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Understanding Length Contraction and Time Dilation in Relativity

This chapter delves into the applications of the Lorentz transformation, covering crucial concepts in relativity, such as length contraction, time dilation, and the twin paradox. It explains how observers in different frames may perceive events differently, emphasizing the lack of absolute simultaneity. We explore the formula for length contraction, relativity of simultaneity and the transformation of velocities. The chapter concludes by addressing the complexities of four-velocity and its implications in relativistic physics, supported by real-world examples and thought experiments.

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Understanding Length Contraction and Time Dilation in Relativity

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  1. EPPT M2INTRODUCTION TO RELATIVITY K Young, Physics Department, CUHK The Chinese University of Hong Kong

  2. CHAPTER 4APPLICATIONS OF THE LORENTZ TRANSFORMATION

  3. Objectives • Length contraction • Concept of simultaneity • Time dilation • Twin paradox • Transformation of velocity • Adding velocities • Four-velocity

  4. Length Contraction

  5. Choice of Units

  6. In this Chapter c =1

  7. Example

  8. Example Measure separation between 2 ends of a rod

  9. Example

  10. Length contraction • Formula for contraction • Concept of simultaneity • Paradoxes

  11. y y' S S' V L0 x' x Length contraction What is length L as it appears to S?

  12. At the same time! xA xB Definition of length

  13. Use of Lorentz transformation • Both are correct • Which is more convenient? Rod is fixed in S', Dx' = L0 always Dx = L when Dt = 0 A moving rod appears contracted

  14. 2 events are simultaneous in S NOT simultaneous in S' What if we use the other equation? (What are 2 events?) • Simultaneity is not absolute

  15. Generally • 2 events which are • simultaneous in S(Dt = 0) • but occurring in different places (Dx 0) • would not be simultaneous in S'(Dt' 0)

  16. A D B C E 2L0 Problem Seen by • S' co-moving with train • S on ground sees train moving at V = b c

  17. L Vt ct Event D Event B Sign? b  0?

  18. A D B C E 2L0 Are they simultaneous?

  19. I'm special We're equivalent Lack of symmetry? • All observers equivalent? • Symmetry SS'? • L <L0???

  20. Paradox

  21. V Paradox • Hole of length L0 • Rod of length L0, moving at V • Push both ends of rod at the same time • Can rod go through?

  22. At rest with hole Rod contracted At rest with rod Hole contracted Observer S Observer S' Goes through Does not go through ??

  23. V Paradox • Hole of length L0 • Rod of length L0, moving at V • Push both ends of rod at the same time • Can rod go through?

  24. S S' At the same time in S At the same time in S' ?

  25. Time Dilation

  26. 2 2' S S' V 1 1' Time dilation • What is time Dt as it appears to S? Dt is the time separation between 2 events. Which 2 events?

  27. Proper Time • Both are correct • Which is more convenient? Clock is fixed to S' (co-moving frame), Dx' = 0 Moving observer measures a longer time

  28. I'm special We are equivalent Lack of symmetry?

  29. Twin Paradox

  30. S' S Twin paradox • Who is older? Is there symmetry? Motion (velocity) is relative • Acceleration is absolute —S' has travelled Clock shows shorter time

  31. 10 ly Q P • According to Q, • According to P, Example • Who has aged more?

  32. Q P Example • Who has experienced acceleration? • Who is the “moving observer”?

  33. Experimental proof: elementary particle

  34. Lifetime appears longer. • Clearly verified.

  35. Atomic clocks Quartz watches Biological clocks Weak decays Strong decays Do these all "slow down" when moving? Other clocks?

  36. Phenomena • Analyze in detail • lnvoke Principle of Relativity Discrepancy not allowed • Study laws of physics (e.g. EM) rather than phenomena

  37. Transformation of Velocity

  38. Transformation of velocity • Galilean transformation • Relativistic transformation • Using Lorentz transformation directly • Using addition of "angles"

  39. V P x x' Vt Transformation of velocity 1. Galilean Same t !! "Addition of velocities"

  40. A. Using Lorentz transformation 2. Relativistic Note +

  41. Cannot add to more than c • If v'or V << c, the reduce to Galilean

  42. Example "0.01 + 0.01" "0.9 + 0.9"

  43. S S' P • Obvious that resultant bsatisfies B. Using addition of angles • Easy to do multiple additions

  44. Four Velocity

  45. V frame v, v' particle Four velocity • Velocity transforms in a complicated nonlinear manner

  46. Displacement is 4-vector Simple case:

  47. 4-vector transforms as

  48. because we divide by , and • is not an invariant, • Velocity does not transform simply

  49. transforms simply; • If we divide by a constant (e.g. 3.14), the result is still a 4-vector • Hint: Divide by a universal time

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