Newonian Relaii A referene frame in whih Newon s laws are alid is alled an inerial frame Newonian priniple of relaii or Galilean inariane If Newon s laws are alid in one referene frame, hen he are also alid in a referene frame moing a niform eloi relaie o he firs ssem Ths his moing frame is also an inerial frame
Consider Newonian Relaii
Newonian Relaii Galilean ransformaion Noe ime is he same in boh ssems 3
Newonian Relaii Noe Newon s laws are alid in boh frames The fore and aeleraion are he same in boh frames There is no wa o dee whih frame is moing and whih is a res d d F ma m m ma d d d d d F ma m m m d d d F F 4
Loren Transformaion We immediael see he Galilean ransformaion is inonsisen wih Einsein s poslaes If he eloi of ligh in frame K, he eloi of ligh V in frame K The Loren ransformaion saisfies Einsein s poslaes and also redes o he Galilean ransformaion a low eloiies A deriaion is gien in Thornon and Re p30-3 5
6 Loren Transformaion / / where β
7 Loren Transformaion Time dilaion reisied Le Δ be he proper ime ineral measred b a lok fied a 0 in K The loks in S read a ime longer han he proper ime. The moing lok in S rns slow. V V V V Δ Δ 0 0
Loren Transformaion Lengh onraion reisied Consider a measring rod wih proper lengh Δ -. The ineral Δ as iewed in S ms hae he posiions measred a he same ime 0 in S. V V Δ Δ / The lengh of he moing obje as measred in S is shorer han he proper lengh V V 0 0 8
9 Loren Transformaion Clok snhroniaion reisied Consider wo loks snhronied in S. Clok B a and lok A a. Wha imes do he read a ime 0 in S? Agrees wih resls from he homework 0 0 0 L V V V V V A B A B
Bea and Gamma β 0
Inarians Inarian qaniies hae he same ale in all inerial frames In he ne homework, o ll show s is he same for all inerial frames s s s s
Inarians Consider wo eens and We define he spaeime ineral as Δs Δ -Δ Three ases Lighlike Δs 0 The wo eens an be onneed onl b a ligh signal Spaelike Δs >0 The wo eens are no asall onneed. We an find an inerial frame where he eens or a he same ime b a differen posiions in spae Timelike Δs <0 The wo eens are asall onneed. We an find an inerial frame where he eens or a he same posiion in spae b a differen imes
Addiion of Veloiies Reall he Galilean ransformaion beween wo frames K and K where K moes wih eloi wih respe o K Consider an obje moing wih eloi in K and in K d d d d d d 3
Galilean Transformaion Noe ime is he same in boh ssems 4
5 Addiion of Veloiies We know he Loren ransformaion shold be sed insead so d d d d d d similarl
6 Loren Transformaion / / where β
7 Addiion of Veloiies Swapping primed and nprimed ariables and leing go o
8 Addiion of Veloiies Eample - le, 0, 0 Eample - le 0,, 0 0 0,, V V V V θ an 0,,
Addiion of Veloiies A roke blass off from he earh a 0.90 A seond roke follows in he same direion a eloi 0.98 Wha is he relaie eloi of he rokes sing a Galilean ransformaion Wha is he relaie eloi of he rokes sing a Loren ransformaion? 9
Loren Transformaion Las ime we arged ha and The mos general linear ransformaion for f, is α α A low eloiies, and α/ V V The inerse ransformaion is he same eep for he sign of relaie moion V 0
Loren Transformaion For a ligh plse in S we hae For a ligh plse in S we hae Then V V V V V V V
Loren Transformaion For he ransformaion V [ V V] V V