Journal of Modrn Physics, 7, 8, 9- http://www.scirp.org/journal/jmp ISSN Onlin: 5-X ISSN Print: 5-96 Using Tangnt Boost along a Worldlin and Its Associatd Matrix in th Li Algbra of th Lorntz Group Michl Langlois, Martin Myr, Jan-Mari Vigourux IRRG, Bsançon, Franc Laboratoir d Mathématiqus Univrsité d Franch-Comté, Bsançon Cdx, Franc Institut UTINAM, UMR CNRS 6, Univrsité d Franch-Comté, Bsançon Cdx, Franc How to cit this papr: Langlois, M., Myr, M. and Vigourux, J.-M. (7) Using Tangnt Boost along a Worldlin and Its Associatd Matrix in th Li Algbra of th Lorntz Group. Journal of Modrn Physics, 8, 9-. https://doi.org/.46/jmp.7.8879 Rcivd: April, 7 Accptd: July 4, 7 Publishd: July 7, 7 Copyright 7 by authors and Scintific Rsarch Publishing Inc. This work is licnsd undr th Crativ Commons Attribution Intrnational Licns (CC BY 4.). http://crativcommons.org/licnss/by/4./ Opn Accss Abstract In ordr to gnraliz th rlativistic notion of boost to th cas of non inrtial particls and to gnral rlativity, w look closr into th dfinition of th Li group of Lorntz matrics and its Li algbra and w study how this group acts on th Minskowski spac. W thus dfin th notion of tangnt boost along a worldlin. This vry gnral notion givs a usful tool both in spcial rlativity (for non inrtial particls or/and for non rctilinar coordinats) and in gnral rlativity. W also introduc a matrix of th Li algbra which, togthr with th tangnt boost, givs th whol dynamical dscription of th considrd systm (acclration and Thomas rotation). Aftr studying th proprtis of Li algbra matrics and thir rducd forms, w show that th Li group of spcial Lorntz matrics has four on-paramtr subgroups. Ths tools lad us to introduc th Thomas rotation in a quit gnral way. At th nd of th papr, w prsnt som xampls using ths tools and w considr th cas of an lctron rotating on a circular orbit around an atom nuclus. W thn discuss th twin paradox and w show that whn th on who mad a journy into spac in a high-spd rockt rturns hom, h is not only youngr than th twin who stayd on arth but h is also disorintatd bcaus his gyroscop has turnd with rspct to arth rfrntial fram. Kywords Li Group of Lorntz Matrics, Li Algbra, Tangnt Boost along a Worldlin, Acclration, Spcial Rlativity, Gnral Rlativity, Thomas Rotation, Twin Paradox, Inrtial Particls, Non Inrtial Particls. Introduction In th fram of spcial rlativity thory, th history of an inrtial particl is DOI:.46/jmp.7.8879 July 7, 7
M. Langlois t al. dscribd by a godsic straight lin in th four dimnsional Minkowski spac, ndowd with th η = diag (,,,) mtric. This godsic is a timlik straight lin and its orthogonal complmnt is th physical spac of th particl formd by all its simultanous vnts. Th passag from on inrtial particl to anothr on is don through a spcial Lorntz matrix, which is calld th boost, and this procss is th Lorntz-Poincar?? transformation. But for a non inrtial particl, all this is lost sinc its worldlin is no mor a straight lin and thr is no Lorntz transformation and boost associatd to it. In ordr to fill this gap w suggst a dpr insight into th action of th Li group of Lorntz matrics (and its Li algbra) on th Minkowski spac. This lads us to a nw dfinition of a tangnt boost along a worldlin. This notion may b usd in both situations of spcial or gnral rlativity thoris. Thrfor w introduc a matrix blonging to th Li algbra, which, togthr with th tangnt boost, dscribs compltly th dynamical systm: acclration and instantanous Thomas rotation. In a first part, w prsnt proprtis of Li matrics and of thir rducd forms and w show that th Li group of spcial and orthochronous Lorntz matrics has four on-paramtr subgroups. Ths tools prmit to introduc th Thomas rotation in a quit gnral way. Thn, w giv som applications of ths tools: w first considr th cas of an uniformly acclratd systm and th on of an lctron rotating on a circular orbit around th atom nuclus. W thn prsnt th cas of th so-calld Langvin s twins and w show that, whn th twin who mad a journy into spac rturns hom, h is not only youngr than th twin who stayd on arth but h is also disorintatd with rspct to th trrstial fram bcaus his gyroscop has turnd with rspct to th arth rfrntial fram []. Lt us undrlin that this formalism can b usd both in Spcial and in Gnral Rlativity.. Th Li Algbra of a Li Group A Li group is a smooth manifold with a compatibl group structur, which mans that th product and invrs oprations ar smooth. Th Li algbra of this Li group can b sn as th tangnt spac T to th manifold at th unit lmnt of th group multiplication. This tangnt spac is a vctor spac ndowd with th Li brackt of two tangnt vctors. xampl: Th Li Group SO( ) and Its Li Algbra Lt s start with th group of -matrics having + dtrminant. As a 9 smooth manifold, it can b rgardd as a -dimnsional submanifold of dfind by th 6 quations rsulting from th orthogonal matrix dfinition: T AA = I. Lt's dnot it, as usual, by SO ( ). Its Li algbra is th -dimnsional vctor spac of skw-symmtric matrics ndowd with th brackt [ Ω, Ω ] =Ω Ω Ω Ω 9
M. Langlois t al. This manifold is obviously isomorphic to th uclidan spac ndowd with th cross product. Th vctors of our Li algbra should b rgardd as tangnt vctors d dt A TA of smooth paths t At on th ( ) T T = A shall b L B AB A SO manifold. Th lft translation on th group by A =. Of cours T if A= B, L ( A) = AA = I. Th linar mapping L A A which by dfinition is qual to its diffrntial DL A maps th tangnt vctor d A from th tangnt dt spac T to th tangnt spac A T : I da da T da Ω= DL = A = A () A dt dt dt Drivating th rlation T AA = I givs T Ω+Ω=, which mans that Ω is skw-symmtric. Of cours it would b possibl to obtain th sam A T, using R B B A A T through I right translation by =. Application to kinmatics of rotation of a rigid body around a fixd point. Kping in mind latr comparisons, w shall apply th rsults of th prvious sction to th study of th motion of a rigid body. W want to show th intrst of looking at th action of SO ( ) on whn this group is rgardd as a, and th maning of its Li subgroup of isomtris of th uclidan spac algbra = T. Lt us associat to a thr-dimnsional point O two coordinat I systms ( O, ) and ( O, ) dfind by two orthonormal basis and rspctivly. And lt A dnot th matrix mapping to. A smooth path t At O, on th ( ) SO manifold corrsponds to a rotation movmnt of around O with rspct to th coordinat systm ( O, ). Lt us dnot by X and X th coordinats of a point M in th, O, rspctivly. nighborhood of O in th rfrnc systms ( O ) and Looking at th movmnt of M with rspct to ( O, ), w thn hav d d d X( t) = At X ( t) X = A X + A X dt dt dt composing by th lft translation A and using () w gt dx dx da dx A = + A X = +Ω X dt dt dt dt This rlation xprsss th drivation rul of th movmnt of a point X in th dx moving coordinat systm ( M, ). Th absolut drivativ dt of X with dx rspct to t is qual to th sum of th rlativ drivativ dt and of th training drivativ dfind by Ω X dx dx = +ΩX dt dt Ω is a skw-symmtric covariant tnsor whos adjoint givs th componnts of a vctor ω in and prmits us to xprss th training vlocity in th wll known vctor form: () 9
M. Langlois t al. ( ω X) Ω X = () An analogous procss starting from th right translation R A would lad us to th sam drivation rul as givn in () but using th componnts in ( M, ). Not that da A dt = Ω Thr is anothr intrsting application of th idntification xponntial map k Ω Ω xpω= k! is a diffomorphism of th opn ball of radius π in subst of ( ) writing ϖ th norm of ω and A = : Th into an opn SO : = xp Ω SO, w hav SO, and w thus obtain an intrsting paramtrization of ( ϖ) ω ( ϖ ) ω ( ω ) AX = X + sin X + cos X X. Such a formula has an obvious gomtrical maning: xp Ω is a rotation through th angl ϖ about th axis having th dirction of th vctor ω. Not that, Ω bing indpndant of t, th function t xp ( tω) : SO( ) dfins a on-paramtr subgroup of SO ( ), and that th matrix product xp tω X is th solution of th linar diffrntial quation dx dt =Ω X with th initial condition X = X. This quation is nothing but () whn M,. Its solution dfins a uniform rotation. writtn in th coordinat systm. Th Li Group of Lorntz Matrics Application to Spcial Rlativity.. Prliminaris In spcial rlativity th motion of an inrtial particl with rspct to an inrtial obsrvr is dscribd by a Lorntz-Poincar transformation. This transformation is associatd to a 4 4-matrix blonging to th subgroup of orthochronous Lorntz matrics of + dtrminant: thy map th subst of timlik futur orintd vctors into itslf. Thy transform a η-orthonormal basis (associatd to th inrtial obsrvr) into anothr η-orthonormal basis (associatd to th particl). Th columns of such a matrix hav a clar physical and gomtrical intrprtation: th first column is th 4-vlocity of th particl (a unitary timlik 4-vctor tangnt to th worldlin), and th thr othr columns dfin an orthonormal basis of th physical spac of th particl. W turn now to th mor gnral situation of a non inrtial particl: th rlativ motion btwn two non inrtial particls, or btwn a non inrtial particl and anothr (inrtial or non inrtial) obsrvr will b dscribd by a tim-dpndnt function with valus in th group of Lorntz transformations. W thus naturally com to th notion of T I 9
M. Langlois t al. tangnt boost along a worldlin, w shall now study its main proprtis... Th Li Group of Lorntz Matrics and Its Associatd Li Algbra Th shall dnot by th subgroup of th Li group of Lorntz matrics consisting of all orthochronous (Lorntz) matrics with + dtrminant. It is a 6-dimnsional submanifold of 6 as dfind by th quations involving th 6 coordinats s ij of th matrix S, obtaind from th rlation T SηS = η. This group acts as a group of isomtris on th Minkowski spac = ( 4,η ). W shall now b intrstd in th tangnt spac = T of I takn at th idntity matrix I. ds Lt t S( t) b a smooth curv on th manifold, and S = its dt tangnt vctor blonging to St simply b: From th rlation T SηS covariant tnsor of typ (, ), T, th lmnt DL ( S ) Λ= = = DL S L S S S T S S I Λ= of = T shall I = η w dduc that S = η T Sη and that th S Ω= T Sη S is skw-symmtric (this is obtaind by drivating th rlation T Sη S), and so th lmnt Λ of th Li algbra can b rwrittn: Λ= = Ω Ω= (4) T S S η with SηS As a conclusion to this subsction, th Li algbra is th linar spac of ηω matrics, whr Ω is a skw-symmtric covariant tnsor of typ (, ). Th skw-symmtric tnsor associatd to th Li brackt [ Λ, Λ ] is ΩηΩ ΩηΩ Th xponntial mapping from th Li algbra to th group dfins a diffomorphism from into radius π in )... Proprtis of th Li Algbra Matrics k Λ Λ S = xp( Λ ) = (5) k! vry matrix blonging to th Li algbra can b writtn 6 (rcall is th opn ball of a a a a b b Λ ( AB, ) = a b b a b b whr A= ( a, a, a ) and B ( b, b, b ) Not th rlation = ar spaclik vctors. ( AB) ( A B) dt Λ, = (6) 94
M. Langlois t al. W thn hav th following proposition about th rducd forms of th matrics: Givn any matrix Λ of such that A B (usual innr product), thr xists an η-orthonormal basis = (,,, ) with rspct to which Λ can b writtn: α α (7) ω ω whr α and ω ar two ral numbrs and whr is a timlik 4-vctor, th thr othr 4-vctors ar spaclik. i If A B=, stting A A= a and B B= b, th thr rducd matrix forms, according to th thr conditions a b >, a b < and b = a rspctivly, shall b a a a a a b b a Proof W shall us following notations: n) Lt (,,, ) b a η-orthonormal basis consisting of 4-vctors, whr is a timlik vctor and whr (,, ) is th basis of th orthogonal complmnt of th straight lin. n) To vry -vctor V = ( v, v, v) a 4-vctor qv = (, v, v, v) can b associatd, which obviously is spac lik. Not that this procss lds us to th dfinition of a linar mapping q. Any 4-vctor V = ( v, v, v, v) = ( v, V) can b writtn V = v + qv. Th innr product of two 4-vctors U and V shall b writtn U,V, and w hav th formula: U,V = uv + UV n) With th aim of mor lgant computations w shall writ C th cross product B A, and abc,, th uclidan norms of th -vctors AB, and C rspctivly. Th following formulas will b usful:, c = B A = a b A B B C = A B B b A n4) With th aim of studying th action of Λ on 4-vctors, w writ Λ ( AB, ) as Λ A +Λ : B a a a a b b Λ A +Λ B = + (9) a b b a b b Lt V ( v, v, v, v ) = b any 4-vctor, w gt th following formulas for th matrix products: (8) 95
M. Langlois t al. ΛA V = A V + v qa B Λ V = q B V () Λ V = A V + q v A + B V To obtain th rducd form of Λ lts start with th study of Λ. Indd its P X = P X whr charactristic polynomial simply factorizs ( ) m m m P X = X A B X A B () P X is th minimal polynomial of th matrix hav th matrix rlation Pm ( Λ ) = O Pm ( X) = ( X α )( X + ω ) m ( α )( ω ) whr α and ω Λ, which mans that w P Λ = Λ I Λ + I = O () ar th two zros of idntity and th zro 4 4-matrics. P X. W wrot I and O for th Not by th way th formulas linking th roots of th polynomial ()): A B = a b = α ω m A B= αω () Th first columns of th matrics Λ α I and Λ + ω I ar th 4-vctors obtaind by computing th product Λ qa rspctivly, using (): Λ qa α = α + A A + B A = a α + C ( ω ) Λ qa + ω = a + + C (4) Th rlation () mans that th columns of th matrix ( ω I ) th ignspac of th matrix ( α I ) Λ + gnrat Λ associatd to th ignvalu α and that th columns of Λ gnrat th ignspac associatd to ω. Lt us writ Π and α Π ths two -dimnsional ignspacs. ω Th ignspac II α associatd to th ignvalu α : Writing Π Λ + ω I and α W =Λ W, ( W, W ) is an η-orthogonal basis of Π. Indd, on th on α hand: W th vctor dfind by th first column of Λ W =Λ W = α W Λ W = α Λ W = α W (5) shows that W blongs to W w gt: W Π and on th othr hand, using (n) to comput α W ( ω ) W ( ω ) = a + + qc = b + qc A C q( a A B C) ( ω ) α ( α ω ) =Λ = + + + = q + a A ba+ ABB = q A+ B Apart from th rlation W, W = w also hav: ( α ω )( ω ) W, W = + a + = N (6) W, W = α α + ω a + ω = N (7) 96
M. Langlois t al. which mans that W is timlik and that W is spaclik. Also not th rlation N = α N. Writing now = W, = W, (, ) dfins an orthonormal N N basis of th spac lik plan Π with: α N Λ = Λ W = W = =α N N N N Λ = Λ W = Λ W = α W = α (8) N N N Th ignspac II ω associatd to th ignvalu ω : Writing V Π th vctor dfind by th first column of ω ( Λ α I ) and V =Λ V, ( V, V ) is an η-orthogonal basis of Π. Morovr ω V and V ar spaclik and Π is th orthogonal complmnt of ω Π. Hr is an outlin α of th computations: V ( a α ) qc (( α ) αω ) = + V =Λ V = q a b A+ B = ωq ωa+ αb Λ V =Λ V = ω V Λ V = ω Λ V = ω V Apart from th rlation V, V = w also hav: V, V = α + ω a α = N Th plan (, ) V, V = ω α + ω a α = N N = ωn Π is obviously spaclik, and writing: ω = V, = V N N constituts an orthonormal basis of ω Λ = ω Π with th rlations: Λ = ω (9) Ths two rlations (9), with th formr rlations (8) linking and, show that th matrix Λ gts th rducd form (7) in th η-orthonormal basis (,,, ). Lt us rcall that is timlik and that th thr othr (,, ) ar spaclik; thy dfin an orthonormal basis of, th orthogonal complmnt of th lin. Th situation whr A and B ar orthogonal: In th cas ( A B= ) w nd to discuss according to th sign of a b sinc th two roots of () ar α = a b and ω = and matrics Λ and Λ ar of rank. Th minimal polynomial () can b simplifid and th rlation () bcoms: ( α ) Λ = Λ Λ = Pm I O 97
M. Langlois t al. ) Assum a > b. Π is dfind by α W and W (6) with ω = : W = a + qc and W = α qa W, W = a α, W, W = a α 4 Π is th krnl of Λ. It is gnratd by V (first column of Λ α I ) and V = qb ( α ) V = a + qc = b + qc V = qb As abov, normalizing th four vctors and writing thm (,,, ) rspctivly, th rducd form of th matrix Λ in this nw basis shall b th first matrix of (8) ) Assum a < b : Noting ω = b a w hav ( ω ) Pm Λ = Λ + I Λ = O Π is timlik sinc it is gnratd by th two vctors blonging to th krnl of Λ : ( ω ) W = a + + qc = b + qc W = qb W, W = ω a, W, W = ω a 4 V Th plan =Λ V : Π is spac lik. It is gnratd th first column ω V of V V V = a + qc =Λ = q a A + B C = q a b A = ω qa V, V = ω a, V, V = ω a 4 Λ and As abov, normalizing th four vctors (th first on bing timlik) and writing thm (,,, ), th rducd form of Λ in this nw basis shall b th scond matrix of (8) ) Assum a = b : whn a = b th minimal polynomial of Λ is simply P ( X) = X that is to say: Noting m Pm Λ =Λ Λ = =, = qa, = qb, = qc a a a (,,, ) form an η-orthonormal basis and (, ) ignspac Π with th rlations: Λ = qa = a O gnrat an 98
M. Langlois t al. Λ = qa ( a q ( B A) ) a a a Λ = a + = + Λ = Λ = Λ qc = q ( B C) = ( b qa) = a a a a W thus obtain th third rducd form in (8). Corollary: Th Li group of spcial and orthochronous Lorntz matrics has four onparamtr subgroups which can b obtaind by intgrating th linar diffrntial quation ds S dt = Λ whr Λ is on of th four rducd forms obtaind abov. t Th solution of this quation is S Λ that is to say coshαt sinhαt sinhαt coshαt S( t) = S cosωt sinωt sinωt cosωt coshαt sinhαt sinhαt coshαt S( t) = S S( t) = S cosωt sinωt sinωt cosωt at + at at at at S4( t) = S at at at 4. Inrtial Particls in Spcial Rlativity Lt O and M b two inrtial particls in th Minkowski spac (,η ). Thir worldlins and O ar two godsic straight lins of gnratd by M th timlik futur orintd unitary 4-vctors t and by V (which dfin th 4- vlocitis of O and M rspctivly). Lt O = ( = t,,, ) b th η-orthonormal basis associatd to O along and lt us rcall that O (,, ) is a basis of th hyprplan of passing through O and orthogonal to th worldlin of O. This hyprplan is th physical spac of O. W not t and τ th propr tims of O and M rspctivly. W also 99
M. Langlois t al. dnot (,,, ) txyz th coordinats of M in th rfrntial fram of O and dm V= ( pqr,, ) th -vlocity of M. Using ths notations, th 4-vlocity V = d τ can b writtn with t h rlations: whr dt V = (, pqr,, ) =Γ +Γ V V, V = Γ V =, V = p + q + r dt Γ= is th Lorntz factor. All ths quantitis ar constants. In ordr to dfin th Lorntz-Poincar transform w may apply th orthonormalization Gram-Schmidt procss to th basis ( V,,, ) obtain an η-orthonormal basis, ( M,,,, ) whr V. W thus = and whr th thr othr vctors gnrat th basis of th physical spac of M. This orthonormalization procss dirctly givs th boost charactrizing th rlation btwn th two inrtial particls: T Γ Γ V L = Γ T Γ V I + V V +Γ In this rsult, V is th column matrix of its componnts and I is th unit matrix of siz. L bing a constant matrix, its associatd matrix in th Li algbra is th zro matrix. All this corrsponds to th classical cas of Spcial Rlativity and can b summarizd as follows: Any constant matrix L dfins a Lorntz transform rlating two inrtial particls. Rmarks:. Th rlation btwn O and M can b charactrizd by an infinity of Lorntz matrics. ach of thm can b dducd from L by a lft or a right multiplication of L with a pur rotation (a Lorntz matrix) R R = A whr A is an orthogonal matrix of siz. A lft and a right multiplication corrspond to a chang of basis in th rst spac of O and of M rspctivly.. Th writing of th boost () can b simplifid by choosing an appropriat basis of ( O, ) (rcall that is th 4-vlocity of O and lt us not qv th 4-vctor associatd to th -vlocity V= ( pqr,, ) of M in (, ) and O ). qv ar two orthogonal vctors in th Lorntz-Poincar?? transform plan. In fact, noting v = p + q + r L = V =Γ +ΓqV L qv =Γ v +Γ qv W can dfin an η-orthonormal basis (, ) () of this timlik plan by
M. Langlois t al. taking = and = qv. W thus obtain: v L =Γ +Γ v L =Γ v +Γ W also know that th two dimnsional orthogonal complmnt is L- invariant. This can b sn by noting that th two 4-vctors W = (, q, p,) and W = (, r,, p) ar orthogonal to and to and that thy ar linar- ly indpndant so that thy form a basis. W can thn construct an orthonormal basis of th spaclik plan which rmains unchangd whn orthonormalization procss is applid: = (, q, p,) p + q pr qr =,,, v p + q v p + q p + q v Ths two vctors ar ignvctors of L associatd to th doubl ignvalu. W thus obtain a nw η-orthonormal basis ( O,,,, ) th transfrt matrix bing th Lorntz matrix Q: p q pr v p + q p + q v Q = q p qr v p + q p + q v r p + q v v Q is a pur rotation matrix ( T Q Q= I ) which only dpnds on th vlocity dirction. Noting ( cos( α) cos ( β), cos( α) sin ( β), sin ( α) ) V = v v v th abov xprssion can also b writtn cos( α) cos( β) sin ( β) cos( β) sin ( α) Q = cos( α) sin ( β) cos( β) sin ( α) sin ( β) sin ( α) cos( α) () To summariz: thr is a basis dducd from through a spac rotation of for which th boost L can b writtn in th following canonical form: Γ vγ vγ Γ = = L Q L Q ()
M. Langlois t al. With rspct to, (,, ) and (,, ) M is th plan of th Lorntz transformation M is th invariant plan of that transformation. 5. Non Inrtial Particls in Spcial Rlativity. Tangnt Boost along a Worldlin Lt us now considr th cas whr O is an inrtial particl and whr M is not. Thn, th wordlin of M is no mor a straight lin and its 4-vlocity V is M a vctor fild along. This lds us to dfin th tangnt boost along M as M bing th boost of th inrtial particl M which coincids with M and th worldin of which is th tangnt straight lin at M. W thus obtain a fild L( τ) τ along whr L is dfind by (). L bing no mor a constant M dl matrix, its associatd matrix in th Li algbra Λ= L is no mor th zro Λ AB, matrix, lt us matrix. Bfor computing th -vctors A and B of th giv som xampls of using this lattr. 5.. Drivation Rul of a Vctor X Dfind by Its Componnts in th Rfrntial Fram of M Lt us considr th two basis = (,, ) and (,,, ) =, bing dfind by th columns of L. Lt X and X b th componnts of th X vctor in and rspctivly ( X = L X ). Lt us drivat that rlation with rspct to t (or with rspct to th propr tim τ of M). Using thn th lft translation, w gt: L dx dx dl dx L = + L X = +Λ. X dt dt dt dt whr th subscripts and corrspond to th basis and rspctivly. Th abov rlation givs th drivativ rul by its -componnts that is th intrinsic vctorial rlation: dx dx = + Λ X () dt dt Lt us now apply that law to th 4-vlocity of M th componnts of which ar (,,, ) in. quation () shows that th first column of ( AB, ) qa (, a, a, a ) = (notation n in paragraph.) of M in. Now, lt W b a 4-vctor dfind by its componnts W ( w, w, w, w) w qw 4-vlocity of M. Noting W th -vctor ( w, w, w ) ration of M with = qa, th -vctor B ( b, b, b ) Λ is th 4-acclration = = + in and lt us rcall that = V is th, and A th 4-accl- A instantanous rotation dfind by its componnts in : = appars to b an dw dw dw = +Λ W = + W A + q wa + B W dw = + W, qa + q ( wa + B W )
M. Langlois t al. dw dw = + WAV, + wa + q B W Changing w into VW, this last quation can also b writtn: dw dw = + WA, V VW, A + q( B W) (4) Not that thr is a minor abus of notation in th last lin: B and W must b undrstood hr as -vctors and no mor as componnts in as in prvious lins. Th trm q( B W) shows that B is an instantanous rotation in th (physical) spac of -vctors. It corrsponds to Thomas rotation. Th matrix Λ thus contains th 4-acclration of M and th Thomas rotation. It thrfor undoubtdly constituts a valuabl tool to dscrib th motion of any physical systm. 5.. xampl of an Uniformly Acclratd Particl In th rfrntial fram of an inrtial obsrvr O, an uniform acclration O of M dos not corrspond to a constant 4-acclration A. In fact, th worldlin of M is not a straigth lin sinc it is not a godsic. At two diffrnt points M M and M, A and A ar not paralll. In th cas of an uniformly acclratd particul, w consquntly only know that th norm of A is a constant a. Morovr, in what follows, w will also considr that, for th inrtial obsrvr O, rmains in a givn plan M (, ). This plan is ncssarily a timlik plan. Th paramtric quation of motion for M and its 4-vlocity dm V = ar thn: whr: ( τ) = ( ( τ), ( τ),,) M t x dx V =Γ (, v,, ), v=, Γ ( v ) = dt ( v ) ( vγ) dγ dv d dv Γ = =Γ v, =Γ (5) Th mr knowldg of V prmits to calcul th tangnt boost L. Insrting V = ( v,,) into quation () w gt: Γ Γv Γv Γ L = Lt us thn calculat its associatd matrix Λ in th Li algbra (6) Using (5) in computing dl T Λ= L = η L η dl dl, th abov quation givs:
M. Langlois t al. a dv a Λ=Γ = dv dv whr Γ = a is th constant dfind abov (whn >, a is th norm of th 4-acclration). Using th drivation rul, w obtain th componnts A of th 4-acclration in ( M, ) Its componnts A in (, ) v A =Λ V =, Γ,, d (7) O ar thn obtaind by a chang of basis A = L A =Γ v d ( v,,,) τλ Calculating L( τ ) = w gt th following conclusions: any uniformly acclratd particl is dfind by a on-paramtr subgroup of th Li group, ( aτ) ( aτ) ( aτ) ( aτ) cosh sinh τλ sinh cosh τ L( τ) = = and th 4-acclration is uniform in th rst fram ( M, ) of M (not again that th basis is dfind by th columns of L). In an uniformly acclratd systm, thr is no Thomas rotation. Lt us now considr two narby particls N and M, N bing at rst with rspct to M and thir coordinats in (, ) X = constant. Lt us calculat don. M bing MN (, X,,) d ON d d d = OM + MN = VM + MN Knowing that X dos not dpnd on τ, th drivation rul givs: Th componnts of don dmn =Λ MN = ( ax,,,) in ( M, ) ar thn don = + ( ax,,,) whr + ax is a vlocity (using c w would gt + ax ). It is important to not that d ON = whr ax c + instad of c is not th 4-vlocity of N and that th propr tim of N is not th sam as th on of M. In fact, th norm V N of th 4-vlocity of N, dfind with its propr tim s bing, w obtain th following rlation btwn τ and s: 4
M. Langlois t al. don don VN = = = (,,,) = ds ds ds + ax This shows that in th cas of a non inrtial motion of M, it is impossibl to synchroniz th clocks in th rst fram of M. Lt us add that N has not th sam acclration as M. In fact, knowing that dvn dvm VN = V and consquntly that M = = AM, w gt A N = dvn dvn d dvn A ds = τ ds = + ax = + ax 5.. Tangnt Boost of a Worldlin and Its Associatd Matrix in th Li Algbra in Spcial Rlativity In th rfrntial fram of O, th paramtric quations of th worldlin ar M dfind by cartsian coordinats whr th paramtr is th propr tim τ of M: (,,, ) τ M τ = t τ x τ y τ z τ Noting V = ( x, y, z ) and A ( x, y, z ) = th -vlocity and th -acclration in th rfrnc fram of O (with its proprtim t) th 4-vlocity and th 4-acclration (first and scond drivativ of coordinats with rspct to t) ar: Th tangnt boost () is: L ( τ ) ( x y z ) ( V) ( V ) V =Γ,,, =Γ + ; Γ = dγ A = ( + V) +Γ qa dγ Γ V = =Γ 4 V A Γ Γx Γy Γz Γ ( x ) Γ xy Γ xz Γ x + Γ+ Γ+ Γ+ Γ xy Γ ( y ) Γ yz Γ y + Γ+ Γ+ Γ+ Γ xz Γ yz Γ ( z ) Γ z + Γ+ Γ+ Γ+ = and its associatd matrix in th Li algbra dl Λ= L is 4 4 4 V A x Γ V A y Γ V A z Γ + x Γ + y Γ + z Γ Γ+ Γ+ Γ+ 4 ( V A) x Γ Γ ( yx xy ) Γ ( zx xz ) + x Γ Λ= Γ+ Γ+ Γ+ 4 ( V A) y Γ Γ ( xy yx ) Γ ( zy yz ) + y Γ Γ+ Γ+ Γ+ 4 ( V A) z Γ Γ ( xz zx ) Γ ( yz zy ) + z Γ Γ+ Γ+ Γ+ M (8) 5
M. Langlois t al. To summariz: using notations (6) w s that Λ givs th complt dyna- mics of M. In Λ ( AB, ) : th -vctor A is th acclration of M in its rst fram (,,,, ) Γ A= V A V +Γ A Γ+ 4 M : (9) th -vctor B givs th instantanous Thomas rotation by its componnts in (,,,, ) M : Γ B=Ω T = V A Γ+ 5... Writing th Tangnt Boost and Its Associatd Matrix in th O,. A first Insight on Li Algbra in a Rotating Fram Thomas Rotation Th rotating basis is dfind in th rmark () of paragraph 4 but, in th () prsnt cas, th rotation matrix Q now dpnds on th propr tim of M. Th tangnt boost L in has th rmarkabl form (). Our aim is to calculat th componnts of th matrix of th Li algbra in th rotating fram ( O, ) in two ways:. Using th dfinition of Λ in th moving rfrntial fram T dl T dl Λ = Q Λ Q= Q L Q= L Q Q and applying th drivation rul to th tangnt boost whr L in T dl T d T dl Q Q Q Q L Q Q L L dq Q d τ = = +Ω Ω T Ω= is th antisymmtric matrix which dfins th instantan- ous rotation of ( O, ). Insrting this rsult in th prvious quation givs th matrix of th Li algbra of th boost ( O, ) W thus obtain Ω and Λ dq dl Λ = L + L Ω L Ω L as sn by th rotating obsrvr Γcos( α) β Γα Γ cos( α) β Γ sin ( α) β Γα Γsin ( α) β T Ω= Q = Λ = v Γ cos( α) β Γ v vγ cos α β vγ α v Γ cos α β v Γ α Γ v Γ+ Γ+ vγ cos( α) β Γ+ v Γ α vγ α Γ+ 6
M. Langlois t al. whr α, β and v ar th drivativs with rspct to t of th thr d d paramtr dfining V (lt us rcall that = Γ dt ).. Using (9) and () which giv th -vctors A and B from V and w gt: ( α) ( β) v vα ( α) ( β) v ( α) β ( β) ( α) ( β) α ( α) ( β) ( α) β ( β) sin ( α) v + vα cos( α) cos cos sin cos cos sin A = cos sin v v sin sin + vcos cos T V = QV = v (,,) (, cos ( α) β, α ) T A = Q A = v v v Using Γ v =Γ, quation (9) givs: 4 Γ A= V A V +Γ A =Γ Γv v v Γ+ and quation () givs th Thomas rotation B = Ω : T ( ) (, cos ( α) β, α ) Γ Γ Ω T = V A = v v Γ+ Γ+ (, α, cos ( α) β ) A. To conclud: from th ( O, ) obsrvr point of viw, th L boost writtn in M with rspct to ( O, ). th rotating basis dfins th rst fram (, ) From th ( O, ) point of viw, th two dimnsional spac of th Lorntz- Poincar transform, as wll as its invariant spac ar not moving. Th matrix Q dfins th rotation of th rst fram of M with rspct to ( O, ). Ths calculations show that w must clarly distinguish btwn th instantanous rotation of ( O, ) (which is dfind from th antisymmtric T dq matrix Ω = Q ) and th instantanous Thomas rotation. d τ In ordr to gt a bttr insight on Thomas rotation, lt us considr th infinitsimal Lorntz matrix rlating M ( τ ) to M ( τ + ) : dl L( τ + ) L( τ) I + L ( τ) L( τ) ( I +Λ( τ) ) A lft-multiplication by L of this rsult givs th Lorntz matrix Li = L ( τ) L( τ + ) = I +Λ ( τ) + o = I + ( Λ A +Λ B) + o = I +Λ I +Λ + o A At first ordr, Li thus appars to b th product of an infinitsimal boost Bi = I +Λ d A τ with an infinitsimal pur rotation (Thomas rotation) Ri = I +Λ d B τ : Li = Bi Ri + o( ) W will s latr that th Thomas rotation is a rotation of th rst fram of M with rspct to th rfrntial fram ( M, ) which is dfind by th tangnt boost. B 7
M. Langlois t al. 5... Application to a Particl in Circular Motion at Constant Vlocity O,,,, of O, th paramtric quations of With rspct to th fram ( t r θ z) th particul worldlin ar thos of a circular hlix with axis ( O, t ) cylindrical coordinats ( tr,, θ, z) t M( t) = ( tr,, ωt,). Using whr R and ω ar constant and whr t is a function of τ.th 4-vlocity is: dm V =Γ =Γ dt (,, ω,) dt Γ= = R ω Noting that th Lorntz factor Γ is constant, th 4-acclration is: dv A= = DVV = RΓ ω (,,,) whr D V is th covariant drivativ in th dirction of V xprssd in cylindrical coordinat. In ordr to calculat th tangnt boost w hav to xprss V in th η-orthonormal systm (,,,, ) M obtaind by applying th Gram-Schmidt orthonormalisation procss to th natural basis ( t, r, θ, z) calculations ar vry simpl sinc th basis = ( Γ,, ΓRω,) = and starting with th 4-vctor. In th prsnt cas, t is alrady η-orthogonal. W gt V. Th tangnt boost is thn dfind by using (): Γ RΓω B = RΓω Γ Lt us rcall that th B -columns giv th rfrntial fram (,,, ) thir componnts in (). Lt us also not that (,, ) Poincar-Lorntz transform, (,, ) = = V of M, and that th 4-vctors α ar dfind from M is th plan of th M bing th invariant orthogonal supplmntary plan of th transformation. db Th matrix of th Li algbra Λ = B = B DVB is RΓ ω RΓ ω ( Γ ) Γω Λ = () ( Γ ) Γω It dirctly givs th -acclration and th instantanous Thomas rotation (which both ar in th physical spac of M). Lt us not that it is also possibl to obtain th -vctors A and B of ( AB, ) (, Rω,) A Rω,, V = and Λ from quations (9) and (): using = w in fact obtain th -acclration and th instantanous Thomas rotation in ( is dfind by th column vctors of L): 8
M. Langlois t al. A= Γ A = Γ Rω,, Γ B=Ω Th = V A = Γ Γ ω Γ+ (,, ) 6. Discussion In ordr to undrstand th maning of Thomas rotation, lt us considr a gyroscop and lt us rcall th dfinition of a gyroscopic torqu along a worldlin as givn in [] and in [] [4]: A gyroscopic torqu along a worldlin th 4-vlocity and th propr tim of which ar V and τ rspctivly is a 4-vctor G dfind along, orthogonal to V and such that its drivativ with rspct to τ is proportional to V, that is to say: dg GV, = and = kv () dv Ths rlations prmit to calculat k. Noting A th 4-acclration, w in fact gt: d GV, = kvv, + GA, = k = GA, Th proportionality condition implis that th 4-vctor G (which blongs to th physical spac of M along th worldlin ) rotats in that spac. In fact, lt us writ th diffrntial quation () with rspct to th componnts of G in ( M, ). Noting G = (, G, G, G) in ( M, ), th componnts of G in th inrtial fram ar thn dfind by: ( ω,,, ) G = B G = ΓR G G ΓG G In th inrtial rfrntial fram, quation () thus bcoms: d ( B G ) (,, R ω,) = kv V = Γ Γ Using th covariant drivativ in cylindrical coordinats and noting drivativs with rspct to τ by accntuatd charactrs w gt: d ( B G ) (,,, ) ( B G ) ω ω ( ω ) = D = ΓR G Γ G + G Γ G + G G V Idntifying this rsult with k = ( kγ,, kγrω,) thr diffrntial quations (not that V givs k Rω G = ): Γ R ω G = Γ ωg G = Γ ωg G = Taking initial conditions G ( τ = ) = (,,, ) diffrntial quations ar = and th A A A, th solutions of ths 9
M. Langlois t al. ( τ ) = (, cos( Γ ωτ ) + sin ( Γ ωτ ), cos( Γ ωτ ) sin ( Γ ωτ ), ) G A A A A A Figur shows th rotation of a gyroscop initially orintd following th x π G =,,, ), in ( M, ) whn τ varis from to that is to Γω axis ( say whn M gos a 8 dgr turn. It shows that in that cas, th gyroscop indicats a half turn plus a rotation which corrsponds to th Thomas rotation (in th clockwis dirction). Th gyroscop rotation in th plan ( M,, ) is also shown whn M gos a complt rotation (6-dgr) in Figur. In that cas, th gyroscop indicats a complt turn plus a part. For th sak of clarity, w only show this supplmntary part. Figur. Th gyroscop rotation in th plan (,, ) turn. Numrical valus ar R = and 5 ω =. M whn M gos a 8 dgr Figur. Gyroscop rotation in th plan (,, ) M whn M gos a complt rotation (6-dgr). In that cas, th gyroscop indicats a complt turn plus a part. For th sak of clarity, w only show this supplmntary part. W usd hr for R and ω th 5 valus R =, ω =.
M. Langlois t al. It is also possibl to highlight th Thomas rotation by applying th drivation rul () to M, ). Noting G (which is dfind by its componnts in (, ); ( G, G, G) ; (, ); (,, ω) G = G G = Ω = Ω Ω = Γ Γ Th Th and using () in that moving fram: w gt Using thn w obtain: dg dg DVG = = +Λ G dt dt dg dg = + AG, V VG, A+, Th G dt dt dg = kv; AG, = k; VG, = ; dt dg + (, Ω Th G) = dt ( Ω ) Th lft hand sid of this quation is th Frmi-Walkr drivativ of G in dg dg dg th V dirction. Using = =, this last quation bcoms dt dg Th = Ω G () Consquntly, th gyroscop rotats with rspct to ( M, ) in th opposit dirction to th instantanous Thomas rotation. ( M, ) taking again its initial orintation aftr a complt priod, this gap shows that th gyroscops also rotat with rspct to th inrtial rfrntial fram ( O, ) It can b notd that th solution of () (with th initial condition G = (, G ) also is th Frmi-Walkr paralll transport of G along [] [4] [5]. 7. Conclusion: Langvin s Twins and Thomas Prcssion Th main rsults of vry dynamical systm ar containd in th tangnt boost L (which givs its 4-vlocity V and th basis vctors of its rst fram ( M, ) ), and its associatd matrix of th Li algbra Λ which givs its acclration and th instantanous Thomas rotation. Th ag of th lctron with rspct to th atom nuclus is thn obtaind by intgrating L, ovr on priod T. In th cas of a uniform circular motion its valu is Γ T Th gyroscop rotation Ψ in th physical spac ( M,, ) can b obtaind by intgrating ovr on priod btwn t =Γ τ = and t =Γ τ = π ω. W gt Ψ= π( Γ )
M. Langlois t al. Figur. Illustration of Langvin s twins in th cas of an lctron rotating on a circular orbit around th atom nuclus. Th twins ar dnotd O and M rspctivly. Th straight lin paralll to th axis of th cylindr is th worldlin of O ; th hlix is that of M. W thus s that in th cas of Langvin s twins, (hr, in th cas of a uniform circular motion), whn th twin who mad a journy into spac rturns hom h is not only youngr than th twin who stayd on arth but h is also disorintatd with rspct to th trrstrial fram bcaus his gyroscop has turnd with rspct to arth rfrntial fram. This ffct is illustratd in Figur in th cas of an lctron rotating on a circular orbit around th atom nuclus. Rfrncs [] Langlois, M., Myr, M. and Vigourux, J.-M. (6) Spcial Rlativity. Using Tangnt Boost along a Worldlin and Its Associatd Matrix in th Li Algbra of th Lorntz Group. Applications, arxiv, 4.754v, 8 Spt 6. [] Straumann, N. (4) Gnral Rlativity with Applications to Astrophysics. Springr, 5-54. [] Misnr, C.W., Thorn, K.S. and Whlr, J. (97) Gravitation. W. H. Frman and Company, p. 7. [4] Rydr, L. (9) Introduction to Gnral Rlativity. Cambridg, 7-8. https://doi.org/.7/cbo978589 [5] Gourgoulhon,. () Spcial Rlativity in Gnral Frams. From Particls to Astrophysics, Springr, Brlin. (Rlativité rstrint: Ds particuls?? l'astrophysiqu, DP Scincs, CNRS ditions ())
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