this rgument "refute" ECE theory? The ltter contins torsion y definition. It is then found in the strct of Hehl nd Oukhov [14] tht lgrngin hs een found tht correctly reproduces the ECE field equtions. We re pprently in greement gin. We re told in Eqs. (8) nd (9) of document two [14] tht the ECE equtions re correct, despite document one [14]. Not only re they correct ut they cn e derived y lgrngin. This omits to mention tht ll of this hs lredy een done [1-1]. In Eq. (10) of document lgrngin is set up for wht is gin erroneously clled Mxwell's theory - in ctul fct Heviside's theory. These lgrngin methods re however lredy widely discussed in ECE theory [1-1]. In Eq. (13) nother irrelevnt lgrngin ppers. We re told tht the electromgnetic field is mssless, wheres in ECE theory the photon must hve mss (highlighting nother filure of the stndrd model). So the criti cism in document two is entirely irrelevnt. We re then told in Section.3 tht lthough A crries spin, s is evident in ECE theory, tht we re not ctully told this y Evns. The ( 1) potentils A X ndsoonfrom A re mny times descried in ECE nd precursor guge theory s crrying spin, for exmple they re descried mny times s propgting plne wves [1-1] tht spin right or spin left s they propgte. It is sserted incorrectly tht A trnsforms s vector under Lorentz trnsformtions.

16 588 M.W. Evns The correct generlly covrint trnsform is: A x = Λ (38) x A nd this is not Lorentz trnsformtion. Agin, A is notvector,itis rnk two mixed index tensor [13]. In eq. (5) we finlly rrive, fter much verige, t lgrngin mul tiplier. We re told tht this is necessry ut we re not told why. Shortly therefter the lgrngin multiplier is discrded nd the correct ECE field equtions recovered. We re in greement gin, so why introduce multi plier? The fct is this is n entirely irrelevnt procedure. When the multiplier is re-instted the tutologicl deduction is mde tht the ECE field equtions re chnged. This is n entirely meningless procedure. Any lgrngin mul tiplier cn e chosen, this one ws chosen to eliminte torsion from Crtn geometry. This procedure is incorrect ecuse torsion is intrinsic to Crtn geometry (see Appendices). In section 4 of document it is incorrectly sserted tht the second Crtn eqution is not the second Crtn eqution known to the textooks since 19. The truth is tht the second Crtn eqution defines the Crtn curvture form: R = d ω +ω ω c c (39) It hs een shown in ll detil [1-1] tht this correctly defines the stndrd Riemnn tensor for ny connection. This is necessry nd sufficient. We rrive finlly t conclusion, we re told tht the torsion hs een removed y lgrngin multiplier tht mkes the correct ECE equtions incorrect. This is pure nonsense for the lrge list of resons discussed in this reuttl (nd in more mthemticl detil in the ttched ppendices, in pulished ooks nd ppers nd on the populr AIAS wesites).

17 Response to the Ppers y Hehl nd Hehl nd Oukhov 589 Appendix 1: The Stndrd Crtn Structure Equtions nd Identities Used in ECE Theory These re stndrd equtions of differentil geometry [13]. The first Crtn structure eqution is: T = D q = d q +ω q (A.1) where T is the Crtn torsion form, q is the Crtn tetrd form, nd uj is the spin connection. Using the stndrd tetrd postulte [13]: D q = v 0 (A.) Eq. (A.1) ecomes the definition of the torsion tensor: T κ κ κ v =Γ v Γv (A3) κ where Γ v is the generl connection of Riemnn geometry. Note tht for the Christoffel connection: κ Γ = 0 v (A.4) there is no torsion. This is the cse in Einstein Hilert theory. The second structure eqution of Crtn is: R = D ω = d ω +ω ω (A.5) c c nd using the tetrd postulte (A.) is equivlent to the definition of the Riemnn tensor for ny connection: R σ σ σ σ ρ σ ρ λ v v λ vλ vρ λ ρ vλ = Γ Γ + Γ Γ Γ Γ. (A.6) Note tht the Riemnn tensor used in Einstein Hilert theory is defined y the Christoffel connection. The first identity of Crtn geometry is: D T : = R q (A.7)

18 590 M.W. Evns nd using the tetrd postulte (A.) is equivlent to the cyclic sum [1-1]: R + R + R := Γ Γ +Γ Γ Γ Γ λ λ λ λ λ λ σ λ σ ρ v vρ vρ vρ v ρ σ vρ vσ ρ + Γ Γ +Γ Γ Γ Γ λ λ λ σ λ σ v ρ ρ v vσ ρ ρσ v + Γ Γ +Γ Γ Γ Γ λ λ λ σ λ σ ρ v ρv ρσ v σ ρ (A.8) for ny connection. Note tht in Einstein Hilert theory there is no torsion, so this identity reduces to: λ λ λ R + R + R = 0 ρ v vρ vρ (A.9) which is equivlent to R q = 0 (A.10) using the tetrd postulte (A.). Finlly the second identity of Crtn geometry is resttement of the second structure eqution: ( ) D R : = D D ω. (A.11) Note tht for Christoffel connection, i.e. if nd only if the torsion vnishes, Eq. (A.11) ecomes: D R = 0 (A.1) which using the tetrd postulte (A.) is equivlent to: κ κ κ DR + DR + DR = 0. (A.13) ρ σ v σvρ v σρ The tetrd postulte [13] is true for ny connection nd is the sttement tht complete vector field e independent of its components nd sis elements. This is lwys true in ny ppliction in physics. In ECE theory the tetrd postulte is proven in mny wys [1-1]. This defines the geometry of ECE theory. It is stndrd Crtn geometry. So if Hehl uses different geometry he is descriing neither stndrd Crtn geometry nor ECE theory. The detiled proofs of these sttements re given in the four technicl ppendices of chpter 17 of ref. [1] nd in numerous documents on

19 Response to the Ppers y Hehl nd Hehl nd Oukhov 591 Appendix : Proof of the Lorentz Invrince of the B Cyclic Theorem One of the most serious errors y Hehl is his uncriticl cittion of clim y Bruhn concerning the B Cyclic Theorem [1-1], ville for fifteen yers. This lone rings into serious dout Hehl's imprtility. The B Cyclic Theorem is stted s follows. Consider the mgnetic flux densities defined y the polriztions (1), () nd (3): ( 0) ( ) ( 1) B i B = i ij e φ, (B.1) ( 0) ( ) ( ) B i B = i+ ij e φ, (B.) B ( 3) ( 0) = B k. (B.3) then: ( 1) ( ) ( 0) ( 3* ) B B =ib B et cyclicum (B.4) is the Lorentz invrint. Polriztions [1] nd [] re trnsverse complex conjugtes, nd Φ the electromgnetic phse. Polriztion [3] is longitudinl nd B (3) is n oservle of the inverse Frdy effect [1-1]. The proof of the Lorentz invrince is simple. By direct sustitution it is found tht Eq. (B.6) is: ( 1) ( ) ( 3* ) e e =ie et cyclicum (B.5) where the unit vectors of the complex circulr sis [1-1] re: ( 1) 1 e = ( i ij), ( ) 1 e = ( i+ ij), e ( 3) = k. (B.6)

20 59 M.W. Evns Under Lorentz trnsformtion the unit vectors i, j nd k do not chnge, nd so the P Cyclic Theorem is Lorentz invrint, Q.E.D. Bruhn flsely clims tht this is not so. The Lorentz trnsformtion of mgnetic field is found in ny textook to e: 1 B = γ B υ E (B.7) c where v is velocity nd where: 1 υ γ= 1. c (B.8) The Lorentz trnsform of n electric field is: E = γ ( E +υ B). (B.9) It is seen tht i, j,nd k do not chnge, they re the sme for B' nd B nd for E' nd E.The B (0) fctor chnges, ut is cncelled out on oth sides of Eq. (B.6) to give Eq. (B.7). This ws first shown nerly decde go [1-1] ut ws ignored y oth Bruhn nd Hehl. Bruhn's unrefereed wesites consist entirely of trivilly erroneous comments such s this, nd Hehl cites these trivil errors without citing the correct reuttls, long ville on This seriously misrepresents ECE theory.

21 Response to the Ppers y Hehl nd Hehl nd Oukhov 593 Appendix 3: Nture of the Constnt ca (0) The lest vlue of ca (0) my e worked out using the fundmentl reltions: B ( 0) ( 0) = κ A (C.l) where К is the wvenumer: ω κ= c (C.) nd where ( 0) ea = κ (C.3) where e is the proton chrge, nd is the reduced Plnck constnt. Using the de Broglie photon mss eqution: κ=mc (C.4) where m is the mss of the photon, it is concluded tht the lest vlue of the voltge ca (0) is ca ( 0) mc = e. (C.5) This is universl constnt ecuse the photon mss is universl constnt. The photon mss is essentil for n understnding of light deflection y grvittion, s is well known.

22 594 M.W. Evns Appendix 4: Derivtion of the Lorentz Force Eqution in ECE Theory This derivtion hs een ville for some time [1-1] ut Hehl ppers to ssert tht it is not ville. His oscurity is such tht his mening is not cler to the present uthor, so the derivtion is repeted in this Appendix. The Lorentz force eqution in ECE theory ecomes n eqution of generl reltivity nd is derived from the trnsformtion properties of the Crtn torsion [13]: T x x = Λ (D 1) x x v v T v v x where Λ denotes Lorentz trnsformtion nd x denotes coordinte trnsformtion. In the specil cse where rottion nd trnsltion re mutully independent the ECE field equtions re: d F v = 0, (D.) d F = J v 0 ρ v. (D.3) in tensor nottion [1-1]. For ech polriztion index these hve the sme structure s the Mxwell Heviside field equtions: d F v = 0, (D.4) d F = J (D.5) v 0 ρ v. It is well known tht F μv in Eq. (D.4) trnsforms s: F =Λ Λ F v v v v (D.6) for ech stte of polriztion. Eq. (D.6) is the limit: x x Λ, x x v v Λ v v (D7)

23 Response to the Ppers y Hehl nd Hehl nd Oukhov 595 for ech fixed. The Lorentz force eqution is otined from Eq. (D.6) s: ( ), E =γ E + υ B + (D.8) 1 B =γ B υ E +, c (D.9) nd: ( υ ) F = eγ E + B + (D.10) Therefore for ech sense of polriztion: ( υ ) ( 1 ) ( 1 ) ( 1 ) F = eγ E + B, (D.11) ( ) ( ) ( ) ( ) F = eγ E + υ B, (D.1) ( υ ) ( 3 ) ( 3 ) ( 3 ) F = eγ E + B. (D.13) The third eqution reduces to [1-1] ( ) ( 3) ( 1) ( ) F = iegγυ A A (D.14) ecuse there is no electric equivlent of the inverse Frdy effect: E ( 3) = 0 (D.15) nd ecuse: ( 3) ( 1) ( ) B = iga A (D.16) where: e g = = A κ ( 0 ). (D.17)

24 596 M.W. Evns The oject A (1) A () is the well known conjugte product of non-liner optics [1-1]. Due to the Lorentz invrince of the B Cyclic Eqution the polriztions =(1), (), (3) (D.18) re lso Lorentz invrint, contrry to n oscure ssertion y Hehl. In the most generl cse the Lorentz force eqution in ECE theory must e defined y eqution (D.1) nd the ECE hypothesis ccepted y Hehl: F ( 0) v = A T v. (D.19)

25 Response to the Ppers y Hehl nd Hehl nd Oukhov 597 Appendix 5: Spinning Spce-Time Hehl exhiits complete lck of understnding of the nture of the electrodynmic tetrd introduced in ECE theory nd defined y: ( 0) A = A q (E.1) nd ppers to ssert tht this is curvture tetrd of grvittionl theory. The electrodynmic tetrd is defined y one frme spinning nd propgting with respect to nother. The two frmes re leled nd / nd re relted to ech other y the tetrd: ( 1) 1 ( ) i( t z) q = i ij e ω κ. (E.) This is new ppliction of the sic definition of tetrd y Crtn geometry. In the grvittionl ppliction the index represents n orthonorml spce () tngent to the se mnifold (μ) t point P. In the electrodynmic ppliction frme spins with respect to frme μ. For plne wve the tetrds re [1-1] s follows in vector nottion: ( 1) 1 ( ) i q = i ij e φ, (E.3) ( ) 1 ( ) i q = i+ ij e φ, (E.4) q ( 3) = k, (E.5) where φ=ωt κz (E.6) is the electromgnetic phse. Here ω is the ngulr frequency of spin t instnt t,nd К is the wve-numer of propgtion t point Z. Therefore the plne wve is one of spinning nd propgting frme. The spin connection is needed to define this frme. Hehl confuses this with the curving of spce-time in grvittionl theory, sic error tht renders his whole document irrelevnt. The spinning nd propgting tetrd in three dimensions for exmple is defined

26 598 M.W. Evns y [1-1]: ( 1) ( 1) ( 1) X Y Z ( ) ( ) ( ) X Y Z ( 3) ( 3) ( 3) X Y Z q q q V = q q q V. q q q (E.7) The column vector V is: V 1 ( 1 i i) e 1 = ( 1 + i) e 1 φ φ i (E.8) nd the column vector V μ is: V 1 = 1. 1 (E.9) The column vector V rottes (i.e. spins) s follows: V cosφ+ sin φ 1 = cosφ sin φ (E.10) 1 1 φ= 0,, 1 π 1 1 φ=,, φ=π,, 1 3π 1 1 φ=,, 1 (E.11)

27 Response to the Ppers y Hehl nd Hehl nd Oukhov 599 nd the μ frme is fixed. The spin in ECE theory is clerly defined in this nd mny other wys [1-1]. The only counter rgument is to ssert tht one frme cnnot spin with respect to nther, which is reduction to surdity. Finlly the interction of fields is defined y comined curving nd spinning.

28 600 M.W. Evns Appendix 6: Criticisms of the Guge Principle There re mny criticisms ville [1-1] of the guge principle, which is extensively used in Hehl's work, nd this is ckwrd step in generl reltivity. In ECE theory the prolems with the guge principle re removed y using the invrince of the tetrd postulte under generl coordinte trnsfor mtion. The guge principle is sed on lte nineteenth century ssumption tht the electromgnetic potentil is mthemticl convenience. This con trdicted the view of Frdy nd Mxwell, nd hs lwys een in dispute, n exmple eing the fifty yer dete over the Ahronov Bohm effects. Such protrcted dete mens tht the question is open. Another well known exmple of the filure of U(1) guge invrince in electrodynmics is the Proc eqution: mc + = A 0 (F.1) where m is the photon mss nd where A μ is the U(1) potentil of the stndrd model. The lgrngin methods of the stndrd model run into insurmountle difficulties when ddressing the Proc eqution, which is not U(1) guge invrint. In ECE theory the Proc eqution is otined from the tetrd postulte nd the hypothesis: ( 0) = A A q (F.) giving the generlly covrint wve eqution of electrodynmics: ( + kt ) A = 0. (F.3) In the limit where the electromgnetic field is free of the influence of other fields (such s grvittion), Eq. (F.3) reduces to: mc + = A 0 (F.4) which is covrint s required. There is no internl conflict s in the stndrd model. The eqution: m mc kt = k = V (F.5)

29 Response to the Ppers y Hehl nd Hehl nd Oukhov 601 defines the finite volume: V k = mc. (F.6) In generl this eqution mens tht there re no singulrities in nture, every prticle occupies finite volume. Therefore the need for renormliztion is removed. Therefore Hehl's ttempt to use guge theory in generl reltivity is ound to fil, it is sed on flwed concept in specil reltivity (the concept tht the potentil in Mxwell Heviside theory is unphysicl). To dopt flwed concept for generl reltivity hs no purpose nd to ttempt to criticise ECE theory with flwed concept is douly inpproprite. The filure of U(1) guge invrince to produce the Proc eqution is ftl for guge theory, the photon mss is oservle to precision of one prt in hundred thousnd in the solr system y mesuring the deflection of light due to grvittion. Without photon mss, there would e no such deflection, nd the Proc eqution is fundmentl to physics.

30 60 M.W. Evns Appendix 7: Refuttion of U(1) Guge Invrince in the Inverse Frdy Effect The inverse Frdy effect is the mgnetiztion of mtter y circulrly polrized electromgnetic field t ny frequency. The mgnetiztion is: ( 3) ( 1) ( ) M = ig A A (G.1) where g' is mteril property nd for plne wve: ( 0) ( ) ( 1) A i A = i ij e φ, (G.) ( 0) ( ) ( ) A i A = i+ ij e φ, (G.3) Under U(1) guge trnsformtion: A A ( 1) ( 1) ( 1) A + x ( ) ( ) ( ) A + x, (G.4) so the inverse Frdy effect is chnged. This is counter-exmple to U(1) guge invrince in the stndrd model ecuse the guge principle sserts tht physicl oservle CANNOT e chnged y U(1) guge trnsform tion. This rule hppens to e true for the electric field nd the mgnetic field, where chnging the potentil under U(1) guge trnsform hs no effect, ut it is no longer true in the inverse Frdy effect, n exmple of non-liner opticl effect. This is one counter-exmple out of mny to Hehl's ritrry ssertion tht the electromgnetic potentil is unphysicl. In ECE theory the inverse Frdy effect is due to the spin connection of spinning spce-time, nd is specil cse of the second term on the right hnd side of: F = d A +ω A. (G.5) When the spin connection is dul to the tetrd, the B (3) spin field [1-1] is defined:

31 Response to the Ppers y Hehl nd Hehl nd Oukhov 603 ( 3) ( 1) ( ) B = iga A (G.6) nd the mgnetiztion of the inverse Frdy effect is: M 1 g = B g ( 3) ( 3). (G.7) When the spin connection is dul to the tetrd, the electromgnetic field ecomes independent of the grvittionl field in free spce, nd the spinning frme tht defines the electromgnetic field ecomes independent of the curv ing frme tht defines the grvittionl field. This condition is defined y the fct tht the rottionl prt of the Riemnn form is dul to the torsion form: R κ = T c c. (G.8) This condition is further discussed in Appendix 8, which summrizes the sic mthemticl dvnces mde in ECE theory.

32 604 M.W. Evns Appendix 8: Mthemticl Advnces Mde y ECE Theory Hehl hs entirely filed to understnd the sic nd y now well known nd ccepted mthemticl dvnces mde in ECE theory [1-1]. His rgument is still sed on the concept of curving spce-time, in which oth torsion nd curvture re spects of grvittionl theory only. In tht theory, there is n orthonorml nd orthogonl [13] tngent spce-time t point P to se mnifold. It is shown in Appendix 5 tht the tetrd my e used for spinning spce-time in well defined mthemticl mnner. In the cse of the spinning tetrd, one frme spins nd simultneously propgtes with respect to nother, the tetrd is defined y two column vectors s usul, ut its interprettion is different from tht in grvittionl theory, nd kin to tht in guge theory. However the strct indices of the fier undle of guge theory re replced y the geometricl understnding of Appendix 5, nd s rgued in Appendices 6 nd 7, the guge principle hs een ndoned in ECE theory. One of the mthemticl dvnces mde in ECE theory is to develop the Riemnn form for ny connection, so tht it ecomes defined for mthemticlly well-defined spinning spce-time s well s curving spce-time. When the tetrd descries spin motion of spce-time (Appendix 5) unf fected y curving of spce-time the homogeneous current of ECE theory vnishes: ( 0) A 0 ( ) j = R q ω T = 0. (H.1) It hs een shown in Appendix J of ref. (1) tht this condition implies: κ c ω = c. q (H.) The spin connection ecomes dul to the tetrd through wve-numer. Furthermore, the Riemnn form ecomes dul to the torsion form through the sme wve-numer: R κ = T c c. (H.3) This is fundmentl mthemticl dvnce of ECE theory. The ECE dulity equtions define hitherto unknown rottionl prt of the Riemnn form:

33 Response to the Ppers y Hehl nd Hehl nd Oukhov 605 R = qqr κ ρ v κ v (H.4) where R ρ κ v is the Riemnn tensor. In Einstein Hilert (EH) theory the dulity (H.3) is not present, ecuse in EH theory the Riemnn form is pure curvture form. There is no torsion in EH theory, so the Riemnn form of EH is non-zero nd the torsion form of EH is zero. So one cnnot e the dul of the other. Hehl's use of torsion is restricted to grvittionl theory where the first Binchi identity of Appendix One implies tht the Ricci cyclic eqution for curving spcetime my e written in terms of torsion. This identity sserts tht the Riemnn tensor for ny connection is identiclly equl to itself. In grvittionl theory there is no notion of spinning spce-time s defined in Appendix 5. The spinning spce-time of Appendix 5 my therefore e descried y new type of Riemnn form. This is not curvture form ut is tensor vlued two-form descriing spin. It is the dul tensor of the vector vlued torsion two-form descriing spin. The form indices (the Greek indices) re fixed, so this is n exmple of n ntisymmetric tensor eing dul to n xil vector, representing spin. In generl the Riemnn form is mde up of comintion of curvture nd spin prts. Prior to ECE theory this ppers not to hve een known in mthemtics, or t lest not clerly defined. The definition of the spin tetrd still uses the concept of one orthogonl nd orthonorml frme (the sttic frme of Appendix 5) nd one dynmicl frme. Ech frme is defined y column vector, so the definition of the tetrd is unchnged: V = qv. (H.5) The tetrd, s in grvittionl theory, is mtrix tht links the column vectors. So the ECE theory is unified field theory ecuse the field sectors re ll self-consistently defined y the tetrd, which is lwys the fundmen tl field. The interction of fields is then defined y the Crtn structure equtions nd Binchi identities of Appendix One. Different mthemticl representtion spces my e used for the column vectors in Eq. (H.5). For fermion they re defined using n SU() representtion spce. In strong field theory n SU(3) representtion spce is used. Another mjor mthemt icl dvnce of ECE theory hs een to derive the generlly covrint Dirc eqution from the tetrd postulte, nd to recognize tht the Dirc spinor cn itself e represented y tetrd elements. The usul method used in the stndrd model is to mke the Dirc mtrix covrint, while still using the Minkowski spce-time for the spinor. This is self-inconsistent. The generlly covrint Dirc eqution of ECE theory is internlly consistent. The fermion wve-function is the tetrd. Therefore ECE hs unified generl reltivity nd quntum mechnics, nd self-consistently reduces to ll the mjor