Lorentz Transformations and Velocity in Special Relativity

phy 712 electrodynamics 10 10 50 am mwf olin 107 n.w
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Explore the theory of special relativity with a focus on Lorentz transformations, energy, momentum, and electromagnetic field transformations. Understand the complexities of velocity transformations in moving frames and the concept of velocity 4-vector in the context of physics.

  • Special relativity
  • Lorentz transformations
  • Velocity
  • Electromagnetic fields
  • Energy
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  1. PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 23: Continue reading Chap. 11 Theory of Special Relativity A. Lorentz transformation relations B. Energy and momentum C. Electromagnetic field transformations 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 1

  2. 3/25/2014 PHY 770 Spring 2014 -- Lecture 16 2

  3. 3/25/2014 PHY 770 Spring 2014 -- Lecture 16 3

  4. 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 4

  5. Convenient v notation : Lorentz transformations v c 1 v 2 1 v Stationary frame Moving ct + frame y y ( x ) = + ' ' ct x ( ' ) = x ' ' ct v = y y x = z ' z x y x y x 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 5

  6. Lorentz transformations -- continued = v : v x For the moving frame with 0 0 0 0 v v v v v v 0 0 0 0 1 - v v v v v v = = L L v v 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 ' ' ct ct ct ct ' ' x x x x 1 - = = L L v v ' ' y y y y ' ' z z z z Notice : = 2 2 2 2 2 2 2 2 2 2 ' ' ' ' c t x y z c t x y z 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 6

  7. Lorentz transformation of the velocity dx dy dz Define : u u u x y z dt dt dt ' ' ' dx dy dz + ' ' ' u u u x y z ' ' ' dt dt dt ' u v = x u x + 2 1 ' / vu c x ' u ' u y = = z u u ( 1 ) 1 ( 1 ) y z + + 2 2 ' / ' / vu c vu c v x v x 1 Where : ( ) v 2 2 1 / 1 v c v Note that the velocity components themselves do not obviously transform according to a Lorentz transformation. 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 7

  8. Velocity transformations continued: v u u / ' 1 + ' u + ' ' u y = = = Consider : . x z u u ( 1 ) ( 1 ) x y z + + 2 2 2 ' / ' / vu c vu + c vu c x v x ' v x 2 1 / 1 vu c = Note that x ( ) ( ) ( ) u 2 2 2 1 / 1 / ' 1 / u ' c u c v c ( ) = + c c u ' ' u v u ' v u x ( ) u ( ) = + = + ' ' u u v u c ' ' ' ' u x v u x u v u ' x v u = = u u u ' ' u y u y u z u z c c ' u u u u ' ' Velocity 4-vector: u x u x = L v ' u u ' u y u y ' u u ' u z u z 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 8

  9. c u u Significance of 4-velocity vector: u x u u y u u z Introduce the rest mass m of particle characterized by velocity u: 2 c E mc u u u p c mu c u x x = = u x mc u p c mu c u y y u y u p c mu c u z z u z Properties of energy-moment 4-vector: ' ' E E E E ' ' p c p c p c p c x x x x = = = 1 2 2 2 2 2 2 - L L Note : ' ' E p c E p c ' ' p c p c p c p c y y y y ' ' p c p c p c p c z z z z 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 9

  10. Properties of Energy-momentum 4-vector -- continued = c mu c p z u z 2 E mc u p c mu c x u x p c mu c y u y ( 1 ) 2 2 2 2 u ( ) 2 u mc u 2 y = = = 2 2 2 2 2 2 2 Note : 1 ' ' x z E p c mc E p c 2 c c c u = 2 p Notion of " rest energy" : For , 0 E mc = + 2 2 2 2 4 2 Define kinetic energy : E E mc p c m c mc K 2 p p = + 2 Non - relativist limit ic : If , 1 1 1 E mc K mc mc 2 p 2 m 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 10

  11. Summary of relativistic energy relationships = c mu c p z u z 2 E mc u p c mu c x u x p c mu c y u y = + = 2 2 2 4 2 E p c m c mc u 2 2 + = + = 2 2 2 4 2 2 Check : 1 p c m c mc mc u u u = 2 Example electron an for : c E 5 . 0 MeV m = for 200 GeV E = = 5 4 10 u 2 mc 1 1 12 = 1 1 1 3 10 u 2 2 2 u u 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 11

  12. Special theory of relativity and Maxwells equations + = J Continuity equation : 0 t t 1 c + = A Lorentz gauge condition : 0 2 t 2 1 c + = 2 Potential equations : 4 2 2 A 2 t 1 c 4 + = 2 A J 2 c A t 1 c = E Field relations : = B A 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 12

  13. More 4-vectors: ct x position and Time : x y z J x c Charge current and : J J y J z A x Vector and scalar potentials : A A y A z 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 13

  14. 0 0 Lorentz transformations v v v 0 0 v v v = L v 0 0 1 0 0 0 0 1 v L = L Time and space : ' ' x x x v v L = L Charge current and : ' ' J J J v v L = L Vector and scalar potential : ' ' A A A v 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 14

  15. 4-vector relationships 0 ct A 1 x ( ) : A ( ) = 0 A , upper index 4 - vector for 0,1,2,3 A A 2 y A 3 z A ( ) = 0 A Keeping track Derivative = of signs operators - - lower : index 4 - vector , A A = , , c t c t 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 15

  16. Special theory of relativity and Maxwells equations = 0 J + = J Continuity equation : 0 t t 1 c = 0 A + = A Lorentz gauge condition : 0 2 t 2 1 c = 2 Potential equations : 4 2 2 J c 4 = A A 2 t 1 c 4 = 2 A J 2 c A t 1 c = E Field relations : = B A 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 16

  17. Electric and Magnetic field relationships 1 t c A z A A ( ) = = 0 1 1 0 = E x E A A x x c t A c t A c t A ( ) y = = 0 2 2 0 E A A y y ( ) = = 0 3 3 0 E A A z z z = B A ( ) y = = 2 3 3 2 B A A x y z ( ) A A = = 3 1 1 3 x z B A A y z x A ( ) A y = = 1 2 2 1 x B A A z x y 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 17

  18. ( ) B F A A Field strength tensor 0 E F E E E x y z 0 B B x z y 0 B E B y z x 0 E B z y x Transformation of field strength tensor 0 0 v v v 0 0 v v v = = L L L ' F F v v v 0 0 1 0 0 0 0 1 ( ( ) ) ( B ) + 0 ' ' ' ' ' E E B E B x v y v z v z v E y ( ) + B ' 0 + ' ' ' ' E B E x v z v y v y v z = F ( ( ) ) ( ) + ' ' ' ' 0 ' E B B E v y v z v z v y x ( ) ' ' ' ' ' 0 E B B E B v z v y v y v z x 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 18

  19. Lorentz transformation of electromagnetic fields: ' x x E E E E B + = = ( ( ) ) ' ' y v y v z = ' ' E E B z v z v y = ' B B x x ( ( ) ) = ' ' B B E y v y v z = + ' ' B B E z v z v y 03/26/2014 PHY 712 Spring 2014 -- Lecture 26 19

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