Incoherence of the relativistic dynamics: E = mc 2 contradicts Special Relativity!

Transcription

1 Inherene f he relaiisi dnais: nradis Speial Relaii! Radwan M Kassir June 15 radwanelassir@dargrup Absra The relaiisi ass nep is red in he prbleai lngiudinal and ranserse ass equains eerging fr he Lrenz ransfrain as presened b insein in his 195 paper n Speial Relaii These equains alhugh aual ues f he Speial Relaii and erified in his paper hrugh bh siplified diensinal analses and nserain f energ priniple had laer been ipliil drpped and replaed b an ad-h relaiisi ass equain needed ainain he nsisen f he Speial Relaii wih he nserain f enu law alhugh i resuls in is ilain f he law f nserain f energ Mainaining he laer law resuls in he sae ranserse ass equain as bained in insein s said paper The relaiisi ass adped in he lieraure is bu an aep neal nradiins in he Speial Relaii and a nenien eans fr arriing a he relaiisi inei energ frula ipling he desired ass-energ equialene equain In his paper he inherene f he Speial Relaii eerging fr is esablished ass frulae is reealed hrugh siplified phsial densrains Depending n he fre definiin and he ing ass equain used fur differen frulae fr he relaiisi inei energ are bained all alidaed fr he Speial Relaii perspeie reaing a derienal inherene in he her All hese frulae are redued he lassial inei energ equain fr << ( eli speed f ligh) I is reealed ha he energ equain is n a alid nsequene f he Speial Relaii Kewrds: Speial Relaii lngiudinal ass ranserse ass relaiisi ass relaiisi enu Newn s send law relaiisi inei energ 1 Inrduin In his 195 paper 1 n he Speial Relaii insein predied (fr he Lrenz ransfrains fr he spae-ie and eleragnei field pnens) he lngiudinal and ranserse ass f ing elern as funins f is eli exended pnderable aerial pin as easured in he sainar sse This was based n defining he fre aing n he elern as being equal ass aelerain (Newn s send law f in) The lngiudinal ing ass bained as suh alng wih he enined fre definiin resuled in he relaiisi inei energ f he aerial pin ing in he lngiudinal direin wih a eli as being ( γ ) ( ) 1 where is he aerial pin res ass he speed f ligh and 1 γ( ) (1 ) Hweer in his nex γ ( ) was n he predied ass f he ing aerial pin whih was raher γ ( ) Thus here was n suh ipliain as he ass-energ equialene whih insein aeped densrae in laer wrs fr he abe inei energ equain In addiin he ranserse ass as well as he lngiudinal ass desn saisf he nserain f enu wihin he Speial Relaii fraewr Thus he Speial Relaii deried direinal relaie ass equains were laer ipliil drpped and replaed b he relaiisi ass γ ( ) required fr he nserain f enu If he relaiisi ass was used in deriing he relaiisi inei energ

2 Radwan M Kassir June 15 γ( ) 1 equain he equain ( ) wuld be bained if he fre was raher defined as he enu hange rae ( fre d ( ) ) equialen he frer definiin ( fre d) if he ass was inarian In suh a ase he inei energ equain bees wih he ass-energ equialene ipliain Hweer he relaiisi ass being equal γ ( ) nradis he aual Speial Relaii prediin f he lngiudinal (as well as ranserse) ass based n he Lrenz ransfrain I is usar nlude he relaiisi ass as being γ ( ) fr he nserain f enu priniple applied lliding pariles fr he perspeie f w inerial fraes in relaie in In he presen siplified apprah he ranserse eli f a bd ing ransersall relaie he raeling frae is redued b a far f γ ( ) in he sainar frae arding he relaiisi eli addiin r as a nsequene f he ie dilain alhugh here is n relaie in in he ranserse direin beween he fraes This will resul in unjusified ranserse enu derease (b a far f γ ) in he sainar frae relaie he ing ne Hene b he eans f he nserain f enu law he ass shuld be saled up b a far f γ ( ) in he sainar frae pensae fr he enu lss The adped relaiisi ass equain γ( ) is herefre an adh ipleened renile he nserain f enu law ha wuld herwise be ilaed b he Speial Relaii i is n a naural prediin f he Speial Relaii and innsisen wih bh he ranserse and he lngiudinal ass predied b he Lrenz ransfrain The ing ass in he Speial Relaii Cnsider w inerial fraes wih rdinae Kxz and ( ) sses ( ) ξζτ in relaie in wih eli The frae K is nsidered be he sainar ne (ie he her frae is being bsered fr i) Assue he fraes are under he influene f a unifr eleragnei field Le ( XYZ ) and ( X Y Z ) be he eleri field [er] pnens as easured in he sses K and K respeiel while ( LMN ) and ( L M N ) are he rrespnding agnei field pnens Le here be an eleriall harged parile in in wihin he field Assue a an insan f ie se as τ he parile is a he K sse rigin ing alng wih rigin a he sae relaie eli A his iniial ie he parile is a res relaie he sse Fr an infiniesial elapse f ie he in f he parile an be desribed fr he perspeie b he fllwing equains dξ ε X (1) d ε Y () dζ ε Z () where and ε are he res ass and he harge f he parile respeiel Arding insein s 195 paper 1 wih he help f he Lrenz ransfrain fr he spae-ie rdinaes ξ γ ( x ) ζ z τ γ( x ) and fr he eleragnei field pnens X X Y γ( Y N ) Z γ( Z+ M ) qs (1) () are ransfred in he sse K as ε γ dx d dz X ε Y N γ ε Z+ M γ whih an be wrien in he fr

3 Inherene f he Speial Relaii Dnais dx γ εx ε X (4) d γ εγ Y N εy dz γ εγ Z M εz + (5) (6) εx εy and ε Z are as insein pu i he pnens f he pnderie fre aing upn he harged parile r sipl he fre aing upn a aerial pnderable pin Therefre ainaining he equain ass aelerain fre qs(4) (6) in whih he send deriaie ers are he parile aelerain pnens as easured fr he sainar sse ipl he parile s raeling ass easured fr he sainar sse an be gien b Lngiudinal ass l ( ) 1 γ l (7) Transerse ass γ ( ) 1 (8) quains (7) and (8) an be erified using siple diensinal analsis In fa a lngiudinal fre f in he sse aing upn a resing ass l has he diensinal fr f ass lengh ie d f l τ whih b he help f he Speial Relaii prediin f lengh nrain and ie dilaain is iewed fr he sse K as D F l d γ l l ( γ τ) Saisfing he requireen f F f q (4) we l l ge γ erifing q (7) l Siilarl a ranserse fre f in he sse aing upn a resing ass fr f ass lenghie h f τ has he diensinal whih b he aid f he Speial Relaii prediin f ie dilaain and inarian ranserse lengh is iewed fr he sse K as h h F ( γ τ) Saisfing he requireen f F f q (5) r (6) we ge γ erifing q(8) Relaiisi ass prediaens 1 Relaiisi ass b nserain f enu Cnsidering ur sses Kxz ( ) and ( ξζτ ) le here be a pnderable aerial pin r a bd f ass raeling a eli w in he ranserse direin wih respe he sse The bd is a res in he lngiudinal direin relaie ( w w ) Suppse he enu ξ ζ p w f he bd is jus suffiien brea a glass hip a res in and inereping he bd s in Arding he relaiisi eli addiin he ranserse pnen W f he bd s eli wih respe K is gien b w γ If he ass was inarian he bd s ranserse enu relaie K wuld be P W w γ p γ insuffiien brea he hip reaing an absurdi Thus he bd s enu relaie K us be greaer han r equal is enu in Hene is ass

4 Radwan M Kassir June 15 4 us be γ Nw assuing he Bd s enu in is jus belw he hreshld required brea a differen glass hip he rrespnding enu P relaie K us hen be equal r less han is enu p in Hene is ass us be γ I fllws ha he sae bd s ass easured fr he sse K uld be eiher γ r γ depending n se areful selein f he glass hip in eiher ase Hene γ us hld in rder fr he w resuls agree n he ass aing he bd s ranserse enu relaie K equal is enu in Hene he ass inariane assupin an hld and he bd s ass in K us be nered 1 (9) The abe nlusin uld be arried a using he fllwing arguen The bd s ranserse eli W as iewed fr he sse K is redued b he far γ pared is eli w in due he ie dilain predied b he Speial Relaii Ardingl he bd s ranserse enu wuld be equall redued b he sae far had he ass been inarian Hweer his hange in ranserse enu an be jusified wih he absene f an relaie in in he ranserse direin Hene he bd s ass us inrease b he far γ in rder nsere is ranserse enu In her wrds he ie dilain shuld resul in a ass inrease in rder pensae fr he inurred enu lss Wha an absurdi! quain (9) required renile he Speial Relaii wih he nserain f enu law is in nradiin wih he prediin f he lngiudinal and ranserse ass bained fr he Lrenz ransfrain as gien b q (8) Relaiisi ass b nserain f energ On he her hand assue he nsidered bd is se up in ranserse in in ( ξ ) b exering a nsan fre f in he ranserse direin Le p be he ranserse enu pied up b he bd wihin an ineral f ie τ The exered fre an hen be wrien as f p If we le he ranserse disane τ raelled during ha ie ineral be he wr dne n he bd b he exered fre will be gien b p w τ (1) Fr he perspeie f K he wr dne n he bd is gien b P W (11) where P is he pied up ranserse enu relaie K he elapsed ie and he rrespnding ranserse disane wih respe K As seen earlier he bd enu us be he sae relaie bh fraes Due he inariane f he ranserse spaial diensin and he ie dilain bained fr he Lrenz ransfrain γ( τ+ ξ ) γ τ sine ξ q (11) leads p w W γ τ γ (1) Hene he wr dne n he bd r he absrbed energ depends n he referene frae whih is in nradiin wih he relaii priniple and he nserain f energ law I fllws ha he inariane f he ranserse enu leads he relaiisi ass equain Ye his inariane leads he ilain f he energ nserain T ainain he energ nserain we us hae using qs (11) and (1) P γ p r W γw ielding w γ γw γ r (1) 1

5 5 Inherene f he Speial Relaii Dnais nradiing he relaiisi ass γ bained in nnein wih he enu nserain e nfiring he ranserse ass eq(8) presened b insein (195) as bained b he eans f he Lrenz ransfrain 4 Kinei energ equains under fre ass x aelerain Le here be a pnderable aerial pin all i a bd f res ass aed upn b a nsan fre F in he ranserse direin Suppse a he insan f ie he bd is a he sse K rigin Le be he eli f he bd a he ie and is ass all wih respe K Relaie he sse raeling a he eli he ass f he bd a he ie insan wuld be (bd is a res in a his insan) The inei energ aquired b he bd a his ie is gien b Fdx Appling Newn s send law f in ( F dx d) he inei energ an be wrien as d ( dx ) Sine dx we ge d (14) idenl if he ass was nsan q (14) will resul in he lassial inei energ equain 1 ( ) 41 Using he lngiudinal ass bained fr Lrenz ransfrain As he fre is ipressed in he in direin appling q(7) fr he lngiudinal ass γ bained in insein s 195 paper we ( ) ge d ( ) 1 1 Hene (15) (16) whih is he relaiisi inei energ frula bained in insein s 195 paper based n he lngiudinal ass ( γ ) bained fr he Lrenz ransfrain and he definiin f fre being ass aelerain Sine he firs er f he righ hand side f q(15) is n equal he lngiudinal ass in he used nex f q (7) he laer equain has n ipliain f ass-energ equialene I is jus a frula fr he inei energ Fr q (16) an be wrien he send rder as whih is he lassial frula fr he inei energ 4 Using he relaiisi ass bained fr he nserain f enu Nw had we used q (9) fr he relaiisi ass ( γ ) an ad-h saisf he nserain f enu q (14) wuld gie d 1 1 ( ) 1 1 (17) This is anher frula (nradiing he ne bained earlier) f he relaiisi inei energ based n he relaiisi ass ( γ ) and n he fre being defined as ass aelerain Again fr q (17) an be wrien he send rder as

6 Radwan M Kassir June whih is he lassial frula fr he inei energ 5 Kinei energ equains under fre enu hange rae If he fre aing upn he bd is defined as F d ( ) whih wuld be equialen ur earlier definiin ( F d) had he ass been inarian he inei energ aquired b he bd bees ( ddx ) Hene ( ddx ) ( ddx ) + d d + (18) idenl if he ass was nsan q(18) will resul in he lassial inei energ equain 1 ( ) 51 Using he relaiisi ass bained fr he nserain f enu Using q (9) fr he relaiisi ass ( γ ) and sling i fr as a funin f we bain Hene q (18) leads d+ d 1 d γ + (19) ( γ 1) quain (19) has he ipliain f energ- ass equialene wih being he bd res energ and is al energ (res + inei energ) As shwn earlier fr he laer equain leads he lassial inei energ equain ( 1 ) Neerheless q (19) bained based n he definiin f fre F dp and relaiisi ass γ is n in agreeen wih he lngiudinal ass ( γ ) and fre definiin ( F ass aelerain) adped in he Speial Relaii riginal paper 1 5 Using he lngiudinal ass bained fr Lrenz ransfrain Had we used q(7) fr he lngiudinal ass ( γ ) as bained in insein s 195 paper q(18) wuld ield d+ d ( 1 ) d γ γ γ ()

7 7 Inherene f he Speial Relaii Dnais whih fr an be wrien he send rder as whih is he lassial frula fr he inei energ quain () being based n he Speial Relaii lngiudinal ass deriain fr he Lrenz ransfrain and n he re general definiin f fre as F dp (raher han F d) i is he s represenaie f he inei energ in he nex f he Speial Relaii Ye i is far ff fr ipling he general energ equain based as being he s rearable prediin f he speial relaii her! Cnlusin The Speial Relaii ransfrain equains resul in lngiudinal and ranserse ass frulae nradiing he relaiisi ass equain ( γ ) required ainain he nserain f enu under he Speial Relaii prediins This relaiisi ass nradis in urn he undesired relaie ass bained in saisfing he nserain f energ law under he Speial Relaii in nfrane wih is ranserse ass equain deried fr he Lrenz ransfrain In defining he fre as ass aelerain eah f he afreenined lngiudinal and relaiisi ass frulae resuls in a differen equain fr he inei energ The frula bained under he Speial Relaii assupins desn auall ipl he laied equialene f ass and energ On he her hand if he fre was defined as being he enu hange rae anher w differen frulae fr he relaiisi inei energ will be bained ne fr eah f he w afreenined lngiudinal and relaiisi ass frulae All hese frulae lead he lassial inei energ equain when << The relaiisi inei energ bained under he relaiisi ass ( γ ) and he fre as being he enu hange rae is gien b he frula ipling he faus equain fr he al energ: Hweer i is n in agreeen wih he genuine Speial Relaii prediins and ipreisel aribued i n frgeing ha he her hree differen bained frulae are all alidaed fr he Speial Relaii perspeie reaing a derienal inherene in he Speial Relaii In addiin he relaiisi inei energ equain is based n he relaiisi ass ensuing fr he nserain f enu law in he Speial Relaii fraewr bu resuling in is ilain f he law f nserain f energ 1 A insein "Zur elerdnai beweger Körper" Annalen der Phsi (1) (195) A insein "Des he Ineria f a Bd Depend Upn Is nerg Cnen?" in A insein e al (195) pp (195) A insein HA Lrenz H Minwsi and H Wel The Priniple f Relaii (Der New Yr 195) A insein "leenar Deriain f he quialene f Mass and nerg" Aerian Maheaial Sie Bullein 4 (195)