apÉndice v. la relatividad especial de einstein y …€¦ · !603 apÉndice v. la relatividad...

28
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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

Upload: others

Post on 27-Apr-2020

5 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 603

APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ

por Juan Auping, con la asesoría de Alfredo Sandoval Villalbazo

"#!$%&$!'()#*+,$!&$#$-.%!/'%!%+01+$#&$%!%$,,+.#$%2!

345!!!6'%!&7'#%8.7-',+.#$%!*$!9'/+/$.!

:45!!!6'%!&7'#%8.7-',+.#$%!*$!6.7$#&;!

<45!!!6'!*+/'&',+=#!*$/!&+$-(.!

>45!!!6'!&7'#%8.7-',+=#!*$!6.7$#&;!*$/!!-.-$#&.!?$#!+#0/)%2!"#"$%&'"(!@!$#$70A'!,+#)&+,'!

B45!!!"/!&$#%.7!-)&7+,.!

C45!!!"/!,1'*7+D$,&.7!*$!/'!D$/.,+*'*!

E45!!!6'%!$,1',+.#$%!*$!F'GH$//!@!/'!81$7;'!*$!6.7$#&;!%.#!+#D'7+'#&$%!I'J.!/'%!!

!!!!!!!!&7'#%8.7-',+.#$%!*$!6.7$#&;!

E4345!6'!&7'#%8.7-',+=#!*$!/'!81$7;'!*$!6.7$#&;!

E4:45!6'!&7'#%8.7-',+=#!*$!/.%!,'-(.%!$/),&7+,.!@!-'0#)&+,.!

E4<45!6'!&7'#%8.7-',+=#!*$!6.7$#&;!*$!/'%!$,1',+.#$%!*$!F'GH$//!

E4>45!6'!&7'#%8.7-',+=#!*$/!(.&$#,+'/!$/$,&7.-'0#)&+,.!

!1. LAS TRANSFORMACIONES DE GALILEO !

!!!"#! /'! -$,K#+,'! #$H&.#+'#'L! /'%! ,..7*$#'*'%! ,'7&$%+'#'%! .7*+#'7+'%! ! %$! &7'#%8.7-'#! $#!

#1$D'%! ,..7*$#'*'%L! ! %$0M#! !"# $%"&'()%*"+,-&# ./# 0"!,!/)L! $#! $/! %1(1$%&.! *$! N1$! $/!

.I%$7D'*.7!%$!-1$D'!$#!/A#$'!7$,&'L!$#!/'!*+7$,,+=#!*$! L!,.#!D$/.,+*'*! !,.#%&'#&$2!

?3O! !

?:O! !

?<O! !

?>O! !

!

2. LAS TRANSFORMACIONES DE LORENTZ !

!!!"+#%&$+#! (7.(1%.! /!# 1%,&+,1,)# ./# !"# %/!"$,2,.".! @! *$! /'! ,.#%&'#,+'! *$! /'! D$/.,+*'*! *$! /'! /1;!

,.-.!%+01$2!

!!!)*+!,-.!/$0$.!1'$!234$%!$/!5-"63#!$%!.3.&$"-.!78.35#.!%#!.#%!-7$5&-9-.!:#2!/-!2$7$2$%53-!-!'%!

.3.&$"-!'!#&2#!.3.&$"-!9$!5##29$%-9-.;!$%!$/!5-.#!9$!"#<3"3$%&#!/3%$-/!'%37#2"$+!

!!!=+!>'-/1'3$2!2-0#!9$!/'?!.$!"'$<$!$%!'%!.3.&$"-!3%$253-/!9$!5##29$%-9-.!5#%!'%-!<$/#539-9!

9$&$2"3%-9-! ;! 3%9$:$%93$%&$"$%&$! 9$/! @$5@#! .3! $/! 2-0#! 9$! /'?! $.! $"3&39#! :#2! '%! 5'$2:#!

$.&-53#%-23#!#!$%!"#<3"3$%&#+A!3E:P!!

!

!!!"%&.%!(7+#,+(+.%!&+$#$#!,.#%$,1$#,+'%!#.&'I/$%!@!.I%$7D'I/$%!$#!/.!N1$!7$%($,&'!/'!-$*+,+=#!*$/!

$%(',+.!@!*$/!&+$-(.L!,1'#*.!1#!.IJ$&.!D+'J'!'!1#'!D$/.,+*'*!-1@!'/&'L!,$7,'#'!'!/'!*$!/'!/1;4!3"#

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!3E:P! Q/I$7&! "+#%&$+#L! B/! C23%53:3#! 9$! /-! D$/-&3<39-9L! 7$(7.*1,+*.! $#! R&$(S$#! T'HU+#0L! E@$! F//'.&2-&$9! #%! &@$!

G@#'/9$2.!#7!H3-%&.!?:VV>O2!:3V!

Page 2: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!604

!"#$%&'(#)$*+,%- ./- 0(#/%"12! ! "#$! %"&%'(! ')*#! '(*+%(! ,$-(! *,&%'%)"! &,.! ,$/('%#! 0! &,.!

-%,*/#1!!!

234! !

!

!!!5(!-6("$7#6*('%)"!&,!5#6,"-8!&,!9"!!$%$-,*(!&,!'##6&,"(&($! !:9,!$,!(.,;(!'#"!

9"(!<,.#'%&(&! !&,!#-6#!$%$-,*(!&,!'##6&,"(&($! =!,"!.(!&%6,''%)"!&,.!,;,! !$,!

6,/6,$,"-(!/#6!.(!$%>9%,"-,!*(-6%8?!

!

2@4! !

!

!!!!5(! 6(8)"! /#6! :9A! =! ! ,$! :9,! ,.!

*#<%*%,"-#!,$!,"!.(!&%6,''%)"!&,.!,;,! =!!6(8)"!/#6!.(!'9(.! !"#!-%,",!'#*/#","-,$!,"! !0! B!

! "#! .#$! -%,",! ,"! ! 0! B! ! "#! .#$! -%,",! ,"! ! 0B! ! ! "#! .#$! -%,",! ,"! ! 0! 1! C%! ,.!

*#<%*%,"-#! 79,6(! ,"! 9"(! &%6,''%)"! (6+%-6(6%(! '#"! 9"(! ! <,.#'%&(&! '#"! '9(-6#! '#*/#","-,$!

2 4=!"%">9"(!&,!,$-($!'#*/#","-,$!$,6D(!',6#=!'#*#!<,6,*#$!*E$!(&,.("-,1!

!

!!!F,!234!0!2@4=!#+-,",*#$?!

2G4! !

!

!!!HI#6(!+%,"=!/(6(!9"(!$,J(.!.9*%"#$(!:9,!<%(;(!'#"!<,.#'%&(&! !&,$&,!,.!',"-6#!&,!9"(!,$7,6(!

I('%(! .(! $9/,67%'%,=! 6,'#66%,"&#! .(! &%$-("'%(! ! 2,.! 6(&%#! &,! .(! ,$7,6(4! ,"! ! $,>9"&#$=! /#6! ,.!

-,#6,*(!&,!K%-E>#6($=!,$!<E.%&(!.(!$%>9%,"-,!(7%6*('%)"?!

2L4! !

!

!!!!!F,!2L4=!#+-,",*#$!2M4?!

2M4! !

!!!!

!!5.,<(*#$!2G4!(.!'9(&6(&#=!/(6(!%>9(.(6!'#"!2M4?!

2NO4! P!

!

!

!!!Q>9(.(*#$!.#$!-A6*%"#$!'#"!'#,7%'%,"-,$!&,!/#-,"'%($!%>9(.,$!,"!2M4!0!2NO4?!

2NN4!! !

Page 3: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 605

"#$%!! !

"#&%! !

!

!!!'()(*+,!-./01+!2)-34)20/,!5!01(,!(-./-2+)(,6!7(!*+7+!8.(!)+,!9/:0/!.)/!-./10/!(-./-23)!;/1/!

1(,+:<(1:/,=!>/7+!8.(!(:!,2,0(*/! !7(!*.(<(!-+)!<(:+-27/7! !-+)!1(,;(-0+!/:!,2,0(*/!! !5!8.(6!

;+1!7(92)2-23)?!

"#@%! !

!!!,(!,24.(!8.(!;/1/!(:!+124()!7(:!,2,0(*/! ?!

"#A%! !!!5?!

"#B%! !

!!!7(!*+7+!8.(?!

"#C%!! !

!

!!!>(!"#C%!,(!7(7.-(!8.(?!

"#D%!! !

!

!!!EF+1/!5/!0()(*+,!-./01+!(-./-2+)(,!"##6!#$6!#&!5!#D%6!;/1/!;+7(1!+G0()(1!(:!</:+1!7(!:/,!-./01+!

</12/G:(,!" 6! 6! !5! %=!H.,020.2*+,!;12*(1+! !;+1! !;/1/!::(4/1!/?!

"##%!! ! !

"#I%!! !

"$J%! ! ! !

!

!!!H.*/*+,!"##%!5!"#I%!5!0/*G2K)!"#I%!5!"$J%?!

"$#%! !

"$$%!! !

!

!!!H.,020.2*+,!"$$%!()!"$#%?!

"$&%! !

!

!!!EF+1/!G2()6!7(92)2*+,!:/!</12/G:(! !-+*+?!!

"$@%! !!!!!!!!

!

!!!>(!"$@%!5!"#@%6!+G0()(*+,?!

"$A%! !

Page 4: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!606

!!!"#$%&'(')#!*+,-!.!*+/-0!#%12'2$#34!

*+5-! !

!

!!!"#$%&'(')#!*++-!.!*+5-0!#%12'2$#34!

*+6-! !

!

!!!728#9)2$#3!*::-!.!;(!8#$%&'($#3!8#'!*+5-!.!*+6-4!

*+<-! !

!!

!!!728#9)2$#3!*:<-!.!;(!8#$%&'($#3!8#'!*+<-4!!!

*+=-! !

!

!!!>?#9(! .(! @#)2$#3! 92@9232'1(9! ;(! 19('3A#9$(8&B'! )2! C#92'1D! 2'! A#9$(! $(19&8&(;0! 8#'! ;#3!

923E;1()#3!FE2!(8(%($#3!)2!#%12'294!

!

*,G-! !

!

!!!H'!2;!8(3#!)2!E'!$#I&$&2'1#!FE2!'#!I(!2'!;(!)&9288&B'!)2!E'!3#;#!2J2!* !.!

-0! ;(3! 8#3(3! 32! 8#$@;&8('! .! 2;! 923E;1()#! 23! 2;!

3&KE&2'12:6+=!!

*,:-! !!!!

!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!:6+=!L#?'!!L(8M3#'0!!"#$$%&#"'(")&*+,-./#0%&$0!328#')!2)&1&#'!*:=6/-4!/N:!

Page 5: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 607

!!!"#$#!%%&'#$!(&!)*+,!#!)*-,!.#/0#!/1/02012$!&3!)*+,!%4/!/151&/04/!(&%!6#/4!5#$0261%#$!)*-,7!#!/#.&$7!

!8! !

!

!!9/0#!:2/:#!23;4$:#62<3!6430&32(#!&3!%#!!:#0$2=!)*-,!8!)*+,!/&!>&!#/?!&3!;4$:#!0&3/4$2#%@!

)*A,! !

!

!!!B43!.#/&!&3!)*+,7!4.0&3&:4/!C1&! !8! !87!#(&:D/! !)**,!8! !)*E,@!

)**,! !!

)*E,! !

!

!!!FC1?!6#.&!4./&$>#$!C1&!%#/!&61#6243&/!)**,!8!)*E,!/&!$&(16&3!#!%#!0$#3/;4$:#62<3!(&!G#%2%&47!/2!

/1543&:4/!C1&! !8! @!

)+,! !

)E,! !

!

3. 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! !/&!/2'1&!C1&@!

)*Q,! !

Page 6: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!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%#:!"#$%#! .).*$%#! #%(! 1.+/.#!9! )-.1%(!*%11%0%1+3%#! .76;1&*+7.($.:!<+!3&/+$+*&=(!

3./! $&.76%! )-.! %0#.12+3+! 6%1! 61&7.1+! 2.>! .(! ?@AB4! .76/.+(3%! 3%#! +#;! //+7+3%#! 1./%8.#! 3.!

C%##0+-.14! ,-.! )-(*&%(+(! *%(!0+#.! .(! )%$%(.#! .7&$&3%#! 6%1! -(! &#=$%6%! 1+3&%+*$&2%!3./! )&.11%:!

D+1$&.(3%!3.!-(+!#&(*1%(&*&3+3!3.!3%#!1./%8.#!*%(!-(+!61.*&#&=(!3.! :!E(!1./%8!#&'-.!.(!

1.6%#%4!./!%$1%!'&1+!1F6&3+7.($.:!!"#$%"&"#'()"*+,*#-%"#".#/0"1$'#"2#".#*".'3#"2#1'+010"2/'#)"#

*"/*,),# 4'2# *")$"4/'# ,.# /0"1$'# &".# *".'3# "2# *"$')'# 4'2# %2# 5,4/'*# "-%0+,."2/"# ,#

4!.#!3.*&14!/%!61.3&*5%!6%1!/+!$.%1;+:?GHB#!

!

!!!"/! /.*$%1! 6%31;+! %08.$+1! .#$+! $.%1;+! *%(! /+! )+7%#+! 6+1+3%8+! 3.! /%#! '.7./%#4! .(! /+! *-+/! -(!

5.17+(%! !#.!.(*-.($1+!.(! !9!%$1%4! 4!!.(! :!I%7%!$%3%!.#!J1./+$&2%K4!/+!2./%*&3+3! !

3./!#&#$.7+! !*%(!1.#6.*$%!+/!#&#$.7+! 4!.#!7.(%#!/+!2./%*&3+3!L M!3./!#&#$.7+!N!*%(!

1.#6.*$%!+/!#&#$.7+! :!"($%(*.#4!#&!%0#.12+7%#!/%#!1./%8.#!3.#3.!./!%1&'.(!3./!#&#$.7+! 4!!6+1+!

!5+01;+!6+#+3%!-(!$&.76%!OPQ!7+9%1!,-.!6+1+! 4!6.1%!#&!/%#!%0#.12+7%#!3.#3.!./!%1&'.(!3.!

4! !5+01;+!.(2.8.*&3%!7F#!,-.! :!"#$+!6+1+3%8+!.#!-(!#%)&#7+4!*%7%!6%3.7%#!.($.(3.1!3.!

/+! #&'-&.($.!7+(.1+:! N-6%('+7%#! ,-.! ?BB! 1./%8.#! 3.! ! .(! ./! #&#$.7+! 4! $%3%#! &(3&*+(3%! /+!

7&#7+!5%1+4!#.!3.6%#&$+(!+!/%!/+1'%!3./!*+7&(%!3.!7&//%(.#!3.!R&/=7.$1%#!1.*%11&3%!6%1! !,-.!

2&+8+! .(! :! S/! )&(+/! 3./! 2&+8.! #.! $%7+! -(+! )%$%! 3./! T/$&7%! 1./%8! 3.! ! 9! 3./! 1./%8! 3.! :!

S6+1.*.1F!/+!3&).1.(*&+!3.!5%1+!61.3&*5+!6%1!/+!.*-+*&=(!LH@M!9!(%!%$1+:!

!4. LA TRANSFORMACIÓN DE LORENTZ DEL MOMENTO Y ENERGÍA CINÉTICA

!

!!!!U.!+*-.13%!+!/+!7.*F(&*+!(.V$%(&+(+4!./!7%7.($%!.#!./!61%3-*$%!3.!7+#+!9!2./%*&3+3W!

LXM! !

!

!!!!!!D+1+!3.$.17&(+1!./!7%7.($%!3.!-(!./.*$1=(!+$1+;3%!6%1!-(!*+76%!7+'(Y$&*%4!#.!-#+!./!1+3&%!

3./! *+7&(%! *&1*-/+1! 3.! /+! 6+1$;*-/+! 3.#2&+3+! 6%1! .#$.! *+76%:! <+! 1./+*&=(! .($1.! ./! 1+3&%! 9! ./!

7%7.($%!#.!3.1&2+!3.!/+!)-.1>+!(.*.#+1&+!6+1+!-(+!1-$+!*&1*-/+1!3.!*-+/,-&.1!%08.$%!+$1+;3%!6%1!

-(+!)-.1>+!*.($1+/W!

LXXM! !

!

!!!<+! )-.1>+!7+'(Y$&*+! ! ,-.! +$1+.! -(+! 6+1$;*-/+! .#! -(+! )-(*&=(! 3./! *+76%!7+'(Y$&*%! 4! /+!

*+1'+! 3.!/+!6+1$;*-/+!!9!#-!2./%*&3+3! W!

LXXXM! !

!

!!!U.! /+! &'-+/3+3!3.! LXXM! 9! LXXXM! %0$.(.7%#! /+! 1./+*&=(! .($1.! ./! 1+3&%! 3.! /+! 1-$+! *&1*-/+1! ! 9! ./!

7%7.($%! W!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!?GHB!ZY+#.![+9!\1.+14!67)04,#!L?@]@MW!?G^!

Page 7: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 609

"#$%! !

!

!!!&'()*!+,-!.'(/'!0'!'12'(/3',4).!5+'!.'!6/7/'(),!0'!89:9!-!898;!.)<('!'=!3)3',4)!"',!/,>=?.@!

!"!#$%&!%!0'!'='74(),'.!0'!-=4-!A'=)7/0-0!"B+76'('(*!C-+D3-,*!E+F'!G!H-A-,76F%*!('D+4-,!=-.!

2('0/77/),'.! <-.-0-.! ',! '.4-.! '7+-7/),'.! ,'I4),/-,-.*! F! 7)(()<)(-,! =-! 4(-,.D)(3-7/J,! 0'!

H)(',4K@!!!!!

"$L%! !

"$!B%! !

!

!!!H-! ./>+/',4'! >(MD/7-8NO8! ./,4'4/K-! =).! ('.+=4-0).! 0'! '.4).! '12'(/3',4).! ('-=/K-0).! 0'! 89:9! -!

898;@!

!!

!!!!!!H-!7('7/',4'!0/A'(>',7/-!',4('!P'I4),!F!Q/,.4'/,*!',!'=!7-.)!0'!A'=)7/0-0'.!-=4-.*!.'!)<.'(A-!

4-3</?,!2-.-! ./!3'0/3).! =-! ','(>R-!0'!+,-!2-(4R7+=-*! =-! 7+-=!S./!,)! .'! ',7+',4(-! ',!'=! 7-32)!

>(-A/4-7/),-=!0'!)4(-!2-(4R7+=-T'.!/>+-=!-!=-!.+3-!0'!=-!','(>R-!7/,?4/7-! F!','(>R-!/,4(R,.'7-!

@!

"$#%! !

!

!!!U-0)! 5+'! ','(>R-! F! 3-4'(/-*! .'>V,! Q/,.4'/,*! .),! /,4'(7-3</-<='.! )! W./3?4(/7-.X*! 2)0'3).!

'12('.-(! =-! ','(>R-! 7)3)! +,-! D+,7/J,! 0'! =-!3-.-*!3+=4/2=/7-0-! 2)(! +,! D-74)(! ! 7+F)! A-=)(!

0'2',0'!0'!=-!A'=)7/0-0!A-(/-<='! !0'!=-!2-(4R7+=-!!F!=-!A'=)7/0-0!7),.4-,4'! !0'!=-!=+K@!

"$##%! !

!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!8NO8!Y)6,!L==0-F*!'&()*+,-.#/%"$+-($0-%1#-234-2($4Z!5#6"$0-#03%3"$!"[::[%@![;!

Page 8: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!610

!!!"#$%&'(')#!*+,-!.!*+,,-/!#%01'1$#23!

*+,,,-! !

!

!!!4#5($1'01!2&!5(!6(7089:5(!120;!1'!716#2#! /!<1=0#'!.!>&'201&'!9#&'9&)1'/!12!)19&7/!2&! /!

! .! ?!@17#! 2& /! 5#2! 712:50()#2!)1! ($%#2!A('!)&A17B&1')#?!+1($#2! 120#! 1'! 15!

9(2#! )15! 07(%(C#! 71(5&D()#! (5! (65&9(7! :'(! E:17D(! ! (! :'! #%C10#/! (91517;')#5#! 6#7! 9&170(!

)&20('9&(! 3!

*,F-! *:'&)()12!G#:512!H! -!

!

!!!41BI'!<1=0#'/!15!07(%(C#!71(5&D()#!123!

*F-! !!

!!!617#/!21BI'!>&'201&'3!

*F,-! !!

!

!!!!4:6#'B($#2! J:1! 5(! E:17D(! 21! (65&9(! (! :'! #%C10#! J:1! 1'! 15! $#$1'0#! ! 120;! 1'! 716#2#!

!.!J:1!(91517(!K(20(!:'(!A15#9&)()! !1'!15!$#$1'0#! ?!>'!1201!9(2#/!

21BI'!<1=0#'/!15!07(%(C#!71(5&D()#!123!

*F,,-! !

!

!!!L!21BI'!>&'201&'3!

*F,,,-! !

!

!!>'!1201!9(2#/!5(!)&E171'9&(!12!715(0&A($1'01!61J:1M(/!617#!2&!21!07(0(!)1! /!5(!)&E171'9&(!

551B(!(!217!$;2!B7(')13!

*F,+-! !!.!

*F+-! !

!

!!!@(7(!A15#9&)()12!9179('(2!(!5(!)1!5(!5:D/!6#7!1C1$65#/! !5(!)&E171'9&(!12!9#'2&)17(%513!

*F+,-! !

*F+,,-! !

!

!!!>'! 0N7$&'#2! '1=0#'&('#2/! 15! 07(%(C#! 71(5&D()#! ! 551B(! (! :'! 58$&01! $;O&$#! 12! !

9:(')#! /! 617#! 1'! 15! 9(2#! 1&'201&'&('#/! 15! 07(%(C#! 71(5&D()#! 21! (9179(! (! &'E&'&0#! 9:(')#! 5(!

A15#9&)()! 21! (9179(! (! 5(! )1! 5(! 5:D?!P&! 9(59:5()#7(! "(2&#! A(! K(20(! /! 9#'! :'!

07(%(C#!71(5&D()#!)1! !.!5:1B#!21!'&1B(!(!9(59:5(7!15!07(%(C#!6(7(!A15#9&)()12!$;2!

(50(2?!Q($%&N'!120(2!671)&99&#'12!'1=0#'&('(2!.!1&'201&'&('(2!2#%71!5(!1'17B8(!9&'N0&9(!K('!2&)#!

Page 9: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 611

"#$%&'%!'!"(#$)'!$*!$+"$(,-$*&.%!/!0'*!%,1.!($2#&'1'%!/!3.((.).('1'%4!($%"$3&,5'-$*&$!3.-.!%$!

"#$1$!.)%$(5'(!$*!6'!%,7#,$*&$!7(82,3'9:;<=!!

!!"#$%&'()*"+,%&&%-.+/).+01-.%'.'/)2)"+3'1%4%/1'/)/-5"+)3')+.+"67')&%.81%&')

!!!!!!

!!!!!>'!$3#'3,?*!@AB!*.%!1,3$!&'-),C*!D#$!6'%!"'(&E3#6'%!D#$!*.!&,$*$*!-'%'4!1$)$*!5,'F'(!3.*!6'!

5$6.3,1'1!1$!6'!6#G4!/!5,3$5$(%'4!#*'!"'(&E3#6'!D#$!5,'F'!3.*!6'!5$6.3,1'1!1$!6'!6#G4!%.6'-$*&$!6.!

"#$1$!0'3$(!%,! &,$*$!#*'!-'%'!3$(.4!".(D#$!$*!&'6!3'%.4! 4!#*'!3'*&,1'1!*.!

1$2,*,1'!".(!6'%!-'&$-8&,3'%4!"$(.!*.!,-".%,)6$!&'-".3.4!%,!%$!3'63#6'!$6!6E-,&$!3.(($%".*1,$*&$9!

H.1$-.%!3'63#6'(!$6!-.-$*&.!1$!&'6$%!"'(&E3#6'%!1$!.&('!-'*$('9!I'1.!D#$!!%$7J*!@AB! 4!

%$! %,7#$!D#$! $*! $6! 3'%.!1$! 4! $6!-.-$*&.! $%! 4! 6.!D#$! %.6'-$*&$! $%!".%,)6$4! 3.-.!/'!

5,-.%4! %,! 4! 6.! D#$! *.! "$(-,&$! 1$&$(-,*'(! $6!-.-$*&.! 1$! 6'! "'(&E3#6'9! H$(.4! 1'1.! D#$! 6'!

$*$(7E'!1$!6'!"'(&E3#6'4!%$7J*!@AKKB!$%! 4!%$!%,7#$!D#$L!

@MMB! !

!

!!!N0.('!),$*4!1$!'3#$(1.!'!6'!&$.(E'!1$!H6'*3O4!6'!$*$(7E'!1$6!2.&?*4!$%!

@MMKB! !

!

!!!%$!%,7#$!D#$L!

@MMKKB! !

!

!!!6.! D#$! ($"($%$*&'! $6!-.-$*&.! @$*! ,*76C%L!"#"$%&'"(! ($6'&,5,%&'! 1$! #*! 2.&?*4! /! 1$! 3#'6D#,$(!

"'(&E3#6'! %,*! -'%'! D#$! 5,'F'! 3.*! 6'! 5$6.3,1'1! 1$! 6'! 6#G4! 3.-.! ".(! $F$-"6.4! #*! 76#?*4! .! #*!

*$#&(,*.9!

!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!:;<=!P.0*!N661'/4!)'*+,-.!/$0&#%-!*%1!&2$!345!3*%59!6$7#%1!$14&4#%!@=QQ=L!<Q!

Page 10: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!612

!!!"#$!%$#&'$!(&!)$*+$'!*$!&,-$,./#!#&012#.$#$!&)!$,&31$'!&*!4&,42!&.#)1&.#.$#2!5-&!*$!%$)$!(&!

-#$!3$'16,-*$7!(&)(&!&*!3-#12!(&!+.)1$!(&*!28)&'+$(2'7!.#,'&%&#1$!,2#!*$!+&*2,.($(7!&)!(&,.'9!

:;;<<<=! !

!

!!!&#! (2#(&! ! &)! *$! %$)$! (&! '&32)2! :*$! #&012#.$#$=! >! ! *$! %$)$! 5-&! .#,'&%&#1$! ,2#! *$!

+&*2,.($(!*$!,-$*!)&!$,&',$!$!.#?.#.1$!,-$#(2!*$!+&*2,.($(!)&!$,&',$!$!*$!(&!*$!*-@!:%&#2)!&*!,$)27!

(&)(&!*-&A27!,-$#(2! =B!

!

!5. EL TENSOR MÉTRICO !

!!!C42'$!(&(-,.'&%2)!&*!1&#)2'!%D1'.,2!(&*!).)1&%$!1'$#)?2'%$(2B!

!

!!E&,2'(&%2)9!

:FG=!! !

!

!!!H&!:FG=!)&!(&(-,&!:IJ=9!

:IJ=! !

!

!!!!!H&!:IJ=!)&!(&(-,&#!:IK=7!:IG=7!:IF=!>!:II=!

!

:IK=! !

!

:IG=! !

!

:IF=! !

!

:II=! !

!

!!!"#$!.(&$!A&#.$*!(&!L.#)1&.#!:&#1'&!21'$)=!?-&!&*!&#1&#(&'!5-&!).!-#!?21/#7!,2#!*$!+&*2,.($(!(&!*$!

*-@! 7!'&,2''&!&#! !)&A-#(2)!-#$!(.)1$#,.$! !&5-.+$*&#1&!$*!'$(.2! !(&!-#$!&)?&'$7!&)1$!(.)1$#,.$!

#2! ,$%8.$! ).! ,$%8.$%2)! &*! ).)1&%$! (&! ,22'(&#$($)! (&! *$! &)?&'$7! &)! (&,.'7! 5-&!

B! M$! ?2'%$! (&! *$! &)?&'$! #2! ,$%8.$7! !"#$ ).! ,$%8.$%2)! &*! &)1$(2! (&!

%2+.%.&#12! (&*! 28)&'+$(2'B! ! L#! &*! "#$%&'()! *#!+(,-).$-(/! *$! (.)1$#,.$! &#1'&! (2)! 3-#12)! &)!

).&%3'&!*$!%.)%$7!$!)$8&'7! !

!

!!!E&,2'(&%2)!5-&7!32'!(&?.#.,./#9!

!

:IN=! !

!

!!!>!1$%8.D#9!

Page 11: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 613

"#$%! !

!

"#&%! !

!

!!!'(!"#)%*!"#+%*!"#,%*!"#$%!-!"#&%!.(!/012(3(4!

!

"#5% 6! !!7!! !

!

!!!'(!"#,%*!"#&%!-!"#5%!/01(3(8/.9!

"#:%! !

!

!!'(!"#;%*!"##%!-!"#:%*!/01(3(8/.9!

",<%! ! !

!

!!!'(!",<%!-!=(>!?(@?/!AB(! !.2CB(3!>D.!.2CB2(31(.!1E(.!2CBD>=D=(.!

",;%! !

",)%! !

",+%! !

!

!!!'(!",;%!7!",+%!.(!/012(3(!AB(9!

",#%! ! !

!

!!!'D=/!AB(! *!.(!=(=B@(!=(!",#%!AB(9!

",,%! !

!

!!!'(!",+%!-!",,%!.(!=(=B@(!AB(9!

",$%! !

!

!!!'(!",)%*!",#%!-!",,%!.(!/012(3(9!

",&%! !

!

!!!F(@/E=(8/.9!

"#,%! !

!

"#$%! !

!

"#&%! !

!

!!!'(!"#,%*!"#$%!7!",&%*!.(!=(=B@(!AB(9!

Page 12: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!614

!

"#$%! !!&!!! !

!

!!!'(!")*%+!"##%!&!"#,%!-(!.(./0(!1/(2!

"#3%! !!!&!!! !

!

!!!45! 06507/-895+! (7! :(5-6;! <=:;806! (5! (7! -8-:(<>! 56! :;>5-?6;<>.6! (-! (@>0:><(5:(! 8.=5:806! >7!

:(5-6;!<=:;806!(5!(7!-8-:(<>!:;>5-?6;<>.6+!>!->A(;B!

!

",C%! !

!

!!!!"#$%&'()&#*+#+,-+# -+&,./#01-/('.# 2#+&#3"#/+3"-(4(*"*#+,5+'("32#,+#/+*%'+#"#,%6(/#7#

6"8"/# 9&*('+,# +&# +3# :3;+6/"# -+&,./("3! &! (@D;(->;! (5! ?6;<>! 06<D>0:>! (7! D;6./0:6! (-0>7>;! .(!

:(5-6;(-2!

!

6. 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!

Page 13: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 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!

"9H%! !

!

!!!I)=2'3'.()!">E%!:!">;%-!2)3!+.!5'()!:-!2)3!)03)!5'()-!"9H%!/,!1)=23+,<'!*+,! !:! !!"!/,!

03'./J)3='.!/,7K.!5'!03'./J)3='14L.!(,!M)3,.0N!(,5!@,10)3!(,!2)/414L.! -!,/!(,143-! -!

Page 14: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!616

"#$!#%&'(!)'*'+&'(,!$#!(-!"#.)#&%'$!"#.#!"/'0&12-"%#&-(3!4'&'!&-.-01'&!-(%'!(1%/'"15$!2'.#(!'!

0-61$1&!-*!(17/1-$%-!"/'0&12-"%#&!0-!2-*#"10'08!

9:;<! ! !"#

"!

!

!!!-$!0#$0-! !-(!-*!%1-.)#!.-010#!-$!-*!(1(%-.'!0#$0-!*'!)'&%="/*'! !-(%>!-$!&-)#(#,!)#&!-?-.)*#!

-*!(1(%-.'! 3!

!

!!@-"#&0-.#(8!

9AB<! ! !

!

!!!C'"1-$0#!*'!%&'$(6#&.'"15$!'*!&-2D(,!0-! !'! !E!%#.'$0#!-$!"/-$%'!F/-! !9*'!)'&%="/*'!

!-(%>!-$!&-)#(#!-$!-*!(1(%-.'! <!(#*'.-$%-!"'.+1'!-*!(17$#8!

9:A<! !

!

!!G-!9:A<!#+%-$-.#(8!

9:B<! !

!

!!!HI#&'!+1-$,!0'0#!F/-!)'&%1.#(!0-*!(/)/-(%#!F/-!*'!)'&%="/*'! !-(%>!-$!&-)#(#!"#$!&-()-"%#!'*!

(1(%-.'! ,!(-!(17/-!F/-8!

9:J<! !

!

!!!G-!9:B<!E!9:J<!(-!#+%1-$-8!

9:K<! !

!

!!G-!9:;<!E!9:K<!#+%-$-.#(8!

9::<! !

!

!!!@-"#&0-.#(8!

9;B<! !!!!!!!!

!

!!!4'&'!/$'!)'&%="/*'!"#$!2-*#"10'0! !.-010'!-$!-*!(1(%-.'! ,!#+%-$-.#(!0-!9;B<8!

9:L<! !

!

!!!G-!9::<!E!9:L<,!#+%-$-.#(8!

Page 15: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 617

"#$%! !

!

!!!&'!()*)!+,'!-./!0,.12)!0)(3)4'41'/!"12'/!'/3.05.-'/!6!,4.!1'(3)2.-%!*'! 7!

"89%! !

!

!!!:!0)4154,.05;4!0)(32)<.2'()/!+,'! !'/!,4!0,.*25='01)2>!?'0)2*'()/7!

!

"@8%! !!6!!! !

!

!!!&.*)!+,'!'4!-./!0))2*'4.*./!'/AB250./C!3.2.!,4!A)1;4!+,'!=5.D.!*'/*'!'-!0'412)!*'!-.!'/A'2.!.!-.!

/,3'2A505'!'/!=E-5*.!-.!.A52(.05;47!

"8F%! !

!

!!!3)*'()/!2''/025<52!"@8%!0)()7!

!

"8G%! !

!

&5=5*5()/!-)/!1B2(54)/!'412'! 7!

"8H%! !

!

?'0)2*'()/!+,'7!

"IJ%! !

!

!

!!!&'!"8H%!K!"IJ%!/'!*'*,0'!+,'7!

"8J%! !

!

!!!?'0)2*'()/!+,'7!

"#8%! ! !

Page 16: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!618

"#$% !

!

!!!&'!"#(%!)!"#$%!*'!+,-.'/'0!

"#1%! !

!

!!!2'3+45'6+*0!

"71%! !

!

!!!!&'!"71%!)!"#1%!+,-'/'6+*0!

"#7%! !!"5.6'/*.8/0!*'9:/5+*%!

!

!!!;<+4=!,.'/>!!"!#$%&'$(")(#$ $*#+#$ $,#)$-)."/-")(',0$1#!'+#,$*#)*2&-/$%&'0$')$'2$',1"*-#$

!'$ 3-)4#5,4-$ 6,-)$ *"+1#,$ 7/".-("*-#)"2',80$ ("+9-:)$ $ ',$ -)."/-")('?! @=! *.5+! /'3'*=4.+!

3+6A4+,=4!'*-+>!A=4=!3+6A4+,=4>!=!3+/-./:=3.8/>!B:'! !'*!:/!3:=54.C'3-+4?!

!

!!!2'3+45'6+*0!

"DE%! !

!

!!!&'!"DE%!+,-'/'6+*0!

"##%! !

!

!!;"!#$%&'$ $',$-)."/-")('0$1#!'+#,$!-.-!-/$2#,$(:/+-)#,$')(/'$ <$

"#F%! !

!!!!

!!!2'3+45'6+*!"77%!

"77%!";%! !!!!!!

!!!"G%! !!!!!!

)! "H%!! !

!

!!!I:*-.-:.6+*!'/!J=!'3:=3.8/!"77H%!J=!"##%0!

"FK%! !!

!

!!!L!*:*-.-:.6+*!"77G%!'/!"FK%0!

Page 17: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 619

"#$%! !&!

!!!!"#$%&#'"()*%&+,#$%&# #-&#.*,/-0"*(,#.&/-"*1,!(&/.&&!!"#$%&'()%$&*#+)'(,$"*-%

%

!!!'()! *+,-+./(! 0! 10! 2+34)4.45)! 2+1! /4+6-(! -*(-4(&! &! ! 7+6(,! 2+! /(60*! +)! .8+)/0! 98+!

,(106+)/+!+,!.(**+./0!+)!+1!+,-0.4(!2+!:4);(<,;4&!2()2+!1(,!(=>+/(,!,+!68+?+)!,4)!0.+1+*0.45)!

.08,020!-(*!01@A)!.06-(!@*0?4/0.4()01B!C0!2+34)4.45)! !2+!10!+.80.45)!"DE%!+,!8)!.0,(!

68F! -0*/4.810*&! ,(106+)/+! ?G142(! .80)2(! )(! +H4,/+! 8)! .06-(! @*0?4/0.4()01! 38+*/+B! I+*(&!

.&,&'(/*&,)&%/#+%#01&)#+%+&%*"&2&,%(3&/&'($#+%#%4'&,($#+%5#'%3(*5#+%.'(26)(36#,(/&+7% /#%!"&%

,#+%//&2(%(%/(%)&#'8(%$&%/(%'&/()626$($%.&,&'(/B!J)!/01!.0,(&!10,!.(6-()+)/+,!2+!10!240@()01!+)!10!

6K/*4.0! 2+! 10! +.80.45)! "LM%! .06=40)! 01! 4)/+@*0*,+! 30./(*+,! 98+! *+-*+,+)/0)! 10! .8*?0/8*0! 2+1!

+,-0.4(N/4+6-(B!

!

!!!

7. LAS ECUACIONES DE MAXWELL Y LA FUERZA DE LORENTZ SON INVARIANTES BAJO LAS TRANSFORMACIONES DE LORENTZ !

!!!J)! +,/+! 0-0*/02(! ,+! 0)014O0! 10! 4)?0*40)O0! 2+! 10,! +.80.4()+,! 2+! :0H<+11! F! C(*+)/O! =0>(!

/*0),3(*60.4()+,!.((*2+)020,!+)!+1!+,-0.4(!2+!:4);(<,;4B!

!

7.1. LA TRANSFORMACIÓN DE LA FUERZA DE LORENTZ !

J)! +1! +,-0.4(N/4+6-(! .80/*4246+),4()01&! +1! /+),(*! 2+1! .06-(! +1+./*(60@)K/4.(! ,+! 2+34)+! -(*!

6+24(!2+!10!,4@84+)/+!60/*4OP!

"$% !Q!/+),(*!2+1!.06-(!+1+./*(60@)K/4.(!!

!

R+.(*2+6(,!10!6K/*4.0!2+!:4);(<,;4!.()!10!.()?+).45)!2+!,4@)(,!S&S&S&NP!

"T%! Q!6K/*4.0!2+!:4);(<,;4!

!

'(6=4)0)2(!"$%!F!"T%&!,+!(=/4+)+B!

Page 18: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!620

"#$! !

!

!!%&'()*&+(,!-.&!&/!'.0*)12&'3()!2&/('1*0*!,&!*&415&!'(+(!,16.&7!

"8$ !

!!!&5!*(5*&!"&'.0'1(5&,!"98$!:!";<$!*&/!0=>5*1'&!*&!/0!)&/03121*0*!&,=&'10/$7!

:!"98$! !

";<$! !

!

!!?5!4()+0!&@=/A'130B!&/!'.0*)12&'3()!2&/('1*0*!,&!&@=)&,0!+03)1'10/+&53&!'(+(!,16.&7!

!

"C$ !:! !

!

D(+E1505*(!"#$!:!"C$B!(E3&5&+(,!/0,!'.03)(!'(+=(5&53&,!*&!/0!,16.1&53&!+03)1F7!

"#$! !

"9$ !

Page 19: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 621

"#!$%&!'#!(#()*#+!$,&-!$.&-!$/&!0!$12&3!

$,& !

$.& !

$/& !

$12& !

!!!

45'!#*)5*67+#'!$,&!5!$/&-!+7'!(5+!85!9)#:;5!(#!47:#+<;!<:6(6=#+'67+58!-!!

:#*7:(#=7'3!!

$11&! !

!

!!"#! 85'! #*)5*67+#'! $,&-! $.&-! $/&!0! $11&!0! ')>7+6#+(7! !>5:5!?5@5'!A#87*6(5(#'-! '#! '6B)#! 85!

9)#:;5!(#!47:#+<;!#+!#8!#'>5*67!<:6(6=#+'67+58-!#+!#8!'6'<#=5!(#!)+6(5(#'!6+<#:+5*67+583!

$1C&! !$5!(69#:#+*65!(#!D5*E'7+!$1///&-!FGH#(6*6I+-!>JBH!KKF-!*7+!#8!'6'<#=5!(#!

)+6(5(#'!B5)''65+7-!(7+(#! &H!

!

!!!45!:5;I+!(#!*5=?67!(#!85!#+#:BL5!(#!85!>5:<L*)85!'#!(#96+#!*7=73!

$1FM&! !N)#!>5:5!#8!*5'7!+7O:#85<6A6'<5!$ &!'#:J3!

$1FP&! !

!

!!!Q7:!87!<5+<7!85!*)5:<5!*7=>7+#+<#!$85!#*)5*6I+!12&!'#!6(#+<696*5!*7+!85!:5;I+!(#!*5=?67!(#!85!

>5:<L*)85!:#85<6A6'<53!

$1R& !

!

!

S#*7:(#=7'!N)#! !0!')'<6<)6=7'! !!#+!$1C&3!

$1F& - -! !

$1R&! - -! !

$1K&! !!!!!$ &!!

Page 20: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!622

"#$% & &! !

!

7.2. LA TRANSFORMACIÓN DE LOS CAMPOS ELÉCTRICO Y MAGNÉTICO !

'()*+!,-!./01,2+/*0! 3!4-!50!,6786-1.-!*01-/09!

"#:%! !

!

;-<+/=-*+,!-5!>./01,2+/*0=+/?!=-!50,!+@-/0<6+1-,!=-!./01,2+/*0<6)1!=-!A+/-1.B9!

"#C% !

"#%!!!! !

!

!!!D8<E+,!.F/*61+,!,+1!<-/+&!0!,0G-/9!

"#H%! 9!

I,.+!20<656.0!-5!<J5<85+&!@+/!-K-*@5+9!

"#L%!

!!

!

M./+,!.F/*61+,!1+!,+1!<-/+&!@+/!-K-*@5+9!

"$N%! !

!"# $%&"'%# %&()*+%# ,"# -%)-.)%&# )*,# /"$(,# 01'$2&*,! @0/0! 55-70/! 0! 50! ,6786-1.-! *0./6B&! O8-!

/-@/-,-1.0! 50,! -<80<6+1-,! =-! D0PQ-55! =-5! <0*@+! -5-<./+R*071F.6<+&! ./01,2+/*0=0,! <+1! 50!

./01,2+/*0<6)1!=-!A+/-1.B&!@0/0!+G.-1-/!-5!.-1,+/!=-!./01,2+/*0<6)1!-1!-5!,6,.-*0!G0//0! 9!

Page 21: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 623

"#$%!

!

!

&'!()*(*+!,-)!.-!(*/(-+!0'!1/!('/2-)!*/(342356()37-+!7-/!()*8*!7')-9! !:'!"$%!2'!0'017'+!*0'5;2+!

<1'!'/!'.!232('5*!=*))*! >!

"##%!!! !

!

!!!:'! "#$! ?! "##%! 2'! 0'017'! !"# $%"&'()%*"+,-&# ./# !)'# +"*0)'# /!1+$%,+)'# 2# *"3&1$,+)'+! ,-)!

'@'5,.-!")'71')0*!<1'! %!

"#AB%! !!!!!!!!!!!!!!!!!!!?!"#AC%!! !

"#DB%! !?!!"#DC%!! !

"#EB%! !!!?!!"#EC%!!! !

"#FB%! !!!!!!!!!!!!!!!!!!?!!"#FC%!!! !

"#GB%! !?!!"#GC%!!! !

"#HB%! !?!!"#HC%!!! !

!

:'!"$G%!?! +!-=('/'5-2!.*!7-))3'/(' !?!.*!0'/230*0!0'!7*)I*! !!'/!!'.!

232('5*!7-/!=*))*! +!'/!0-/0'! !'2!.*!J'.-730*0!?! !.*!0'/230*0!0'!7*)I*>!

"#KL%! !!!!!!!!!"#KC%! !

"AML%! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"AMC%!! !!!!

"A$L%! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"A$C%!! !!!

"A#L%! !!!!!!!!!!!!!!!!!!!"A#C%!!! !!!

Page 22: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!624

"##$%! !!!!!!!!!!!!!!!!!!!!!!!!"##&%!! !!!

!

7.3. LA TRANSFORMACIÓN DE LORENTZ DE LAS ECUACIONES DE MAXWELL !

$'()*+,-.! (+! /0+1.2-0,+*)31! 45! 6-051/7! "5*8+*)31! 9:;<%! +! (+.! *8+/0-! *--0451+4+.! '+0+!

=5(-*)4+45.!+(/+.>!

";<% !

?-,+14-! 51! *851/+! @85! A! 5./5! 5B50*)*)-! 1-.! 4+! (+.! *--0451+4+.! 51! 5(! .)./5,+! A!

5C'05.+4+.! 51! /D0,)1-.! 45(! .)./5,+! :! ?+,E)D1! '-45,-.! '0-*5450! +(! 05=D.A! F! 5C'05.+0! (+.!

*--0451+4+.!45(!.)./5,+! !51!/D0,)1-.!45(!.)./5,+! A!(-!@85!),'()*+!81!*+,E)-!45!.)G1-!"45!H!

+!I%>!

"#J$%! !!!!!!!!!!!!!!!!!!!!!!!!"#J&%! !

"#K$%! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#K&%! !

"#9$%! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#9&%! !

"#<$%! !!!!!!!!!!!!"#<&%! !

"#L$%! ! !!!!!!!!!!!!!!!!"#L&%!! !

!

$M-0+! /-*+!4548*)0! (-.!-'50+4-05.! !F! :!N+0/),-.!45! (+.! 5*8+*)-15.! "#J&%!F! "#L&%:!O1!

+,E+.!281*)-15.! A!(+.!=+0)+E(5.!45'514)51/5.! !F! A!05.'5*/)=+,51/5A!.-1!81+!281*)31!45! !F!

:!$'()*+14-!(+!05G(+!45!(+!*+451+A!-E/515,-.>!

"#P%! !

"JQ%! !

!

!

R50)=+14-!"#J&%!F!"#L&%!.-E05! A!-E/515,-.>!

"J;%! !

"JS%! !

Page 23: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 625

!

"!#$%&'()#*!+(,!-&,-(,!./)0&*)$,!,*1%$! 2!*13$)$-*,4!

5678! !

5668! !

!

9/,3&3/&-*,!56:8!;!56<8!$)!57=82!>(%(!*13$)$%!$+!*>$%(#*%! 4!

56?@8!! !!;!!!56?A8! !

"!,/,3&3/&-*,!!5678!;!5668!$)!56B8!>(%(!*13$)$%!$+!*>$%(#*%! !

56C@8!! !!!;!!56CA8! !

D*%!%(E*)$,!*1'&(,4!

56F@8!! !!!;!56FA8! !

56G@8!! !!!;!56GA8! !

!

H$0*%#$-*,! (I*%(! +(,! $0/(0&*)$,! #$! J(KL$++! $)! $+! ,&,3$-(! M! 9&! +(,! $0/(0&*)$,! ./$%()!

&)'(%&()3$,!$)!$+!,&,3$-(! 2!*13$)#%N(-*,4!

OPQM! !!!!5+$;!#$!R(/,,8!

OOPQM! !!!!5+$;!#$!

&)#/00&S)!#$!T(%(#(;8!

!

OOOPQM!! !!!!!5+$;!#$!R(/,,!$)!-(U)$3&,-*8!

Page 24: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!626

"#$%&! !!!!!

'()*!+)!,-./0)1!

$2! )(! 3456! +)! ! ""$%! ! )! ! "#$%! ! ! 34+4! 726! +)! (65! 36-.62)28)5! +)9)! :4()0! 3)06&! ;7)0)-65!

36-.06940! <7)! (45! )3743=62)5! +)! >4?@)((! 0)4(-)28)! 562! =2:40=428)5! )2! )(! )5.43=6! +)!

>=2A6@5A=B!)-.)C42+6!362!(4!()*!+)!D4755!'"$%1&!E758=87=-65!'FGH1B!'FIH1B!'FJH1!*!'GGH1!*!'IJH1B!

'IKH1!*!'ILH1!)2!'IM1!.404!698)2)0!'JN1O!!

'IM1! !!!!

'JN1! ! !!!

!!!!!!!!!!!!! !

!!!!!!!!!!!!!! !

!

%4+6! <7)! ! '()*! +)! D4755! )2! ! 5=58)-4! E1! *! !

'36-.62)28)! !+)!(4!()*!+)!,-./0)!)2!5=58)-4! 1B!5)!5=P7)!<7)O!

'JQ1! !!!"#"$"%&'()%*#$+%

!

,R604!.063)+)-65!362!(4!3600696043=S2!+)!(4!R=.S8)5=5!+)!<7)!(4!()*!+)!=2+733=S2!+)!T404+4*!

)5!=2:40=428)&!E758=87=-65!'FGH1!4!'FLH1!*!!'IUH1!)2!!""$%!.404!698)2)0O!

'FGH1! !!!!!!!!!!!!!!!!!!!!

'FIH1! !!

'FJH1! !!!!

'FUH1! !!!!!!!!!!!!!!!!!!!

'FKH1! !!

'FLH1! !!

'IKH1!! !!!*!'IKV1! !

Page 25: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 627

"#$%&!! !!!'!"#$(&! !

"#)%&!! !!!!

!

*+,-./!012-+1.!+3!012-+1.!4+!3./!51+/!6.-0.7+75+/8!9:+1+-./!6.-01.;,1!<:+=!

">?& !

">@&! ! !

">#&! !

!

A,4.! <:+! ! "BBBCA&! ! '! ! ! "BBCA&D! /+! /2E:+! <:+! +3! 012-+1!

6.-0.7+75+!4+! !!+/!6+1.8!F7G3.E,-+75+!/+!6.-01:+;,!3.!-2/-.!0,1,!+3!/+E:74.!'!

5+16+1!6.-0.7+75+8!H.1!3.!5,75.=!

">>&! !!!!!"#"$"%%%"3+'!4+!274:662I7!4+!J,1,4,'&!

!

A+!"@$&!,!"#@&!.;5+7+-./!3,!51,7/K.1-,62I7!4+!3,/!+6:,62.7+/!4+!L,MN+33=!

"@O&! !!!!!

"#P&! +Q+! !! !!!!!

+Q+!! ! !!!

+Q+!R! ! !!!

"#S& !!!!!

"#?&! !!!!!"3+'!4+!F-0T1+&!

!

!

!

7.4. LA TRANSFORMACIÓN DEL POTENCIAL ELECTROMAGNÉTICO !

FU.1,!;2+7D!1+6.14+-./!<:+!3,!K:+1R,!+751+!4./!6,1E,/!+/=!

Page 26: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

!628

"#$%! !

&'(!)*+,-,.,/-0!*1!.234'!*15.6(,.'0!*-!*1!7,76*32! !*78!

"##%!*-!+'(32!9*.6'(,218 !!"1*:!)*!;'<1'3=%!

>*.'()*3'7! 623=,5-!?<*! *-! *1! 7,76*32! 0! *1! 9*.6'(! ! ")*1! 4<-6'! ! *-! *1! 7,76*32! ! 2! 12!

.2(@2! !*-!*1!'(,@*-!)*1!7,76*32! %0!:!12!32@-,6<)! !)*!*76*!9*.6'(!7'-0!(*74*.6,923*-6*8!

"#A%!! !

"#B%! !

C<76,6<,3'7!"##%!2!"#B%!*-!"$$%!2!"$A%!42(2!'=6*-*(!*1!.234'!*15.6(,.'!3*),)'!*-!*1!7,76*32! 8!

"#D%!! !

"#E% !

"#F%! !

!!!!

>*7<3,*-)'!"#D%!2!"#F%0!'=6*-*3'7!*1!.234'!*15.6(,.'! !*-!*1!7,76*32! 8!

"AG%!*-!+'(32!9*.6'(,218!! !

H2)'!?<*!"##%!:!"AG%!7'-!,@<21*70!.'-762!?<*!*1!.234'!*15.6(,.'0!3*),)'!*-!*1!4<-6'! !4'(!

'=7*(92)'(*7!*-!1'7!7,76*327! !:! !*7!,)5-6,.'I!!

&*('0! *1! .234'! 32@-56,.'! *-! *1! 4<-6'! ! -'! *7! *1! 3,73'0! 9,76'! )*7)*! ! ! '! )*7)*! I!!

>*.'()*3'78!

"$B%!! !

"$D%!! !

"$E%!!! !

!

!!!J-!127!*.<2.,'-*7!"$B%!2!"$E%!(*.'-'.*3'7!*1!4(')<.6'!.(<K!"4(')<.6'!9*.6'(,21%! 0!.'3'!

7*!.'34(<*=2!2!.'-6,-<2.,/-8!

"AL%! !

Page 27: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

! 629

!

!

"#$%&!'()*+!%),$%-).$/!01)2!

3456! !

!

7$.'(*&*-$!3486!9!3456+!$':)*).$/;!

34<6!)*!=$%.&!>),:$%(&?2! !

@),$%-).$/!01)2!

34A6! !

B$%!?$!:&*:$+!

34C6!! !

D$.&*-$!)*!,1)*:&!01)2!

3446! !

E!,$.'(*&*-$!!34<6+!34C6!9!3446!$':)*).$/2!

34F6! !3?)E!-)!G($:HI&>&%:6!

!

!!!!"#$% &'('2! D$-&/! )/:&/! ,1&:%$! ,$%%$'$%&,($*)/! -)! ?&! (*>&%(&*J&! '&K$! L$%)*:J! -)! -(=)%)*:)/!

M%1N$/!-)!),1&,($*)/!>),:$%(&?)/!N$-%O&*!&N&%),)%!(**),)/&%(&/!&?!?),:$%!)PN)%:$!)*!.&:).Q:(,&/+!

-&-$! 01)! ?&/! ),1&,($*)/! )01(>&?)*:)/! )*! =$%.&! :)*/$%(&?! M&%&*:(J&*! !"#$ %"! 01)! /$*!

(*-)N)*-()*:)/!-)!?$/!/(/:).&/!-)!%)=)%)*,(&!(*)%,(&?)/+!-&-&!?&!?)E!-)!:%&*/=$%.&,(R*!-)!$'K):$/!

:)*/$%(&?)/2!

34S6! !

Page 28: APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y …€¦ · !603 APÉNDICE V. LA RELATIVIDAD ESPECIAL DE EINSTEIN Y ALGUNAS TRANSFORMACIONES DE LORENTZ por Juan Auping, con la

630