603 ! APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la asesoría de Alfredo Sandoval Villalbazo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LA DILATACIÓN DEL TIEMPO ! !!!H&!%4!#30&$24$!54(&:4/!64:5$4.#$!!"#$%!"&"'%()#$*!#&%*+,-#C1&!/&!643/0#0<!&:5?$26#:&30&!&3! %#!/&662<3!I7!64:4!>&$&:4/!#!6430231#62<3J!! ! !!!K#!)*E,!/&!&>#%L#!5#$#!13!02&:54!23262#%! !8!13!02&:54!54/0&$24$! !&3!%#!:2/:#!54/262<3! @! )*M,!! !!!"4$!%4!0#304@! ! )*N,!! !! ! ! ! !!!H&!)*E,7!)*M,!8!)*N,!/&!4.02&3&@! ! )*O,! ! ! !!!H&!)*O,!/&!4.02&3&!&%!$&/1%0#(4!C1&!/&!4.01>4!&:5?$26#:&30&!&3!%#!/&662<3!I!(&!&/0&!%2.$4@! )*I,! ! !!!"4$!&P&:5%47!/2! )*Q,! ! !/&!/2'1&!C1&@! ! 608 ! !!!"#$%! #&'(&)&*+! ,-.! ./! %0#.12+3%1! *%(#$+$+! ,-.4! *-+(3%! 5+! 6+#+3%! -(! 7&(-$%! .(! ./! 1./%8! 3.! /+! 6.1#%(+!,-.!2&+8+!*%(!-(+!2./%*&3+3!3.! 4!*%(!1.#6.*$%!+/!#&#$.7+!3.!*%%13.(+3+#!.(!*-9%! %1&'.(! #.! .(*-.($1+! ./! %0#.12+3%14! .(! ./! 61%6&%! 1./%8! 3./! %0#.12+3%1! 5+(! 6+#+3%! -(! 7&(-$%! 9! ,-&(*.! #.'-(3%#:! 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EL CUADRIVECTOR DE LA VELOCIDAD ! !!!!E/D65F><6-!1/(! !(-!(7!G(0:6;!.(!G(7608.>.!065!1/(!(7!-8-:(<>! !-(!</(G(!065!;(-D(0:6!>7! -8-:(<>! +!(5!7>!.8;(00895!.(! !&!1/(!/5>!D>;:H0/7>! !-(!</(G(!065!G(7608.>.! +!<(.8.>!(5! +!&!G(7608.>.! +!<(.8.>!(5! 2! ! !!!I(06;.(<6-B! "JJ%! !! ! !!!K65!A>-(!(5!"JJ%!!.(?858<6-!7>!G(7608.>.! ! ",L%! ! ! !!!M6;!7>!;(F7>!.(!7>!0>.(5>B! ",N%! ! ! !!!'(!",L%!&!",N%!6A:(5(<6-B! ",J%! ! !!!M6;!.(?8580895B! ! 615 ! "#$%! ! ! !!!!&'()!*+,-!,.!,/0,!1'/)!2'3041+5'3-!! ! "#6%! -!/,!/47+,!*+,8! "##%! ! ! !!!&,!"#9%-!"##%!:!";$%!)<0,.,=)/8! "#>%! ! !!!! !!!?,1)3(,=)/!*+,8! "9$%! ! ! !!!&,!"9$%!)<0,.,=)/-!(,34@'.()!/)<3,! 8! "#A%! ! ! !!!&,!"#A%!)<0,.,=)/-!(4@4(4,.()!5)/!0B3=4.)/!,.03,! 8! "#C%! ! !! !!!?,1)3(,=)/D! "##%! ! ! !!!&,!"##%!:!"#C%!)<0,.,=)/D! ">E%! ! !!!F.G5)7'=,.0,8! ">;%! ! ! !!?,1)3(,=)/8! 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