The Special Theory of Relativity Chapter II

Transcription

1 The Speial Theory of Relaiiy Chaper II 1. Relaiisi Kinemais. Time dilaion and spae rael 3. Lengh onraion 4. Lorenz ransformaions 5. Paradoes?

2 Simulaneiy/Relaiiy If one obserer sees he eens as simulaneous, he oher anno, gien ha he speed of ligh is he same for eah. Conlusions: Simulaneiy is no an absolue onep Time is no an absolue onep I is relaie

3 Time Dilaion How muh ime does i ake for ligh o Trael up and down in he spae ship? a) Obserer in spae ship: D proper ime ν b) Obserer on arh: speed is he same apparen disane longer Ligh along diagonal: D D / 4 1 / This shows ha moing obserers mus disagree on he passage of ime. D 1 / Cloks moing relaie o an obserer run more slowly as ompared o loks a res relaie o ha obserer

4 Time Dilaion Calulaing he differene beween lok iks, we find ha he ineral in he moing frame is relaed o he ineral in he lok s res frame: 1 / is he proper ime (in he o-moing frame) I is he shores ime an obserer an measure wih 1 1 / hen Appliaions: Lifeimes of muons in he arh amosphere Time dilaion on aomi loks in GPS (4 km/s; iming error 1-1 s)

5 On Spae Trael 1 ligh years ~ 1 16 m If spae ship raels a.999 hen i akes ~1 years o rael. Bu in he res frame of he arrier: 1 / 4. 5yr The higher he speed he faser you ge here; Bu no from our frame perspeie!

6 Twin Parado Quesion: On her 1 s birhday an asronau akes off in a roke ship a a speed of 1/13. Afer 5 years elapsed on her wah, she urns around and heads bak o rejoin wih her win broher, who sayed a home. How old is eah win a he reunion?

7 Twin Parado Soluion: The raeling win has raeled for 551 years so she will be 31. As iewed from earh he raeling lok has moed slower by a faor: 1 1 / ( 1 /13) 5 So he ime elapsed on arh is 6 years, and her broher will be elebraing his 47 h birhday. Noe ha he raeling win has really spen only 1 years of her life. She has no lied more, her lok iked slower. Time really has eoled slower.

8 Twin Parado Where is he real parado? Think abou he problem from he perspeie of siser who sees he arh moing in her frame of referene, wih he onsequene ha he ime in her brohers frame should eole more slowly. Why isn he broher younger? The wo wins are no equialen! 1) The siser is no in an inerial frame of referene! Well she is, bu wo imes in a differen one (on Laue) ) The spae ship urns around whih requires aeleraion (Langein) Can be eplained by a Minkowski diagram

9 Lengh Conraion Disane beween planes is: Time for rael: arh obserer Time dilaaion 1 / Proper ime Spae raf obserers measure he same speed bu less ime Conraion only along The direion of moion

10 Lengh Conraion Only obsered in he direion of he moion. No onraion, or dilaion in perpendiular direion

11 erise The Barn and Ladder Parado There one was a farmer who had a ladder oo long o sore in his barn. He read some relaiiy and ame up wih he following idea. He insrued his daugher o run wih he ladder fas, suh ha he ladder would Lorenz onra o fi in he barn. When hrough he farmer inended o slam he door and hold he ladder fied inside. The daugher howeer poined ou ha (in her frame of referene) he barn, and no he ladder would onra, and he fi would be een worse. Who is righ? See: he fanasy rain in 36.6 & 36.7

12 Lorenz Transformaions In relaiiy, assume a linear ransformaion: ( ) y y z z as a onsan o be deermined (1 lassially). Inerse ransformaion wih - ( ) Consider ligh pulse a ommon origin of S and S a measure he disane in and : ( ) ( ) ( ) ( ) ( ) ( ) ( ) fill in Transformaion parameer 1 1 /

13 Lorenz Transformaions Sole furher: Time dilaion and lengh onraion an be deried From hese Lorenz ransformaions ( ) ( ) ( ) Leading o he ransformaions: z y z y 1 1 ( ) y y z z

14 Tes inariane of Mawell equaions under Lorenz Transformaions ( ) Coordinaes Parial Deriaies Spaial, firs Spaial, seond 4 aluae similarly he emporal erm and es inariane of Mawell s wae equaion

15 erise The addiion of eloiies in referene frames I. longiudinal Obserer in frame S deermines speed u of obje in S (,,u ) ( ) Then: u d d Deriaies: of oordinaes d d u d d d d 1 u u 1 d d d d u 1 d d u u 1 d d d d d d [ ( )] [ ( )] ( u ) u u 1 1 u 1

16 erise The addiion of eloiies in referene frames II. Transersal Obserer in frame S deermines speed u of obje in S (,,u ) y Then: y u y Deriaies: dy d dy d u y dy d dy d d d u y u y dy d 1 u 1 d d 1 d d 1 u 1 So also u y and u z ransform; his has o do wih he ransformaion (non-absolueness) of ime d 1 d

17 erise Lorenz Transformaions Calulae he speed of roke wih respe o arh. u u. 88 u 1 This equaion also yields as resul ha is he maimum obainable speed (in any frame).

18 Faser han he speed of ligh? Cherenko radiaion A blue ligh one Cf: shok wae Pael Cherenko Nobel Prize 1958 Appliaion in he ANTARS deeor Parile raels a p β Waes emied as (spherial) miane one: os θ 1 nβ e n