h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional fourth rank antisytric tnsor, ε4, and so of its uss h ε4(i,j,k,l) tnsor is a vary clos rlativ of Mathcad's ε(i,j,k) tnsor First th tnsor itslf will b xaind Aftr that its us to valuat 4 by 4 dtrinants and to calculat th 4 dinsional quivalnt of th cross product will b xplord Blow is th dfinition of ε4, th antisytric tnsor of rank 4, for a 4-dinsional vctor spac h coponnt's valu is if any of th indics ar qual, and if th indics ar (,,,) h tnsor coponnt is qual to - if th indics ar an odd prutation of (,,,), and if th indics ar an vn prutation of (,,,) pr4 A cnt nd for for if last A j nd i j A i > A i If th indics ar all diffrnt thn ε4(i,j,k,l) uss pr4(a) to count th nubr of prutations ndd to arrang th indics in ascnding ordr, (,,,) t A i A i A i A i t cnt cnt cnt ε4 i, j, k, l if i j j k k l i k j l i l A od i, 4 A od j, 4 A od k, 4 A od l, 4 cnt val val pr4 A val if od cnt, val othrwis othrwis If any of th indics ar th sa, thn that coponnt is qual to zro If thy ar all diffrnt, xain th prutation count If th count is odd, thn th coponnt is qual to - If th count was vn, thn that coponnt of th tnsor is qual to
Blow ar a fw of th 56 coponnts of th fourth rank antisytric tnsor ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, ε4,,, h rang variabls; i,j,k,l rang ovr zro to thr for th indxs of th ε4 tnsor i j k l Proprtis of a 4th rank, 4 di tnsor h nubr of coponnts in a 4th rank, 4 dinsional tnsor is: 56 fro contraction of δ th dinsion of our spac is: δ i, i 4 i Coordinat ransforation Law is: i, j, k, l d c b a a, b, c, d λ a, i λ b, j λ c, k λ d, l So of th proprtis of th 4(i,j,k,l) tnsor h nubr of non-zro coponnts of ε4 is: ε4 i, j, k, l 4 h nubr of coponnts of ε4 qual to is: ε4 i, j, k, l h nubr of coponnts of ε4 qual to - is: ε4 i, j, k, l h nubr of coponnts of ε4 qual to is: ε4 i, j, k, l
Now that w hav cratd th 4th rank totally anti-sytric tnsor, what can it b usd for? h ε4 tnsor can b usd to valuat th dtrinant of a 4 x 4 atrix, as shown blow o valuat th dtrinant on convrts th coluns of th atrix to colun vctors Nxt apply th tnsor to all four vctors yilding a singl nubr, th dtrinant Blow ar so xapls, using th ε4 tnsor to calculat dtrinants and th built in Mathcad calculation for vrification of th rsults i j k l D4 ε4 i, j, k, l i j k l 4 D4 D4 4 4 D4 5 9 6 7 D4 48 48 4 5 Sinc th ε4 tnsor can b usd to dfin th 4 by 4 dtrinant, on should b abl to dfin th ε4 tnsor in trs of th 4 by 4 dtrinant and th δ function or δ tnsor h 4x4 dtrinant of dlta functions is usd to dfin th ighth rank δ tnsor with 4 contravariant indxs and 4 covariant indxs h coponnts of th ε4a tnsor, dfind fro th 4 by 4 dtrinant, ar a subst of th ighth rank δ tnsor's coponnts If you ar still using Mathcad- you would hav to us th dfinition blow for th 4D, 4th rank anti-sytric tnsor δ i, r δ i, s δ i, t δ i, u δ4 i, j, k, l, r, s, t, u δ j, r δ k, r δ j, s δ k, s δ j, t δ k, t δ j, u δ k, u ε4a r, s, t, u δ4,,,, r, s, t, u δ l, r δ l, s δ l, t δ l, u Latr in this papr both ε4 and ε4a will b usd to calculat th sa 4 dinsional cross product to vrify thir rsults
h scond intrsting thing you can do with th 4th rank totally antisytric tnsor is to prfor th 4 dinsional quivalnt of th cross product in Euclidan -spac, R First lt's rviw so of th proprtis of th cross product in R h R cross product is gotrically dfind as: C A B A B sin θ E E E is prpndicular to A and B h dirction is givn by th right hand rul in cartsianl coordinat systs A E and B E Vctor C's agnitud is qual to th lngth of A tis th lngth of B tis th sin of th angl btwn A and B h cross product of A and B givs a noral vctor to both A and B, who's lngth is qual to th ara of of th paralllogra constructd fro A and B A cross B is anti-sytric so: A B B A givn, th atrix of basis vctors: h non-zro lnts of th cross product of th basis vctors in R ar: < > < > < > < > < > < > < > < > < > < > < > < > < > < > < > < > < > < > h ultiplication tabl to th lft suarizs th action of th cross product on th orthonoral basis vctor of Euclidan -spac Fro this tabl on can s that th rd rank, dinsional antisytric tnsor, which is Mathcad's built in ε(i,j,k,) tnsor, applid to two indpndnt -vctors, can b usd to valuat th cross product h cross product iplntd with th dinsional anti-sytric tnsor is: i j k CP A, B < k > ε i, j, k A i B j h k suation is usd to collct th coponnts, fro th application of th ε k j i tnsor to vctors A and B, into th rsults vctor Not athcad's ordr of oprations is usd to always forc a vctor rsult A B 4 5 6 wo indpndnt -vctors for th xapl blow Cp CP A, B Cp 6 A B 6 Cn CP B, A Cn 6 B A 6 it is anti-counitiv A Cp B Cp h cross product vctor is orthogonal to both A and B CP A, A A A h Cross product of dpndnt vctors is zro
h ε4 tnsor will now b usd to gnrat th four-dinsional quivalnt of th cross product his function taks thr linarly indpndnt 4 dinsional vctor that span a D hypr-plan and rturns a 4-vctor prpndicular to all th input vctors Not: A fourth rank tnsor ust b applid to thr vctor in ordr to produc a singl vctor as its output i4 j4 k4 l4 id4 h id4 atrix has th four dinsional orthonoral basis vctors as its coluns h four dinsional xtnsion of th Cross product using th ε4 tnsor and thr 4-vctors is: CP4 V, V, V l4 id4 l4 k4 j4 i4 ε4 i4, j4, k4, l4 V i4 V j4 V k4 Again, Mathcad's ordr of oprations is usd to forc CP4 to always rturn a vctor h zro 4-vctor is rturnd instad of just scalar zro CP4a V, V, V l4 id4 l4 k4 j4 i4 ε4a i4, j4, k4, l4 V i4 V j4 V k4 CP4a is bas on th 4th rank, 4 dinsional antisytric tnsor gnratd fro th dtrinant and dlta function dfinition his had bttr yild th sa rsults as CP4 id4 id4 id4 4 id4,,,4 ar st to th four orthonoral bais vctors Blow ar th rsults of th four dinsional cross product applid to sts of thr basis vctors at a ti O4 CP4 4,, 4 4 O4 O4 O4 h 4D cross product of thr of th four basis vctors yilds th fourth basis vctor that's prpndicular to th othr thr O4 O4 CP4,, 4 O4 h 4D cross product is anti-counitiv O4 CP4, 4, O4 just lik th dinsional cross product O4 CP4,, 4 O4 O4 CP4,, 4 O4 O4 CP4,, O4 h 4D cross product rturns th zro O4 vctor if any of th input vctors ar linarly dpndnt 4 4 4 O4
Blow ar thr linarly indpndnt tst vctors and thir four-dinsional cross product calculatd by both ε4(i,j,k,l) and ε4a(i,j,k,l) his was don to chck th rsults of both th prograd vrsion of th ε tnsor and th vrsion gnratd fro th 8th rank δ tnsor W gt th sa rsult fro both calculations A 4 indpndnt tst 4-vctors A 5 6 7 8 A Rsults of th 4 dinsional cross products ar qual 9 CP4 A, A, A 7 44 7 CP4a A, A, A 7 44 7 h rsults of th 4D cross product, R4, is orthogonal to A, A, and A as xpctd R4 CP4 A, A, A R4 7 44 7 R4 A R4 A R4 A Closing Nots: h Author would lik to point out that th rsults of th four dinsional cross product ar xactly corrct only for Euclidan vctor spacs with orthonoral basis vctors In ths spacs contravariant vctors, or just plain "vctors", and covariant vctors, or on-fors hav idntical coponnts his is bcaus th tric is qual to th idntity atrix h ε4 tnsor dfind in this papr is of typ (,4), or it oprats on four (contravariant) vctors and rturns a ral nubr if this tnsor oprats on thr vctors, as in th four dinsional cross product, th rsult is a (covariant vctor) on-for Only in a Euclidan spac with orthonoral basis vctors will th on-for's coponnts, rturnd by ε4, b qual to its dual vctor's coponnts o turn an on-for into a vctor, ust apply th invrs (contravariant) tric tnsor to it h invrs tric aps on-fors into vctors and th (covariant) tric tnsors aps vctors into on-fors h tric allows th apping of vctors into on-fors and visa-vrsa his stablishs th duality btwn vctors and on-fors in gnral tric spacs A or gnral dfinition of th vctor cross product with non-orthogonal basis vctors would b: CP4 G ε4 a, a, a G is invrs of th tric ε4 is th 4 dinsional anti-sytric tnsor a, a, a, and CP4 ar contravariant vctors Although th on for rsult of ε4 a, a, a is just as usful
Rfrncs MA Akivis & VV Goldbrg, "An Introduction to linar Algbra & nsors", Dovr Publications, Minola, NY, 977 AIBorisnko & IE arapov, "Vctor and nsor Analysis with Applications", Dovr Publications, Minola, NY, 979 Richard Shiffan, "h Cross Product, Dot Product and fun with nsors", 995, (URL: http://wwwathsoftco/pub/apps/crossdotcd ) 4 Alfrd Gray, Modrn Diffrntial Gotry of Curvs and Surfacs, CRC Prss, 99 5 Brnard F Schultz, "A First Coars in Gnral Rlativity", Cabvridg Univrsity Prss, 985