The Lorentz Transformation

Transcription

1 The Lorenz Transformaion Relaiviy and Asrophysics Lecure 06 Terry Herer Ouline Coordinae ransformaions Lorenz Transformaion Saemen Proof Addiion of velociies Parial proof Examples of velociy addiion Proof of conracion along he direcion of propagaion Reading Spaceime Physics: Chaper 3 & Special Topic: Lorenz Transformaion Homework: (due Wed. 9/6/09) 3-, 3-7, and 3-0 (maybe more on Friday) A90-06 Lorenz Transformaion A90-06

2 Moving beween inerial frames Evens and inervals beween evens define he physical world. This defines he physics We define isolaed evens using a laicework (inerial reference frame) of recording clocks. Bu we will need o move from clock-laice frame o anoher (for insance, lab o rocke frame and visa-versa) Lorenz ransformaion Name for he ranslaion beween inerial frames I may useful because we may wan o ag he locaion of a number of evens in our lab The ransformaion allows o compue he space and ime separaions beween evens in differen inerial frames A90-06 Lorenz Transformaion 3 Speed example Suppose I ravel in a rocke ha you observe o be raveling a v = 4/5c. I fire a bulle ha I observe o fly forward a 4/5c. v = 4/5 v = 4/5 Wha velociy do you see for he bulle? The velociy is no 4/5 + 4/5 =.6! I is 40/4 which is deermine via he Lorenz ransformaion. Suppose he bulle srikes a arge 4 meers away from me and my clock measures he ime of fligh o be 5 meers. Wha do you (in he lab frame) measure for he space and ime coordinaes of he wo evens? We use he Lorenz ransformaion o figure his ou. A90-06 Lorenz Transformaion 4 A90-06

3 The Spaceime Inerval is no enough Rocke Frame: x = 4 m and = 5 m Bulle Frame: x = 0 m, use spaceime inerval o ge he ime x x => 0 5m 4 m So ha, = 3 m (proper ime) Lab Frame: primed = rocke frame double primed = bulle frame unprimed = lab frame Can use inerval because i is no sufficien o deermine x or separaely 3 x m The Lorenz ransformaion allows us o deermine hese separaely A90-06 Lorenz Transformaion 5 Lorenz Transformaion The Lorenz ransformaion allows us o move beween inerial reference frames x, are in he lab frame x, are in he rocke frame v is he aive velociy beween he rocke and lab frames For convenience le he posiive x-axis be along he direcion of moion of he rocke. The ransformaion equaions are x v x v v x v y y z z The LT is powerful bu is no fundamenal in ha i doesn expose deep new feaures of spaceime Bu i is useful someimes wan o know he lengh of a yach, bu a oher imes you would like he posiions of he bow and sern aive o norh A90-06 Lorenz Transformaion 6 A

4 Proof of LT par Requiremens of he ransformaion The lineariy, ha is, space and ime coordinae mus appear he firs power, no squared or cubed. - Since we require ha we can choose any even as he zero of space and ime. Mus preserve spaceime inerval beween wo evens Le us define x, are in he lab frame x, are in he rocke frame v is he aive velociy beween he rocke and lab frames Sep : There will be no change in he ransverse direcion y y z z Sep : Consider a sparkplug ha sis a he origin of he rocke frame and emis a spark a ime. Wha are x and in our lab frame? Spark mus occur a locaion of sparkplug so ha x v Since x = 0 a = 0 by he way we chose he frames. A90-06 Lorenz Transformaion 7 Proof of LT par Now use he spaceime inerval o ge he aion beween and. x 0 x v Thus v / or v Le us define,, a quaniy which occurs ofen in Lorenz ransformaions Then we have v and x v These equaions allow us o move from space and ime coordinaes in one inerial frame o anoher bu only apply when x = 0. We need o exend o he more general case when x 0. A90-06 Lorenz Transformaion 8 A

5 Proof of LT par 3 Since he Lorenz ransformaion mus be linear he general form should look like: Bx D x Gx H We wish now o find B, D, G, and H. These coefficiens should depend upon he rocke speed bu no he coordinaes of a paricular even. The ransformaion mus agree wih our previous resul for x = 0. Bx x Gx v B and G will be se by requiring ha he Spaceime inerval is he same in he rocke and lab frames See pages 0-0 of Spaceime Physics A90-06 Lorenz Transformaion 9 Lorenz Transformaion This gives he ransformaion equaions x v vx x y y v v The inverse ransformaion from (x,) o (x, ) Transforms coordinaes he oher way. To derive we noe ha he laboraory moves wih he same speed bu opposie sign (negaive-x direcion) We ge he reverse ransformaion by changing he sign of v and swapping and. x x v v v x v y y z z z z See pages 0-03 of Spaceime Physics for long proof. A90-06 Lorenz Transformaion 0 A

6 Addiion of Velociies We can derive how velociies add up from he Lorenz ransformaion. Wriing he LT using, x x v vx Taking he differenial of boh equaions dx dx v d d d vdx Now dividing he wo dx dx v d d d v dx => v v v vv This is call he Law of Addiion of Velociies See page 05 of Spaceime Physics for a non-calculus derivaion Your exbook prefers he erm Law of Combinaion of Velociies For our earlier example using v = 4/5 and v = 4/5 we ge v = 40/4 = A90-06 Lorenz Transformaion Anoher velociy example For insance, a boaer hrows a ball wih velociy v (as seen by he boaer) and he boa is raveling owards he shore wih velociy, v : v v Sanding on he shore we expec o see he ball moving wih velociy = v + v. This is called a Galilean ransformaion (afer Galileo). A90-06 Lorenz Transformaion A

7 How do we add velociies? The old law of Galileo and Newon was v v v If v = (flashligh), hen v v Wrong!! Can have v >. Bu we have he Lorenz ransformaion of velociies o save he day. v v v vv For v = => v v v A90-06 Lorenz Transformaion 3 Anoher example 0.9 A B 0.9 C Person C sees A and B moving a 0.9. How fas does A hink B is moving? Choose A as lab frame, C in rocke frame (moving wih v ), and B as he bulle (wih velociy v in rocke frame). So we are asking for velociy of B (bulle) in lab frame. v v v vv A90-06 Lorenz Transformaion 4 A