Simplified proofs of the Cartan structure equations

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1 382 M.W. Ens Journl of Foundtions of Physics nd Chemistry, 2011, ol. 1 (4) Simplified proofs of the Crtn structure eutions M.W. Ens 1 Alph Institute for Adnced Studies ( The two Crtn structure eutions re proen strightforwrdly through use of simplified formt for the tetrd postulte. In so doing new generl condition on Crtn s differentil geometry is discoered nd illustrted with respect to the tetrds of propgting, circulrly polrized, we. Keywords: Crtn structure eutions, simplified tetrd postulte, new eution of Crtn's differentil geometry. 1. Introduction The first nd second Crtn structure eutions define the torsion nd curture forms of differentil geometry nd re euilent to the Riemnnin torsion nd curture [1 10]. They show the rigorous nd complete internl self-consistency of Crtn geometry nd illustrte the role plyed y the Crtn tetrd nd the Crtn tetrd postulte. It is therefore importnt to e le to proe the structure eutions s simply nd s clerly s possile nd to e le to proe their euilence to nineteenth century Riemnn geometry. The role of the inde of Crtn geometry is lso clrified nd well illustrted y these proofs, which re gien in Section 2. In Section 3 the proof of the second structure eution is illustrted with reference to the tetrd of the propgting nd circulrly polrized we. If geometry is internlly self-consistent it stnds s such nd my e used in the philosophy of generl reltiity s in ECE theory to gie new physics nd unify older concepts of physics. 2. Proofs of the Crtn structure eutions In the stndrd nottion [1] of differentil geometry the first Crtn structure eution is: T = +ω 1 e-mil: EMyrone@ol.com (1)

2 Simplified proofs of the Crtn structure eutions 383 where T is the torsion form, ector lued two-form, is the tetrd form ( ector lued one-form), ω is the Crtn spin connection, nd d is the eterior deritie. In tensor nottion this nottion trnsltes into: T = +ω ω (2) The proof is now gien of the euilence of E. (2) nd the Riemnnin torsion: T κ κ κ =Γ Γ (3) K K where T is the Riemnnin torsion nd where Γ is the Riemnnin connection. The dntge of E. (2) oer E. (3) is tht E. (2) uses n inde tht my e deeloped with ny sis elements such s the Puli mtrices of Dirc theory. In Pper 141 (see the current issue of the journl) it ws shown tht the inde my e used to denote the comple circulr sis, where it hs profound mening in tht it etends the Helmholtz Theorem. The euilence of Es. (2) nd (3) origintes in the fundmentl property tht the complete ector field is independent [1 10] of its coordintes nd sis elements: V = V e = V e (4) This is lwys the cse in physics nd generl reltiity. Without the property (4) three dimensionl ector field for emple would e different in Crtesin or sphericl polr coordintes. The property (4) leds to the eution: D = +ω Γ λ λ (5) This is confusingly nd oscurely known s the tetrd postulte [1]. Eution (5) is fundmentlly true in physics nd in lmost ll of mthemtics, nd is not postulte. It follows from the property (4) [1-10]. Using the rules [1]: ω =ω Γ =Γ λ λ, (6) (7) the tetrd postulte simplifies to: Γ = +ω (8)

3 384 M.W. Ens nd the first Crtn structure eution simplifies to: T =Γ Γ (9) It follows immeditely tht the Riemnnin torsion is: T = T =Γ Γ (10) κ κ κ κ Q.E.D. The Crtn nd Riemnn torsions re oth ery fundmentl concepts of geometry, in ny spce, of ny dimensions. The Riemnn torsion is lwys present in ny spce, nd is defined y the commuttor of corint derities in ny spce of ny dimension [1 10]: D D V = R V T DV ρ ρ σ κ ρ, σ κ (11) Here V ρ is ny ector, R ρ σ is the Riemnnin curture nd DVρ κ is the corint deritie of V ρ. From E. (11) it is oious tht the Riemnnin connection must e ntisymmetric: Γ = Γ κ κ (12) ecuse the indices u nd re the sme on oth sides of the eution nd ecuse the commuttor is ntisymmetric y definition [1 10]. The connection cn neer e symmetric ecuse the commuttor cn neer e symmetric nd t the sme time non-zero. The second Crtn structure eution [1 10] is: R = d ω +ω ω c c (13) where R is the curture form ( tensor lued two-form). In tensor nottion E. (13) is: c c R c c = ω ω +ω ω ω ω (14) nd it is now proen tht this eution is euilent to the Riemnn curture: R ρ ρ ρ ρ λ ρ λ σ σ σ λ σ λ σ = Γ Γ +Γ Γ Γ Γ (15) Strt the proof y noting tht:

4 Simplified proofs of the Crtn structure eutions 385 ω = ω =Γ =Γ (16) Note tht: = (17) ecuse the tetrd multiplies rnk three mied inde tensoril untity. Therefore: R ( ) = Γ Γ +Γ Γ Γ Γ c c c c Γ Γ +Γ +Γ c c c c c c c c = Γ Γ +Γ Γ Γ Γ c c c c (18) Finlly use: R = R ρ ρ σ σ (19) nd Γ Γ = Γ Γ =Γ Γ (20) ρ c c λ ρ λ ρ λ c σ λ c λ σ λ σ to otin E. (15), Q.E.D. This is much simpler nd clerer proof thn one preiously gien in n ppendi of Pper 15 of the ECE series. By the rules of Crtn geometry it follows tht: = (21) so using the Leiniz Theorem: = + (22) The tetrd in Minkowski spcetime is unit digonl: = (23)

5 386 M.W. Ens so: = 0 (24) From Ens. (17) nd (22) it follows tht: = 0 (25) nd this is new condition or constrint on Crtn geometry ecuse the ltter uses the inde to denote Minkowski spcetime [1 10]. In ECE theory the potentil is defined in terms of the tetrd [2 10], so, for emple, the electromgnetic potentil is: ( 0) A = A (26) Therefore: A A = 0 (27) nd this is new generl constrint on the electromgnetic potentil in ECE theory. Similrly, in grittionl theory: Φ Φ = 0 (28) where Φ is the grittionl potentil. Eutions (8) nd (25) show tht: Γ =ω (29) so the euilence of the Crtn nd Riemnn curtures follows immeditely. Note tht the metric tensor is defined in generl y: = g (30) where is the position four-ector: = (ct, X,, Z) (31) In Minkowski spcetime:

6 Simplified proofs of the Crtn structure eutions g g = = (32) Eution (30) cn e epressed s: = g (33) where: g = (34) In Crtn s differentil geometry the tetrd is defined in generl y [1 10]: V = V (35) where V is ny ector field. E (33) is specil cse of E. (35): = (36) so it follows tht: = g = (37) Q.E.D. To eemplify E. (25), consider the following plne we tetrds: 1 ( i j) i = i e ϕ 2 (38)

7 388 M.W. Ens nd ( 2) 1 ( i j) i = + i e ϕ 2 (39) Eution (39) is the comple conjugte of E. (38) nd φ is the phse of the plne we. The comple circulr sis is defined y: ( 2) ( 3* ) =i (40) *in cyclic permuttion Consider elements such s: 1 iϕ i i X = e, = e 2 2 ( 2) 1 ϕ i ( 2) i X = e, = e 2 2 ϕ ϕ i (41) (42) nd use the rule: 1 = (43) of Crtn geometry to find tht: X ( 2) X ( 2) = 1 (43) A solution of E. (43) is: X 1 ϕ i i ϕ i = e, = e, 2 2 X 1 iϕ i iϕ ( 2) = e,( 2) = e 2 2 (44) Consider n emple of E. (25): X = ( 2) + (45) For: ν = 0 (46)

8 it follows tht: Simplified proofs of the Crtn structure eutions 389 X X 0 ( 2) 0 ( 2) + = 0 (47) For: ν = 3 (48) it follows tht: X X 3 ( 2) 3 ( 2) + = 0 (49) Consider second emple of E. (25): ( 2) X = + (50) nd it follows tht: ( 2) X = 0 (51) nd: ( 2) X = 0 (52) So the generl constrint (25) hs een tested with the tetrds of the circulrly polrized plne we, nd self consistent result found. This procedure illustrtes the importnce of simplicity nd clrity in mthemtics nd shows tht within its own terms of definition, the two structure eutions, Crtn s geometry is perfectly self consistent. Similrly, within its terms of reference, Euclid s geometry is perfectly self consistent, or Riemnn s geometry is perfectly self consistent. It my e possile to he n ultr strct geometry in which ector field is not independent of its coordinte nd sis elements, ut tht type of geometry merely dds strction nd contrenes the most fundmentl principle of generl reltiity, generl corince. There is no eperimentl eidence tht such geometries he ny role in nture. For emple it my e possile to define mnifolds other thn tht of Riemnn, ut gin there is no eidence tht these mnifolds re relent to nturl philosophy (physics). The construction of such mnifolds without ny eperimentl eidence is contrry to the most fundmentl rule of nturl philosophy, Ockhm s Rzor. In other words they re

9 390 M.W. Ens just mthemtics for the ske of mthemtics. String theory flls into the sme ctegory, nd ny theory tht contins unoserles. The twentieth century is pockmrked with such filed theories. ECE theory on the other hnd hs een tested eperimentlly in mny wys [2 10] nd is fundmentlly ery simple, eing sed directly on Crtn geometry. It should not surprise nyone tht ll the eutions of physics come out of geometry using Ockhm's Rzor. References [1] S.P. Crroll, Spcetime nd Geometry: n Introduction to Generl Reltiity (Addison Wesley, New ork, 2004), chpter 3. [2] M.W. Ens, Generlly Corint Unified Field Theory (Armis, 2005 onwrds), in seen ol umes to dte, olume 7 with H. Eckrdt, D. Lindstrom nd F. Lichtenerg. [3] L. Felker, The Ens Eutions of Unified Field Theory (Armis 2007). [4] The ECE wesites nd [5] K. Pendergst, The Life of Myron Ens (Cmridge Interntionl Science Pulishing, 2011). [6] M.W. Ens, H. Eckrdt, S. Crothers nd K. Pendergst, Criticisms of the Einstein Field Eution (Cmridge Interntionl Science Pulishing, 2011). [7] M.W. Ens (ed.), Modern Nonliner Optics (Wiley 2001, second edition). [8] M.W. Ens nd S. Kielich (eds.), iid, first edition (1992, 1993, 1997). [9] M.W. Ens nd J.-P. Vigier, The Enigmtic Photon (Kluwer, 1994 to 2002), in fie olumes. [10] M.W. Ens, ppers on B(3) nd ECE theory, Omni Oper section of