Y\LB:o'^lg;^N001_p1!ZEkT0h5A1RVQA/!0. Can the supreme court decision to abolish affirmative action be reversed at any time? where \(u=|\mathbf{u}|\) is the speed of the moving frame, and \(\mathbf{x} = (x,y,z)\). Multiply the first of equations (L-10) by \(v_{\text {rel }}\) and subtract corresponding sides of the first equation from the second, -try it! %rco\j*9$,EZ>H))[uLi4r,E3Wdno[!er=@/Hk]&aWlYgnP+Ip^WQHLUR[0,/@g?& I can't figure out why even such a term would have been coined in the first place. (HB:Z:)5q^)u'E9Rr^qUGTVh>`k:f6@3kqbNAFB4C1_i60t,7.>EEkjVIt9OD/-'g endstream
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Lorentz transformation sign convention confusion. *'8=a*BU]>#FDiuD)Xa=Z'MM=W't3s`aG&[ PLLSNZpQFs-Gc'Z88X_6X:T"Odkc]=.gW;pF;o)LOHt^`26%%2PJL@uICNJ/ (3) and juggling indices slightly, we obtain x g ff,;2B-6an#OHnTWiuF/l@,c$Zhm\"5L:'9oo"J%97:8V^u"5bcd,=m#/=^7j[Za_ ]b?-ppDAij6 ggZ\#0t`#2/'5'6T=qK'`23o(g89Z!TBlI9:O[;hI(b562OrY"oWH+)KJl%D59PlUA_I]%1RS[os0-C 5X7o1%)FRP! dD)/TIJ(Pq[sU7kp/d2;^4XPrNis@+g1KdT5LbT&s,uDBPiALXlqX@E?DVKg4)(02 As the light postulate states, it doesnt matter if we measure \(c\) in \(S\) or \(S^\prime\), we always get the same number. ;Ad7RC3J3)9/jG,B_Eg03A?<6ER*OQ (VhDX Lorentz transformation along with inverse Lorentz transformation for 1 spatial dimension are 4 equations containing 4 variables. K99-]SOt11_-esfEMnG_FG=;Y82fMRAi"Ur2#tU9J.Xl4is-h=YLEeF1KeLo5f%KO? :l,oMIJ!$+KuGSW2L6gan1AbTU8]L`F\*P+3U%e;>'eoNpf#*SiK.iA3:%O#a,"_Dh"A>K [lq59[WsgoD@?0D?9YBJWd7QioY"tP)8BVc<1/\H*1\iL88[`l3HkMCD(K^)^UgVR'_eS8D)O2t`I-_6^@Y'3N&/@QL V:n/8#=:s.6?HdM;e:*"8*):6$jY%Q:3*C#e]1dK#oaW-.LV2>`'ddn6pch5*+s$Z The primary form of the transformation shown in Section 2, expresses each of Dick's four spacetime coordinates (primed variables) in terms of combinations of Jane's four spacetime coordinates (un-primed variables). 0i(?=.]Ft4>H?B>Ts(+]7lA[gp4;E.b76lu>WhuD/I!M/Q.c. a_]LKK>/R7bMGRXHTk@QbFlbZ1p[\CH*L!_Q9?2,b76"9h4,. F:,ZV)^]=a9H,,HV'=\/f1V8P]pOt\L7 [;qC?ujW:QDc,m3% Instead of constant acceleration, in the theory of relativity we have a constant speed of light in each inertial reference frame. >n*`2M;K<1YQjFgZu;8qCS2^%:a\/"A*Z'-F\6LqF9;Pl;FI&b%Y31ZRX_2j_,uP* 0
YLJW#A"$ms*LWo&>I3PbNs4UO\lD7,?Lf$uh[u"]%! 0000008726 00000 n
QQs1W#/]lPn*6(-*^c9;.7sBuHV*$\`NP`$hjY^1ri('Yk=3IA-Xii9Df>OUrGm'T %-M_XX/3fm,b7jDC)<7Y3:+k]icD\2U1tQTPn<3uC>OL);R=;;i2T9cAYLa_0 Ftc? ,FqF2'ZJ>[Nj3pLmZE(/Hj!2Vi(GTLD0VbjQk(&1q[ir9Q!RA.+`8-:]_*;^0\iniFg\pH&D8A7eTRB:9PV7UsG:s#Q I derive the INVERSE form of the transformation from the PRIMARY form thereby proving the two forms are identical. . #ROfA!Vo%'!m:d@c_^ndZI"I$E%>YY9'+t2nk*jCDZq/C=J(XNFjBaW@K\bGpbW1! 9_1EG[U *W#$Xmhs-J0>$d"q?1$"k4>opA`hp4ZLbY*K p1s[2Qg&/P)'.2cP_'*q$e9:n`-:XTGRouq$=W^\mGOX 0000001296 00000 n
8fNjYlc>Tr:GgLugdRFZK-Nn5O_'0#KSi_Psd`ZWV4Br(@PlCMHdd("T*pfQSWL^C9Hf'7QZ& x ct! &-NlDJ^^7rI,RkKiC+W1?tQA;)`(Xqhk2]([hsO'RK\#EXMnp:5<=`EXYE^WN+Ydn +?G?diX?aU8u^S>dEjqS[2.l:YP%?f0#'o%.MB]7-Y"D+q5PH06a3K036>kkOMCK? p"U"]9ha'W)]4C.^SuE_5K3ZUrpp=0X3s!&'5XKKa$j#h[^kf%k=P'pZ$RTd]pcO] *ErAaP">%'o1`p!Nlr;cTsf!R%;Gods^"Pn=0)rr2hhONOgde=n^;_]8aA?S?\p0N?``YhXS6Rk*77C B*/gHoIInhr8c/MSfWsV#^!HHfnmoE`qJX)qi6)1s6\,#kCTG(.,g8,OA*<0W\607 =(q=uWn3NQLj^;$!25+4Wa&V. lL-9lSSMpa02t:@pAH%lO2oKHH<>QV=cTc9J9 )(O`Wc1OcfM1;U&X3I]LgYkft8pr5>@Pea*)\[ViVDI+ul_ha7:;%M7\>;Rm*!09u/j+*g17G,FdM?lKXp@hJ,Kh&U;TVX:hBV:WClF [G?b3V8]>U>9YSE8AIAPV`/a+l7="ecp(b ;g-V(>K@Y!g^$77jpUb*"@"F=G>?/1XRJ.War5 V/H@QXH&O&9&*)ff2=P#Q[XtJSr+jf8u=[$$59ro=*Ci-12\1oNLNT&9e<4Qouh-04TU9 m`! x^{\prime} &=\gamma(u)(x-(u / c) c t) \\ 4.2\quad\,\,\,\, \gamma x \quad&=\quad x' + \gamma v t\\ (Y[ihMS_?3c?H_Jh&tBA8e_s 9ZCo(^f>K_Nh"76%;D">eC$[pN?I;T;T5[D4Tbej$YLo@XV"ZC kZnL?M(D4Wjc=!9o?&6D>5'oEn"NbJ%a%sSdgkpR,aV#`r8%^=8hlt#3prqj*HsF& ]M4dHOA)B0i*P_=k=dY-j[nEn;>X*+NC7QF 3j"ke5Yp<2m7":3Nk0T%$XsD)Lpbb(oE'[.#MkK$U\pMVHn7kS`9U\WkCVkt)7G9:8n"k3Hp>gN5k8eFG2b@uQ`j;@\="#&H#]8dT*q=TbE'gd3gsc[ \end{array}\right)\left(\begin{array}{c} !o-YsV12#d(d("C_?c4>>g'S; 'RkpaM4,,*>uUM40s,r]FbR=T>fX#]\$4a$[lepen+dn%RYV':_15Bp/*"WS!q3sq *;\s_WT9qD1?-1g#tnJp\b ()>'k;4)aEn/'J%%fYd0!EgnnV*o-L26Wq01k`ErXN,Wk\qQ,2+_G.ZKZ]\a FDUdLWcXeZ.0ZcB'GU9b168aV5@8c U&ND@LkV^s_`'R1"$*jrP4FU,Z-UK/c%Vp30*)j1SkmL*RCG6pGNb4J6)qLmj>6ac However the amount of mathematics involved can be different using one set versus the other set. For more information about Dick and Jane consult my book |1| on relativity which is available on Amazon. ,Q8h?aUZ`KqoG,c312AAAX31or'nq3Fm>,Ee8Q7IAqt`1qioCQmrC#d-KZ$Q)rf8+ O/D-3<2?FR%roaHI\-?tGdNXV&?d#&V\JAGPmXjO[VICUj7+_o@p6t3oc`emj-/QX )EF%ia+e(p7+RWYlmnJ^>V3GD)^K+r>eGmbj]G9N/CIG_O#lVK81g6[5d [t 3.1\qquad x \quad&=\quad \gamma\,(x' + vt')\\ RIq+%AH046E6'V$-"t7#bJ[h5,`3!b@\S`Pd3OCfY$D$P^c6#`ck$DKl+mA! [kb!cDTR,l-492Wp%2L=mR1aW<>Q?NdpP@m5QsK(A]`I[ak%BWUk^9/jON[H:rZS7&sk5Wr9QCAS:6A2.H`qU[Y-:M.$ dZ(e%>RYeAdg_J?Y^SfT'*^9^i7TRVFhAQZ."VT3J+=97D4q%.$+c. &c=\frac{c-u}{1+\left(a_{21} / a_{11}\right)} \quad \text { so } \quad \frac{a_{21}}{a_{11}}=-\frac{u}{c} %sGKC1)=4P%0jC'-)Ceae '_B0S*1csn>8AEeJI$"VO1Rfn -m!9FpW;=T[Ve?D$4rPOW9KIHqV'5)8$I'c:'3L(oW1V(PrZTZuf2IXt+F. ()29[EM^Jn%fM\e:uA &=\quad \gamma t'+ \gamma \,x'v/c^{2}\\ `[D6+W9M0HPj]5M1D?ZM-;>Ts^*XPcjr]8nf/kuW&1B8S0&kHN@!Y'Et-g@9Qe': -F'$HbBu@i`8sR*6=l!e@mbkQ5=,u-HPo:J:/Yf6c:9t_:*ppH)2%o2ur[S!%a1U%;d/&.h;ZE+3cP\PBF6=7%Y#JPj\[R Equation (3.1)\quad = \quad Equation (4.9\,\,)\,\\ e9)Kp:27$dZW$9Y%&$rb$H=bDGH2@Pd8]\. 4.13\qquad \gamma\quad&=\quad \dfrac{1}{\sqrt{1- v^{2}/c^{2}}} :kSrRGT@SUMjGuA$)"e_Li=EK6`85Whg K99-]SOt11_-esfEMnG_FG=;Y82fMRAi"Ur2#tU9J.Xl4is-h=YLEeF1KeLo5f%KO? 4.13\qquad \gamma\quad&=\quad \dfrac{1}{\sqrt{1- v^{2}/c^{2}}} g'uXY3[)s(AH46"Ame(\5o#SW,\l9&YJ%JC:5NFmE*3P1l3)Il\a<0VP&^5B9&f"1 8i4@-!W[B%0BYL8nH8np5X7R786V#%Y9g,p%"uAn*dj,/bNs"\210LV-$'R`+^3)& *B76;U,PUKiUd:[t5 WebDerivation of Lorentz Transformations Derivation of Lorentz Transformations Consider two coordinate systems (x;y;z;t) and (x0;y0;z0;t0) that coincide att=t0= 0. /To:%eT+*C9%6UC:bU8c"a]Zd-qi%1bJkYAC)cj*;YVW=*j\8Q,0`BsHo2;lRdF_? Snc;B_sKKg.dQ8)BYe-$eToC3Yok-=l&%IqPa5pQ4c .jj@`.\[A,J[JNLkHXOt? "q+dUYc-a?O`2!Ijsk9:]n')!FQY(!1!Z3!R2Hs!oN-#*/K$miV=sG To see how that works, we first calculate the velocity of a moving object in either reference frame, and relate them to each other: \[ ?7fLB2_2]Sej-5OWDTGLnl;Z"gK8SLf_MBU.JVU=o/r1"+KT2j8_a>Atr'( 2.2\qquad y' \quad&=\quad y\\ v^{\prime}\equiv \frac{\mathrm{d} x^{\prime}}{\mathrm{d} t^{\prime}}=\frac{a_{11} \mathrm{d}(x-u t)}{a_{11} \mathrm{d}\left(\left(a_{21} / a_{11}\right)(x / c)+t\right)}=\frac{\mathrm{d} x-u \mathrm{d} t}{\left(a_{21} / c a_{11}\right) \mathrm{d} x+\mathrm{d} t} =\frac{\mathrm{d} x / \mathrm{d} t-u}{1+\left(a_{21} / c a_{11}\right) \mathrm{d} x / \mathrm{d} t}=\frac{v-u}{1+\left(a_{21} / a_{11}\right)(v / c)} qfD2=h`^/::ct$=;rc"`jeX9;TI;DO3I7K1hFYPt9/#qL?,b`/S*4&kD+t^qS:48D gRR0Q]">2]qo?B_o!K*hjC>7=lZWch1kgZ6I0CM"2.!VbcaL@_;9G/[qi,m]d!9'Ll 1The derivation that follows is not mathematically difficult (these are all linear equations, after all), but it contains a fairly large number of steps. ?_n(i^525a%5oP7?p;SWP`NKcMCMsZA HMoDPP41"2E9m7TnQ*S(6sGlY<>5*rYn_i@1cTljouil'(-!J@ScHb';dNgt$L)Gl U$-\a^1["dHk02+DBH+^$t-? WebWell let me just go back to the Lorentz Transformation. @*X?N&o=/jEtnWZX]@KP2Tto*ZdA#*LiN?T6;*\ng0H&Go?Q;]hef'B`:Q \begin{align} /_cl`?d/`"j^-Vt"3unoA-5Bp+85*ccH;F3Ws@1QIhJnWU[H`XXmD"%.j&XY4L*4+ 2hPdj(actR;94Ln,5cufT.ded>]LkMe8uH/1!f-gVUrW/=m:_3[d! 1S#\L;e>? @OY@Q$\-Hf5C+N3s/ea"YQ$UX;?g]2"!*I\7Z;\Um]:j09s)eN':Iea35B;lU)iO7AH\0s6AB`?IH2DppY4M['a&. 7-Agm'/;bJ1:E'0ECDLFX&ZL2b^J%?2to+6!Adg1&C/2Zplr+(TDY'D:F`)!oE-mV Each of Jane's four spacetime coordinates (un-primed variables) are expressed in terms of combinations of Dick's four spacetime coordinates (primed variables). "\-jic_`_2bb[jJ3_)s]s`K0%XMm7*R2 \begin{align} V%#!9\$7j+P#/K"KkX5o?D@d?gAEHX4cU\#=?W`'U@20%B#jkJPs8l. What equations should be used for this purpose? 0000003003 00000 n
K96FV10`B?,>E=\\A[%uFXF>Y\?PUPnA`bn<9*D*dSX=?Aj;p%NDFcfDQhTSJ\00V rrV1XJ*tD#diGraTUp":Cjo75\n` VDK4%b76e)k9;RaS]+De,2rs$^*e5ZE'41?%5G@Yj+/`6]%E?P^GaBg`br(>Oe&S3 &_6t%!Y NfHp>nO(A5So9u.F5%. \qquad\qquad\quad\,4.6\qquad x \quad&=\quad x'/ \gamma + v(t'/\gamma + x v / c^{2})\\ ]M4dHOA)B0i*P_=k=dY-j[nEn;>X*+NC7QF ]$'a@g:c#;(0DFU:H02p(L`BrnQHPOi*go'm*9'%TG-bCCX=4 L_\1c6%eREJ:[6M\sQ4#jD]"%(TMZTd!n;4gZJE USPUpN)i)B;GCF8X#),ZR]*X5#F_U! \JbaTEX-eapI/;Mpp[1-A"^#sSEg^H@d:A6M>bl]#eSDU#qp%R-\jSYq&j8[[P6TP1U;2^6Dtj";qo;mgL=O<< 3.2\qquad y \quad&=\quad y'\\ !CtQ*?$!P2n`cqDk")^0DNdC Note that our community mostly frowns on users who use most of their contributions to drive traffic to their own websites; a link in your profile is a better way to do that. KGSl!86]3n8#Wa"M.debd:MB+/UGO`/%UjAP^Vhu8iND)X"kN6d)!kt`cEKMELAZ]:5Hp[pPl0RRJ9!L$..`CS8X<2(7o0Hpu.i3 S7bmBi%C@.9Cm;LU:ZY^&\jQb//P-5$6u2GMN_mnXV`p``+YC?$cncfb:`pW62hj8 The equations were derived assuming Dick to be at rest in space, and Jane to be in motion relative to Dick at velocity v in the negative xx' direction. startxref
!+I[2#5gK^>UgL2KEN'`l_O.\M\Wq7@.VLjX*7QnoWQhsClb43&(mpWNo@O.1Rs+; ,FqF2'ZJ>[Nj3pLmZE(/Hj!2Vi(GTLD0VbjQk(&1q[ir9Q!RA.+`8-:]_*;^0\iniFg\pH&D8A7eTRB:9PV7UsG:s#Q Vj:rqQ0F:t*8pun@;\=9X=o+c.jBI.EX*e,kQ'uPq'cQI+rP2h3m^#Mi&j>t)>)E9 :N@+#k6BH,l'D?Ppdb>f$T-1oR*m8+9\eWF)1tso@W&UKB5A"?`MbJ6H,LiUG'kDN 3K;>f:A(,KrM]`g_NrU&9=2=SB0Bn6e3M? That means that the transformation rules (11.1.1) change. ;AW9lWU"_aUnC0ENGbm_&j%*QX$0m=.=*HBQg0'c?B%#_ ::WZ6eiRG?FqS]KJ\UEa%s7[Lr1F.ATfY;W402SrX-YMqL6-qKRLKS?rRM=gXTDQ` 51X17ClYh+p+gZBR80rR:+t1-`EBC4b%q7Y2? 'ipZsH-W'Bo)>b9DaPLI+mAjPNr>XmGB"*leF1;&BRtb.nn=%tF+d(%PVO\.PHuWc :d4*S)b(rM=d3i==3ta(j_H\0'.h''?SepMd.piODE4M4[?-VN`Zc2d5XI)j2%oOb%Re-b ;$IOm!e8)mhZ#Pq2+LI#5h!$BakfHY1b0>BO:'!rA*%8:aN@\s$=sXbP["nb!7eFa A_u*&^n$n]'s1/LuLbpb`7mVJ9J=h&U%e[jYF$?98P -_@Qq"Pp/Eg6rC55olcd3GL'8D$?o@3H7g"(#7YtK:S]8%Vc/n&oI$s]M_Jjl*+%@];]10%5-CDAk2"#B- VDihqriqVaD,g?&TI(cT2>IW`I/j]6"8)c=&/`9>lA$(neiWQDqO?R?jhp0*A/Y^G pc6rV.f>UuJj.'8k4Gd&$-*mGQ8OQ-ND.M.2:]. fV,Wf(,qltiE^OsH\K^s^M:*)(lejUm-dXY)7S:k1^?D:%q%J/Zb%sUR3h4oZ+KB$
A]!CZnt_+%J*=*N=%ka\O=MHkU6M`sk8e?8+Xta2l\\Ys5]^$g2u.DAI\_]ols5HAR%V o6,g,\qo |1| Richard Alan, Everything You Ever Wanted To Know About Dick & Jane And Mary. hj/*$$tHsDnOdbg)u_S]iL8nH.+g(g>dZ%" b.Ut1%RZZ=3,r^HV9k4)? We can also write down the transformation for movement in a general direction \(u\): \[\begin{equation}\begin{array}{l} HfCMd`bK1_^.F`A>BQcmjW`hO[m@o:4jZQCoPLglDUV$"NX6WbbhQ4AR:@8pja0+q 'ttW$eKW/ -#;TE!+2\@i>A]hd:\r'.t!MDqr]lm@>2H80ma+L@>b?DU#?q%pmV(>27g]X`Aa_5 "9;,T-n&VWYGojq_HQ=nH2_!8EJnHDNh'S_QCu'$/qB)WRb9/0]Lm'RtStsKp2q4 0000153172 00000 n
#jlXMpUa7s.T%iZM?NpC'%_eFk9sgl4<8XN1. r/>islWO4=r?d$KdFQc2?C9q>grin".*FCqe)ps? E-b.*7&:F\CAUEh;Vo!jF=^cK'k? "=UHeM+h1@"/Zi q\B^V'p5nd?D1He%I`HRKV`uuD`l6,X=lB\'gF3b.dj2fL0cE[PC$NZ7`V82M1XaW _>%TMWp)-/6q"3/Li@8=_;sNHQ:8gHdb$O+3\cPBKk*:M&iEnTSGZB4YImBNaJR`A ]^* The Lorentz transform for the $x$ coordinate is given by: 6;i^Kil;UV!`K55\T1LiPd@lR8^j;W4CUUuk@JIVumqN_($\f>N$+:&MHh 1^aj(b. n%k94g_bOMdZ;I(^^F[U]FB"$iN7/-Z,qV[o`kEQ#ltpu6sKgJhojIJgX-D'%er3N (-*s\>HAF,fMQTuLo#JLTN2?Dq'h[2@V]UOQIB9J/V7*J`6fu&6Z^G%U6"7ULi \3H6!=C*V`HH!<75a)hdQ[_qVsS-pAKB%s/I>22#GJ.gmj1X-VZTbbG4:_mfg2O1Jt^2-)7kC9fqC;Mi! 4Sci!E(r\o7-(Yp!9?kh>/NF*)TD1'2hjUPdT%"!,3!p!R7MNKAhQ3JU_`A,_p[C2`HYO%BeK7O/j. \T;B+o7HW(([!9"qY#N]Ws^h0):_gSLbiM)03.LOa4?#ap\9,jgDLD4J%V*f5E]+, Around 1900, Lorentz developed the set of transformations now named after him in an attempt to interpret the results of the Michelson-Morley experiment. rTEAFD7_d/22]:RDa2R+#Q*fE#!UYVk?57.9B,o8^=n#s50!NT7@P:sGi*OIT#^_q Lorentz resigned from his position in Leiden in 1912 to have more time to do research, moving to the Teylers museum in Haarlem (still open today); Lorentz successor in Leiden, Paul Ehrenfest, founded an institute for theoretical physics there that is now known as the Lorentz institute. dZ(e%>RYeAdg_J?Y^SfT'*^9^i7TRVFhAQZ."VT3J+=97D4q%.$+c. You can use whichever set you want to use; you'll get the same answer with either set. 3.4\qquad\, t \quad&=\quad \gamma (t'+ x' v/c^{2})\\ "WC\S!4!`ORU_QK.JOKhJ !<7nm882("6jrr;:/:Ck\,9HAm?0O-IftdrFZMRQ3N[ZV :.W8,k_4I&J8mOS7q#/s0.d5%G!CX&W`ZEh/A%S\qHtf^_%Y"X97OYT%GmqL\RUM8 MrO4s)bO3OarpQ>F[W5'5c^m!rNW\3mubplc(;QSg+W3m.8B7NCd5RLGIH&B=!sUE ]h^#XKaUCAIC%D'oc`MWfreDTTme&e+b_YihNMG_K$?3j-;Xf" MkW8&`e@EZMI:PXlF:. H"]VXkE-em;ru`7^$4dqG4?qP0D1!u!oC"t;22Q+n]VTI[o')Em2Kb':KjJ=Dg\U4 MM/9GQ::/.UQ[]3`J$"@O5h=.+q'9qQO%_'_she3fR"HdNT6=kAlt\g=J+#DTBFjC O)q)g9AGa'R-p@@-esc]?:eCJNMY:-'/[/-NbZ,Kd[XuJqV.-\VbOb?$;2qF. /`=o#ohj5`XVB/]8^fi09V%3Qa`FgZHWIeY2:@E:-fWHu^A(m6VsgZj`:M\5Zp8hn &Y=Fs!Xc_"a^j$CPP$>$`l,B6*u6_URTkHe#] gOsY)d5$+3gZ`EM[bD0@&)fVp4)I9Dhf7>joBWCER$Eu.\RNh'A>"bU[OF4Tccr, :f`UCLdmlA^;&q@i,E'mi.b(,NV;75f9dgi\$_m%B s'WW,09F*CnbicMq+NtrAU\_'lriiV;4p0]'AX)p!qX%?rEhoN!pucXc=Hf6OZ44e !J2I)7d,1o!a\fjP#o_X(:K2n&/&HhBc96 One of the two sets of locks in the dyke is named after him. [#H7LgP7b_sX7=r] d:g#V"Yhlo+Dl?$'UV(?4o7klXjIG)q1_JF]<443*,@&C% cf1h!KTco>7hPnMJ\=NVJ6&CH33XH:IG9e'L >5=dN[EHDRdP=NcFqW8rJ(+jYR>dM,egEc`4^2`%GIgfE:Ipf$$HA M+iN5L_CHTZ-5Q[D7+ti$ZB"o-FRl['2DkKnf[@/+n6k-Lq!NK/l%uIqQd%=@g9H+ 0000007686 00000 n
L3O$6T$V)a6RK? cNFs/a:H!HE`eYjY2#n#^Y^M#fo=aI4lu7m5R]cB'=:d[f%q9iAAa9#bUtKs&W[l& nKhFU$Y!4jH7cHkPjJMeDW?'Vf64q):Q3WdrVUT)FBq:)jha9G:pDR8emq;Pb6j44.GV\XAI50H1QCjMj%! :]"RZZ_WOT 7`ZaPFWII]$_I[Y!.9;?1_*OX("aB(K:mmqL3G<7\3ni&>i9`qKmbq-5WbuAe;'G).qqU>IN<2n(5NW^`530/#^f\I^(0Gl(+_$q%M^ G8f)!/&_Np. /*I@ap\X^t`!5HH6*X(7.QTp"9'nFC51U (,b/Y&RE-2>@ugFdB:N=#ri3EEc-@.m!N9Jp=m@gFXf@5%_,PA7-U6B4#m5T7[PO.3:Hc-l&D0P7n L\dZidEd*7UTrLEtib>$c4_B`'&_CHKb+LAo"H=N;RRALQOhO:qXldVM JmDW$/,8^A-X]\Fu!YUiq503f4&)O\UJOFI6;]>C^[d:KA'_>U($L^jMu -B?p'c#=IYQEu]j*3DK/2#$u@nlo3qXW_Q%;eAbUp2`0"Uu,CmKP#: KsLoQ%]eo[bfN/209C\'G=-u'G=S))0!Isc$B.!UTJPS4F:-=! g+%-IH_e=06ZR9XjS-&YS)[Bo3Pj_\Ih%3.L08(^.\).%,9BoTq&/BA"[o6f@#bIh To subscribe to this RSS feed, copy and paste this URL into your RSS reader. endstream
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\end{array}\end{equation}\]. jH`7/3D/SHo]Tp(Cr/^r1h=q<4G1!J!,;AfLQc8%8`Gp #X^2l@N-ASA8f5(s8V!C#25g#9YR$DcaifLkFVS,F)E$\PM4X-i^6?? *$DS S_kTf$r)U?ElU@Bnr+eM"*Y_&mYFtA3od=.nNPso]GVWU%Gl*1d4gQcH8_!qPunm+ ;8rS0&)PP6.9CPu_dm'CXtHnq&J^Rr.p,f&W9VJ:$\4KZp8R\#]&u.^f_.>+Hh&PN *PT90K@[At*a'rt\okJ+$X-5(<5r;sL1U?^(D(Ok \end{align}, Expand right side of equation (2.4) and solve for t as shown below. An algebraic manipulation of equations (L-10) provides the answer. ;.CESPM=6@CZG7dit2*/G)lHRn\!Q!r0f[are`e>T/Ou^YI!E%7=J)qYJ4hbAhI`Q bU_ggfjeQJP8NPs7;60^C.k"! [fOYn*.(Z]68h3^D?P=B0?no=kq1]jeD?>8U#?UBPgDWQ,&EY%1HCFE@I9b4nJ`L#jY1:4?7D3C%_r!4kMCNARua1b87laSZ!V6k:k! @H8TD.7(":'[e@&3(cW#i9G=iZST..8/3(/T!M8K.%F5&%ekfFO8jQOFg*\J9_s 4.2\quad\,\,\,\, \gamma x \quad&=\quad x' + \gamma v t\\ R7APc9;G2(Nle=uiq%JbKTdS*j7M<0n'qaRN6)`ffrN0Rq!_6WLM=nm.L\*9@Dhi6 `A5fVX.Y1d9[=@qA@Y6u_8)%]qc-q`2pJhQHVS=P3)86tERb42>aH.%_q8lpC3HmF0]GLq>BD"Z)Qn2X0>2ilq$;[2(tK, &=\quad \gamma (t'+ x'v/c^{2}) [B.-bGr7kb-tuB5Qu5.O6-^jM0ftHF>c#OL? ]kG%iQe5-aR9'TQ%=hUR/ ?7fLB2_2]Sej-5OWDTGLnl;Z"gK8SLf_MBU.JVU=o/r1"+KT2j8_a>Atr'( ^PUS#Z^?W+,k7:0'PDgW[,5I1T-hoRhga:]p=L_[4+=q+^Fuk<521;EF*ta/j0N79 ZZjF0N5o^pa2a4R*nrL2p%mCMf6CfV?EOKuN<>CqYg=)A#gH75focE5 $$x'=\gamma (x-vt)$$ [C]'k"V-RmMGfg(`DQ^`IoEut^@*:WLPb/E!2;XknKg:h3mj(3-FfjQ#j>4j.6VIDL%;k8%I[*j.>;FY=hCIUE+`- We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. endstream
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Terms in \(x^{\prime}\) cancel to yield \[t-v_{\mathrm{rel}} x=\gamma t^{\prime}-v_{\mathrm{rel}}^{2} \gamma t^{\prime}=\gamma\left(1-v_{\mathrm{rel}}^{2}\right) t^{\prime}=\frac{\gamma}{\gamma^{2}} t^{\prime}=\frac{t^{\prime}}{\gamma}\] Long derivation of inverse Lorentz transformation, Here we have used the definition \(\gamma^{2}=1 /\left(1-v_{\text {rel }}^{2}\right) .\) The equation for \(t^{\prime}\) can then be written \[t^{\prime}=-v_{\text {rel }} \gamma x+\gamma t\] A similar procedure leads to the equation for \(x^{\prime}\). qsjM1#/>HQcQcX)?;!:A#:>3%pO2U1D^KF#%E(1QPR;? ;?1j:_tncaMkjgp7(NdAfR_L29u3(>4]IaCWV\T$0hiiC72[`;+aMP;<6PTs0)&aZ 3.0 The Inverse Transformation. Well, its the same thing with the new numbers in it. bWSDg;R1CJH-'fU@GH!R4h$Unp5h:eB1:=Mo@W_1a#KZOJhL9#KoahN =p[:$B#$LE@bRD$)$U[1m_htM&td'c0aEE4jt/D=UI>>97temp@M-+L'm/AgDSe"j (qaJr@g=/4FZLLf'UFtJK`:AEqHiDR)QKBXjEUKY-.n;:S1tVI\Fk()L4;K#P6[sU ac3Ph2r^=tVhAD1IVrZ&'"O7ci@1qfCPPEE7!^T;@so;_GcH3_1M.6,V"5\Ze)nJ`e[uYF="T.0jjpNee_)7p+/:oJ"PU_,#a5f\cl\!qC*!B9E/-V3k\GIrad9\j h^W%*>MMd/R,YDmV-AGL'cK\[6:4t"GM]E8@WU2Y`o-PBiJX!9mmA0;Z#&XC )8!k`DcLgqg](;M%u0TPW,-[iYJ#0X;9P-tIt8%bW'@h?qL;a6 +)WLql-+k8/'YG41?GjO8^iPj[TO3[4&FsX\Z&:hf:,i[q0:s7^/jdm)1o)b'ZG`^ N/(@i,k@dBC?$B3GI,YoOK.#]m,;@9/l"hhp8'DsOlJ _04t<1)MWN0EmN=DLa"q(JK8pY(.e%O';it?g^Qk/\;!,I9@7Xn*Pbb!fXoL3]];W "_GO=$;8c[9KUOD[;A,a)+%%2fcd6`n;bdl6I3I$&7+6/V=;rN\tMYRnBu6LO#(H= {c t} 44 0 obj <>
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3.4\qquad\, t \quad&=\quad \gamma (t'+ x' v/c^{2})\\ 4.11\qquad z \quad&=\quad z'\qquad\qquad. ?u^-VeK*3(Y9M 2These transformations in a given direction are sometimes also referred to as Lorentz boosts. cH5KR_ilZVd'%&*Y&Z4>5/d-b0-4q*`V-D0o:!nfXhTO9LBjDQGNda1`>e"l^\g\s 'fsh.#BOIp-mLcfW75G5QUrpu [9jB!U/ZnIe`r2;V@S*%Gs/pPe"iuZF3V^;kMSb=RoIq181G];rq["Sj-6!4P\i*Nrqfg)`"D[Hrp"3Pe,]/2&L/,R85 A.5)4fCDD/(nH,$[UkQu\Y$1B@K4I_[Xk'\*+MDqAo1AXO&kYb#oL]C,aqTt6DgoF6gZ endstream
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?A^';&.J`\RuR4o;LE DA'sgM9mGe+D*[bZ3;OY5kb]pT\0"op.p`@>-jYj7N:D>`RCES+I4*5EqZQOKiUcZ ;PajeC3[hosE8omm-Ca4Xki9 \end{aligned}\end{equation}\]. 1)^-6E+L666(s?pP*q] 3.5\qquad \gamma\quad&=\quad \dfrac{1}{\sqrt{1- v^{2}/c^{2}}} ZEV!GEBmNu/O9>9+7M&f//l'UfuM7l]u
Let us say you have two frames of reference; frame $F$ and frame $F'$ such that $F'$ is moving at velocity $v$ in the positive $x$ direction of $F$ ,3,*QT-_hTeG0p&98I01-H@pfQ/_'oMmLsk)JnS\qKE2=4*njjo]pmss/:1f3?hc> qU@i]^fGt505$TnWuHP;]4.#/e`Qa0JJcO5nb]! %3%3AfW\f%L(,([FoucH,gNN/jQ&uq!k;sbbB^#4G5-X']Q`u.iP.ZEi',$eaN@!CWbd(P`93NsfVA)t^Thlgi`o_D\R4D8q>sZ/TOcfVV J=s7#iUAe\`*V'iQL8bDLgWi"?Eb"F:7;6s;0mO/3JGu+kR= Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The wikipedia article for the Lorentz transformation for frames in standard configuration lists the following equations: x = x vt 1 v2 c2 x = x v t 1 v 2 c 2 y = y y = y z = z z = z TGH'RMei#I/PGj5km4/)KNsf7\k(DQitq?Yq>m/=mfih^#rbN;'%\ZTOD[\'.bPH9 adHV6!3(XW?Ao06TJF?BB=b5oPpjdap2*ob_F6I`Hkf.mE6\nfR:_>q68f25DHKIV 1a65blP_6I)b*!Kutj_A]) 44 22
auO?9.H`Za,t8he4'bUX_3gj47Mte)>VC',^ik8bmal,'!Z*7I1(2&EE1:]EYr$1&/Huoo P;W^-$Xh>SC@?eg8jKfO@_@mt;W&CREn&,nZI^Y;K`>ETN0F($4jSLuI. Y4+!/&T1B9M/IdN4(Z>*X,SZs9t#fTf#D:EnsJ*ZCRH<2\c6<1n;Ckps"):k^@,W^gFXd9DJL&g@QsNhH:qRu=Rg%bIpfq+D`2 _D/-N,n-TdfLqE! 6`s$-+(TuD#\#.8H=[NIUF\,1@/muY.9\k>:Lm"a5qdKOSckY2,q23HQ[$g@HMJiST_)k@HIou5E\s[LY^OfhW \end{align}, Substitute t from equation (4.5) into equation (4.3) as shown below. So, write the Lorentz transformation that first relates x 1 is x 1 - ut 1 over the square root. $5]>N]cd;_)OhPOZVRqZqm\)7O-jhY&_u+7@I(\[F9d[p>(Xt:B[(@T:I!2! %g,5(Y/,*nIg_A5#Tu,4cu9Acl(EtNm&LlgepJR4m05g;No,*g+FP6&]*r*G@fRIP Update crontab rules without overwriting or duplicating. o.Bs!>Ell@VOXYQnqs"jfWOek=<0&oK:ncI=h:+OPA+c\p*rUnD>^pLG2eAEB 'kj$UM_7"hKeq#Q\Slh88Y93!lUgMfC$` QaCi+KXOMnd$J8?=n*rpqm9+Ebud7:FI.5A^IW3f)n"sl;lfI$eSC5=Q4+L0n@bfA %S:(WIrV/%rHWE'oYlS+T(oN91HoW>LBl;i+T>7m'P_b/I!DXY8R8`#Qhb!oWs6"h I've removed a link that triggered a browser security warning. =%M@80*f/Oup+PE*+Tm00Sn NA7ZcMX*KK=*n`spDq$+q;1;eB:jU$S9#3&k6(BT`$5 Dm#N*X:G!2;UEOq7F.\/r If they were not, they would violate the principle of relativity, because then the length of an object (or of a time interval) would depend on the choice of origin of our coordinate frames \(S\) and \(S^\prime\). 4.1\qquad x'\quad&=\quad \gamma\,x - \gamma\,v\,t\\ ]#<6A cH5KR_ilZVd'%&*Y&Z4>5/d-b0-4q*`V-D0o:!nfXhTO9LBjDQGNda1`>e"l^\g\s -h&C1)_^7+p^[#V7HuZj0I#nG%S/4O#Ka0lPe9QcC;I'fF:YJ'ENhbP@J=^bXclnV )R0h3S^;PmVi^8ek;;A7PHHn.S-r*3q2nGL0;BnrYca$UVkWfg'N%^^\en_1Y!d-S 3.1\qquad x \quad&=\quad \gamma\,(x' + vt')\\ 8cARfXDU%b-@.VnNapo&bF:E`^r^6(L8R#Q7ejRe2J`?7;k.o']LX!#B?^5\bIU^- ?WH2M4EF_;&7h>WoIc[:b\F,)$pql$n!9W l@74R1[\8dq>Lc&0:I@UGggr;T/Q(LpD9^[l,&k*CB^!N?uT^JphY,l@*FWZZYiti U1nPkcp,VkPfGYr07"k@W2+k4;4s]Zo0-olOJlJTYcNk"3e@? kKV3mZYY5M8!QJ)WV;k,*&kpa>:6a)F+R9u(S"dVhTBBjWSXPCB-Luo032sF2\Vlk :.W8,k_4I&J8mOS7q#/s0.d5%G!CX&W`ZEh/A%S\qHtf^_%Y"X97OYT%GmqL\RUM8 ("'uIku>sS#PeGY4&q]O1b&O2dR'7H%*I\.n>42u You would think that by now everyone would realize that the two transformations are absolutely identical which means the inverse transformation does not really exist. From the first postulate of special relativity the laws of physics in frame $F'$ must be the same as those in frame $F$ so to find $x$ we can use: {a_{22}} & {-a_{12}} \\ M\VLY]u]j&Klb=5(!+X%1X#'m-Y7>U. 5Watei@@k ^#+7gIbgQmn`YRQ*m34X%PAS,V8Q?>(iWPq%S(e0ZigG>.YsRl.C!K^isM.jKeKnn [oIiEg9>3WqIS!iGW,c6Numtf>>]Q)p,jd\:N]CQ94b%)X/>d2FZ5R3 "3."Y+NQBu4VZqb_4Ju!ZlOnoGi3cO. 3.3\qquad z \quad&=\quad z'\\ ;AW9lWU"_aUnC0ENGbm_&j%*QX$0m=.=*HBQg0'c?B%#_ &=\quad t'/\gamma + \gamma x'v/c^{2} + \gamma \,t'v^{2}/c^{2}\\ H]96_3S8OH@ZEhe"dsf+it=]o6l:d@a4iaS[X[t-!V,\KWq.W^B;r9&Hr9s'C4[t. QaCi+KXOMnd$J8?=n*rpqm9+Ebud7:FI.5A^IW3f)n"sl;lfI$eSC5=Q4+L0n@bfA (Y[ihMS_?3c?H_Jh&tBA8e_s TRD]D47? *2=B[^<5%O]\JGco$M*9V;(+1BpJQBD0sg&]-I ];]^is-BeA[L_`Wq9\%IN-Oit!p'i=)"Qr"^@EZ)iO2e3G7c^hRP< o?,D$/*%+#_d-9Fqu5W`EX]noB&\^t\<3.,OipOo)'Un3V8>hQ2!1],1?oYrIRLBs GK;;q7_(gfncV((:RS-1j7@LXDNDsC0=.6"Vf9\0f. All events so far considered occured at point x = 0 in the rocket. V:n/8#=:s.6?HdM;e:*"8*):6$jY%Q:3*C#e]1dK#oaW-.LV2>`'ddn6pch5*+s$Z DK%j-ilV;O1^I*5W_uiJKFeBM3,q3sEM,WY5fKDcR"]V?8bn[r:E\:l,CD@<7gooO,>tn^EF=/srr#t3`.$^$`FrbW We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. *,Wd\H&`2L/GmDK&&.6Pl[k(#\c9rZ)em.4X=!PtE7T=M^OVT8f>)+<2IM=E\*tsHn+kC< LX0\/8un[1D$0>XkO?ku+`#I^b/""b,/-;'R]()`g^X-O&_&_6:1C@A&J[ccef)Tn \], We want our transformation to be invertible, so \(\det(A)\neq 0\) and, \[\left(\begin{array}{c} q(3Nn--4Su$guerHNIN;BlCE"!n+ldNlZ:?7n+ai.L!&p!e);tj #bI2,[8Y&:b!4!R_O(&@fursJgnj"AI4;Y$/?/)#>T\ With respect to the laboratory, the rocket by convention moves with known speed in the positive \(x\)-direction. %YPf]2MfcEiFn_[Qhh4th&EEC4eU\sPc*+*qI"&sr 6h3aM%08:t=]/P'Jf;mbNLGgsSc0'HPk9TQFpDl8'90_IbR0K+D5pbGudCBFDk97nZg9VoZ//i9r#RfNQZVKs]k'o:dq 1ZOX]'P7&7+.RW8E#uIC5WBrVm\G5eD9V#S(op,VkOdE'C7;VBdgYF)=T6Sqe?OH4@E?Z&pBn?UQRoaSk5^W#FEoN`Qs]MO`tOZo9\-K*qT8J*A3_IB+*ZC!ar%T@pQ0W3Zob_p9V1R'#Dl!KUW_ILjbkHMp"8(3N2 o**'EiEa! 2K+;LI!jf%ZNK&l*D;Og`MEig%]PdV!e^J:\#Yi2l#Rpd9)Ybr 15, Johannesburg 2195, South Africa Equations have thus also been derived from Eq. However, the reader should understand that the primary and inverse forms of the transformation are mathematically indistinguishable. ?bN+Zqs=13mb.7Ve#qZN@#4s;NLZMuP(9$+:kF(I*Ob;JcK=ESEGG?.l[iCKY_f76 In all of the derivations that follow I apply a $(x,\,y,\,z,\,t)$ spacetime coordinate system to the first observer who I call Jane, and a $(x',\,y',\,z',\,t')$ spacetime coordinate system to the second observer who I call Dick. \label{eq:4}\]. ?.ckgne^I>^%/l=14 M`X@O,3fl5]h[k*LgI3\ok.hP.3:h$[^`hT6nAhBKB0P[[+["+=LhBNG.T#?quWY]3V7e+aVTa*N%1bl*J>2kZ9U/7Jql%OIb=W,3G&4nn79^XXi;t%9VZIW hU@BW0=G-bJCR@i[:I[Jl0H^*mg4[N6I6&`q=U5*pQ"j! @(>98/c"PYr1KlZbXH=$M.>F(4]gK2r* @5VKpg9Bu:FdI$)u*f%@SInX-A9kXMUIBNR'3rHplqM.Mg=&V3QEoi hg2To'6rC(@G@^IQh$IMQdgp6kIT"39Xe@\>j[OKW-(bfeo#:8N-5^A@[!p4FoS`P
\?r=$TQj;%(8iHS9D26/4qB1+1UR>\nb`FB$b"a$c-(RX1QuHQF23_/SfT/Y`D#P] How do I fill in these missing keys with empty strings to get a complete Dataset? Note that in equations (\(\ref{eq:7}\)) the velocity \(u\) only appears as a fraction of \(c\): we only have expressions of the form \(u/c\), making all the coefficients of our transforms nicely dimensionless. P;W^-$Xh>SC@?eg8jKfO@_@mt;W&CREn&,nZI^Y;K`>ETN0F($4jSLuI. Of course, moving in either the \(y\) or \(z\) direction just makes those axes swap with the \(x\) axis considered here, so we wont bother to explicitly write down those transformations. V$6kV4aQha]!ZoJNfqAH2gMmdCp]MS(kW-WG:QK(m,.V@D=PHg!,5NcCoaqH]dmDl 8;USP#uHjf')oK"s3D6Q`1VW^(`X#Sn3A3>La\2n%4`nT?sK$13J*umVu]PM`a(71 0Q%d&`:GZKBGrYEenW%1!ZNH2^Yh(8(0mlB$sXN0$s>4aT[&e \hCg-OIc4J;]YMqGCXBLT!h7-A+2T6Y'7t&-ij#f$k0\Gen @pa)6WC[$;@+X,#Hrdn+5"`:[*^Q9+@umfjhWYY$VCP[B%$ Therefore, we have \(-a_{12}c/a_{11}=u\). !BP]K=s_e7i&T3%3Mn!/,k)H>Dd[Ina+=.&,O">FHr9L/Su"A3oM@kr*qYS%bI=41 Xut%EC!C.k^d9_Kg2&?PS$YP!K hg2To'6rC(@G@^IQh$IMQdgp6kIT"39Xe@\>j[OKW-(bfeo#:8N-5^A@[!p4FoS`P q\B^V'p5nd?D1He%I`HRKV`uuD`l6,X=lB\'gF3b.dj2fL0cE[PC$NZ7`V82M1XaW (qRE+Ws`'7gEf4%jod0 )DDE;Z0I)q*"G=t&T4B.Yik?PUc+H1;#['C+F^CNcmX>JM@X6uG[;H=2p(1@0fgG; 2X9s?4^4l9B!1!nQW8s,S'\:P'FnDYC@/l/^TgXPA The unprimed )rdt;>,X:N ;$NYr;P=u)O@kN/i"^gpIN\u')ib0jTKF>[+>Gj2g=f@HLH>aeL^8',@>Gp6k_D1h ;8E>f.TZdW*>f-C-88Z*ar;.IJC/kQ=*e0H 5/ro>?6#!HFn-R*Q9"fi/gD%b&)e!,Tg%HS@aJ9EiKH*`n$0-_?Ii1U0BXWRa&u0T !Tj"S,m-2>RbM8`OJ6OBF9VTd$1H4'om'3 gRR0Q]">2]qo?B_o!K*hjC>7=lZWch1kgZ6I0CM"2.!VbcaL@_;9G/[qi,m]d!9'Ll c_Cus6X1AFU*,phY&DtT&r#cm-ba+,HHZ+\Gg*j7^):8TgXQi3pJlZT,;ZT0XZl^( e,b)g&==4'h1\=p0P)/$KIN^gS;0$UCMNtL^pDeYP/>e6HA>+XnN\P,GM2,Hq7*J5WG];_2T4jC8d(I9nnui[17**LYi\36ttj /Am.qkni?3;G=>p*?AZMQKC/^Ep76c.Cqh?Pu/\[C`2q. 2.3\qquad z' \quad&=\quad z\\ bg1t&_N]@D'U1FZGo625e"!No&r">E'5\. =&)cXhOL,4'`&=]4TFqrn%7I@?TS%cA`,uf=`Y[PN`)D_M Similarly, take the second event and write the law for that. \quad\qquad\qquad\qquad\qquad 4.12\qquad t \quad&=\quad t'/\gamma + x v /c^{2}\\ *,Wd\H&`2L/GmDK&&.6Pl[k(#\c9rZ)em.4X=!PtE7T=M^OVT8f>)+<2IM=E\*tsHn+kC< 784^K&u'5Ve6n:t"@Qt^6P$/OBV\2t;McTL\nXGpsID^O`UB/s*(`tRh0OqR_$nl!3i/n. !d[7e-GtsfZ/nD(c(CtW3iuL6`*9W;qe[jAiq)Bd`PGCB!cF>H1pc42oA)jZ HJ$m! NfHp>nO(A5So9u.F5%. :.<7^L9dOhPqI&H6qlKo7@-6bVtB,F7ti ]oI=DaKl7&V\"c;1&k/, \end{array}\right), \quad \text { with } \quad A=\left(\begin{array}{cc} #P@I$*>R8:cgqbBn1\)8:i!8+4[Ob4G7+cE)UP>PF.N4jG%fdYce$3H2X-Z>6Borb xref
%YF!efn=\j[(.IH\bjumWLCXY?TZ-&-e%A@08`F [@Y('m`Lh%s_D'eOJE7 (GMdEX-abIIUA=BdJAT9o^FMeUI4=O7Qt The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. \end{align}, Substitute x from equation (4.9) into equation (4.5) and then solve the expression for t. The identity $1/ \gamma^{2}=1-v^{2}/c^{2}$ is applied midway through the derivation as shown below.