The Full Mathematics of: SPECIAL Relativity. By: PRASANNA PAKKIAM

Transcription

1 The Fll Mahemais of: SPECIL Relaii : PRSNN PKKIM

2 CONTENTS INTRODUCTION 3 TIME DILTION & LENGTH CONTRCTION 4 LORENZ TRNSFORMTIONS & COMPOSITION OF VELOCITIES 6 RELTIVISTIC MSS RELTIVISTIC ENERGY 3 ibliograph 5 GLOSSRY 6 PPENDIX of SYMOLS 7

3 INTRODUCTION This domen ses a sraighforward roe o ndersand he deriaion of erain oneps of Speial Relaii. Noe ha his domen does no se he idea of 4 Veors in order o derie an of he eqaions efore Einsein s heories, i was belieed ha ligh was a wae ha reqired a medim. This medim was belieed o be Eher. Ligh did no hae a limi. Howeer, Mihealson and Morle onded an eperimen ha was repeaed again and again in he aim o aain a resl ha showed ha iewed he speed of ligh o moe faser. Howeer, he eperimen ielded a onsan ale, no maer how faser he obserer was moing. So Einsein ame p wih Speial Relaii o eplain his; he had o modif onenional iews of ime and spae o aommodae for his onsan of he speed of ligh. 3

4 TIME DILTION & LENGTH CONTRCTION Consider he following apparas: DEVICE ha: Emis Ligh Cliks when i reeies ligh MIRROR The ime for lik is gien b: d Now; if he apparas moes o he righ, he ime for Clik from he iewpoin of a Saionar obserer is gien b (noie he diagonal pah aken b he ligh beam): d h d h h lso: d Inegraing wih respe o d Soling Mahemaiall: 4

5 : Phagoras Theorem d h Sbsiing This ields he TIME DILTION EQUTION. Now onsider he same apparas eep wih a bar adjoining he mirror and he deie: le: h eqaing he resls Original Lengh (of he bar) in a Saionar frame of referene Lengh (of he bar) in a Moing Frame of Referene ar This ields he eqaion for LENGTH CONTRCTION. d h 5

6 LORENZ TRNSFORMTIONS & COMPOSITION OF VELOCITIES The Loren Transformaions an be hogh of as onersion of erain Phsial Qaniies inoling ime, beween Frames. For eample, in lassial phsis, if an obje s eloi is being measred, he eloi of he referene frame is added o he eloi seen: Obje s Lengh: d Viewer s Veloi: The iewer measres he lengh, d d d UT if d d d as: The Loren Transformaions are obiosl more omple, b he firs ake he same form: where : disane onsans inoling he referene frame... In he beginning:,. This is bease he Frames sar from he origin. The origin (,) of eah Frame of Referene oinides ogeher in he same plae in - Dimensional Spae. The siaion, js when he Cloks are sared... Soling Mahemaiall: b Sbsiing ino Noe ha his is er similar o he Classial heor, eep for he mliplier. lso, sine he laws of Phsis look he same in all referene frames : Sbsiing: a 6

7 7 The ime aken of a erain een in reorded in he frames is gien b: If i were an insananeos een: he Time Dilaion Eqaion: Sbsiing: or Sin e: Sbsiing: a Sine d, he Disane Transformaions and he Time Transformaions shold be modified like in an siaion, where ales are aken for a rae of hange. Hene, he smalles infiniesimal ales ms be sed o epress hese eqaions:

8 8 Now all he eqaions are faorised b a ommon faor of. Sine: Sine: Sine: One of he famos eqaions deried from he firs of he aboe eqaions is known as he VELOCITY COMPOSITION LW:

9 9 w

10 RELTIVISTIC MSS Consider he following diagram below: saionar obserer obseres a ollision beween, Parile and Parile : Saionar Now he same ollision is iewed from a moing referene frame. This frame moes a he same eloi as Parile : Moing Using he Veloi ransformaion Eqaions: Sine If and he masses of he pariles were he same i wold ield: P P m m s opposed o Newonian mehanis

11 Ths Momenm is no onsered in relaie frames. So if he masses were named: and M : M M Sine: Sbsiing: M M This shows ha if he priniple of relaii is o appl; he mass ms hange b an amon sh ha he onseraion of momenm is re. NOTE: The following ses of eqaions will se he Veloi Transformaion Law ha applies for horional eloiies: In he saionar referene frame (refer o diagram) i is lear ha: : Now an be epressed in erms of ompleing he sqare... M

12 4 4 The pls is ignored as i is nneessar laer M Now his eqaion is sbsied ino M : M M M M M M Now Obje is a res (in he moing frame), hs is mass is known as he Res Mass wih he smbol m. Now in he beginning he masses were idenial (i.e. when here was no eloi). as he sared o moe, Obje wold pereie Obje s mass differenl, his is de o he relaie eloi i is raelling in his is known as Relaiisi Mass smbolied as: m. Sine he eloi gien in he aboe eqaion orresponds o an obserer who is saionar wih he Obje (i.e. moes wih i ), i is fair o label i:. m m The more onenien noaion is: m m

13 RELTIVISTIC ENERGY There are man was o reah he final Eqaion some inoling he inomial Theorem of Epansion., here is a proof ha js ses simple differenial alls. I sars off wih he eqaion for Relaiisi Mass: m m m m m m m m 3 m m m 3 3 m Sbsiing m m m 3 wih m m m m m m Sine: m m m m m F m Sbsiing m m m m m m m m F ma m m m F b hain rle m m F 3

14 Seperaion of Variables F m m F m m Sine: E F d F E m E m E m E m E m m m Sbsiing he eqaion of Relaiisi Mass E m E m if - i.e. a res: E m 4

15 Wiki-ooks Speial Relaii Dnamis ILIOGRPHY 5

16 GLOSSRY DISTNCE: Disane is a basi Fndamenal SI Uni. I anno be defined. Howeer, i an be hogh of as he measre of he amon of spae beween poins. EINSTEIN S FIRST POSTULTE: The Laws of Phsis are he same in all Referene Frames or There is no Uniersal Referene Frame EINSTEIN S SECOND POSTULTE: The eloi of he speed of ligh in free spae is a onsan in all Referene Frames ENERGY: Work is done when a fore as pon an Obje. Energ is he apai o do work. Howeer, Einsein showed ha an obje need no do work in order o possess an energ if he obje has an mass, hen i an hae Energ. FORCE: The aion or inflene ha ases ha aeleraion or hanges in shape of an obje. INERTI: proper of Maer ha ases i o resis an hange in Veloi or Direion. LENGTH CONTRCTION: The shrinkage in lengh obsered when an obserer in a saionar frame obseres a moing obje. MSS: Mass is a Phsial Qani ha measres he amon of Ineria a bod onains. MTTER: nhing ha opies free spae and has aribes of Grai and Ineria MOVING REFERENCE FRME: Saionar Frame of Referene is when an een is being iewed from an obserer who is saionar. I is smbolised b js he smbol of he qani being referred o: e.g.: wold js be wrien as REFERENCE FRME: Referene Frame old be hogh of as an obserer iewing an een: see Moing Frame of Referene, Saionar Frame of Referene. RELTIVISTIC MSS: The inrease in mass of a mass as he eloi inreases STTIONRY REFERENCE FRME: Moing Frame is when an een is being iewed from an obserer who is moing a a erain eloi. I is smbolised b he smbol of he qani being referred, followed b an aposrophe: e.g.: wold js be wrien as or. VELOCITY: Veor qani ha desribes he Rae of hange in displaemen per ni s of ime:. when here is a lear one in a posiie linear plane in -Dimensional d Spae, i an be epressed as:. TIME: Time is a basi Fndamenal SI Uni. I anno be defined. Howeer, i an be mahemaiall defined parameriall as a forh dimension. Time an be hogh of as he seqene of inerals in whih he eens or in 3-Dimensional Spae js like a Parameriall defined fnion ha is drawn for eah ineral of. 6

17 PPENDIX of SYMOLS, lengh, ime m, mass, eloi, speed of ligh, d, disane F, fore E, energ,, momenm, This Eqaion will be laer sed in a Sbsiion, This Eqaion is he Final Eqaion, n infiniisimall low nmber - as sed in Leibni Noaion For a Phsial Qani, q q, q, s obsered in a Saionar Referene Frame q, s obsered in a Moing Referene Frame 7