第5章ローレンツ変換と回転 - sp.u-tokai.ac.jp

1 1/10 29 3 24 1 37 I. 3 2222, 11ctxxctccctxcc--==-- vvvv( ) 2222, 111ct xx tcctxcc-- ==-- vvvv( )2222, 211ctxx tcctxcc + +==-- vvvv( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , ct x 22222222111111ctctccccctxxccUx - == -- --- vvvvvv( ) ( ) ( ) U ( ) U 2 2222, 111 CcScc==--vvv( ) C SUS C- = - ( ) ( ) 22C S- 221C S- =( )x[m]v0 vt [m]v0 t x [m]x[m]vt [m]t t t x [m] 12 v2/10 29 3 24 1 37 ( ) 2222si1snco1 SCqq+=- =( ) + - ()2 1i i= - ( )( )

2 22222222211 CCC SSSiSiC- = += + = +=-( ) s ncosiiSCqq= = ( ) 221CS- = ( ) ( ) cos sinieiqqq=+( ) ieq- cos, sin22iiiie ee eiqqqqqq--+-==( ) ( ) ( ) 221C Sqq- = C S Cq Sq ( )( )2= 2e eC Ce eS Sqqqqqq-- += - = ( ) ( ) 221C Sqq- = ( ) ( ) 221C Sqq- = , 22e ee eCSqqqqqq--+-== 221C Sqq- = ( ) ( ) 222cos , sinx ry rx y rqq== + = ( )3/10 29 3 24 1 37 Cq Sq 222, x rCy rSx y rqq== - = ( ) q 22rq 22rq = 22rq ,C Sq q Hyperbolic function ()()coshypabolic sinhypabolic CineSeqqqq==h cosh sin ( ) 22cosh sinh 1qq-=( ) ( ) ( ) ()()2222cos, sin cos sin 122cosh, sinh cosh sinh 122iiiie ee eie ee eqqqqqqqqqqqqqqqq----+-==+=+-==-=( ) ()()cosh cos , sinhsinii iqqqq== -( ) ( ) iq tanq tanhq sinhsintanhtancoshcosqqqqqq = = ( ) ( ) coshsinhsinh coshUqqqq- = - ( ) ( ) 4/10 29 3 24 1 37 22221cosh, sinh11cccqq==--vvv( ) ( ) sinhtanhcoshcqqq == v( ) ( ) ctctUxx = ctctUxx = ( )coshsinhsinh coshUqqqq- = - ( )

3 V q tanhcq=v( ) ( ) 22222222221coshcosh sinh11sinhcoshsinh111ctctccctcccx tcxcccxxxxtctqqqqqq - = ==- - - --= ===- - -- vvvvvvvvv( ) U ()()21tanh , cosh1cbq gqb= ===-v( ) ( ) ( )( )()2222ctxct x - =- = ( ) II. 0m E (): , ,x y zp p p p ()22 22 40 :E cm c c-= p( )5/10 29 3 24 1 37 p (): , ,x y zv v v v 0m=vp( ) ( ) 2Ec=vp( ) ( ) ( ) 2222222022 = + +1xyzm cEcbb == - vv v v v( ) 2b 22 2222ccEb = = pv( ) 3 (): , ,x y zv v v v cc E=vp( ) ( ) 2Ec=vp( ) 20 v ( ) 2002EE m cmc = pv v( ) ( ) 2Ec=vp ( ) 02222220211m cEcmccc===--pvvvvv( ) 22020222022222211= + +1xyzmmmm cEcmcccc Emc == = - -= == - vvvvvv v v vvpp( ) II.

4 ( ) ( ) 6/10 29 3 24 1 37 III. (),E c p ( ) 22 22 40E cm c -=p( ) x ()(): ,0,0 , : ,0,0pp pp( ) 22 222 2E c p E c p -= -( ) ( ) ( )( )2222ct xct x - =- ct Ex cp ( ) ( ) ()20,0m c v (),E cp ( ) 2222, 11ctxxctccctxcc--==-- vvvv( ) ( ) , ctEctExcpxcp ( ) 2222, 11 EcpcpEccEcpcc--==-- vvvv( ) v 20, 0, Em c pEE pp == = == v v( ) ( ) v ()20,0m c( )

5 , with cpE cpc E=v7/10 29 3 24 1 37 202222200022222222111111 Ecpm ccEEcccpEm cm cmcccpppcccc - = = -- ---- = == = ---- vvvvvvvvvvv( ) 2002222, 11m cmEpcc-==--vvv( ) cp E cpEc= -v( ) ( ) ( ) cpE c=v 2 2 1 v v v ( ) 20, , 0EE ppEm c p== = == v v( ) 20222211 EcpE cpccEm ccc--= =-- vvvv( )2222011cpEcp Ecccpcp Eccc--= = =-- vvvvv( ) ()20,0m c(),E cp vvv8/10 29 3 24 1 37 ( ) cpE c=v ( ) 2222022222222211111111 EEEE cpccccpEccm cEccccc --- - ===== -----vvvvvvvvvv( ) 20221m cEc=-v( ) 02220222211m cmpcEccc-===-vv vvv( ) ( ) 2222, 11 EEppEc cc cpccc-- ==--vvvv( ) ( ), , ,Ect xpc ( ) IV.

6 1v 2v v 12= +v v v( ) v q tanhcq=v( ) ( ) U v ()Uq ( )coshsinhsinh coshUqqqqq- = - ( ) ()Uq ( ) 2v1v???=v9/10 29 3 24 1 37 ( ) 1111tanhctctUxxcqq = = vv( ) ( ) 121222tanhctctUxxcqq = = vvvv( ) ( ) ( ) ( ) ( ) ( ) 121221ctctctUU Uxxxqq q == vvv( ) 2v ()()21U Uq q ()()21U Uq q ( ) ( )()()22112122111212122112211212coshsinhc oshsinhsinh coshsinh coshcosh cosh sinh sinhsinh cosh sinh cosh sinh cosh sinh coshcosh cosh sinh sinhU Uqqqqq qqqqqqqq qqqqqqqqqqqq q-- = -- +-+ = -++ ()()()()12121212coshsinh sinhcoshq qq qq qq q+-+ = -++ ( ) ()121221sinhsinh cosh sinh coshq qqqqq+ =+ ()

7 1212coshcosh coshq qqq+ =12sinh sinhq q+ ()()()() 1212121212coshsinhsinhcoshctctxxq qq qq qq q +-+ = -++ vv( ) v ( ) ()12tanhcq q+ =v( ) ()12tanhq q+ 12tanh , tanhqq ()121212tanh tanhtanh1 tanh tanhqqq qqq++ =+( ) 1212tanh, tanhccqq==vv( )10/10 29 3 24 1 37 ()121221sinhsinh cosh sinh coshq qqqqq+ =+ ()1212coshcosh coshq qqq+ =12sinh sinhq q+ ()12tanhq q+ ( ) ( ) ( ) 121 221c+=+v vvvv( ) ( ) 12,cc v v ( ) 2c=v c=v( ) ( ) v A B 1v 2v v ( ) A B B()12+= -vv v A ()12- +v v 21 2211c-+=+vvvvv BA A B 1v 2v A B 121 221c+=+v vvvvBA 121 221c+=+v vvvv 1vB2-vA v vAB1vAB2-vB2vB1-vA