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EPPT M2 INTRODUCTION TO RELATIVITY. K Young , Physics Department, CUHK The Chinese University of Hong Kong. CHAPTER 4 APPLICATIONS OF THE LORENTZ TRANSFORMATION. Objectives. Length contraction Concept of simultaneity Time dilation Twin paradox Transformation of velocity
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EPPT M2INTRODUCTION TO RELATIVITY K Young, Physics Department, CUHK The Chinese University of Hong Kong
Objectives • Length contraction • Concept of simultaneity • Time dilation • Twin paradox • Transformation of velocity • Adding velocities • Four-velocity
In this Chapter c =1
Example Measure separation between 2 ends of a rod
Length contraction • Formula for contraction • Concept of simultaneity • Paradoxes
y y' S S' V L0 x' x Length contraction What is length L as it appears to S?
At the same time! xA xB Definition of length
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
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
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)
A D B C E 2L0 Problem Seen by • S' co-moving with train • S on ground sees train moving at V = b c
L Vt ct Event D Event B Sign? b 0?
A D B C E 2L0 Are they simultaneous?
I'm special We're equivalent Lack of symmetry? • All observers equivalent? • Symmetry SS'? • L <L0???
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?
At rest with hole Rod contracted At rest with rod Hole contracted Observer S Observer S' Goes through Does not go through ??
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?
S S' At the same time in S At the same time in S' ?
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?
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
I'm special We are equivalent Lack of symmetry?
S' S Twin paradox • Who is older? Is there symmetry? Motion (velocity) is relative • Acceleration is absolute —S' has travelled Clock shows shorter time
10 ly Q P • According to Q, • According to P, Example • Who has aged more?
Q P Example • Who has experienced acceleration? • Who is the “moving observer”?
Lifetime appears longer. • Clearly verified.
Atomic clocks Quartz watches Biological clocks Weak decays Strong decays Do these all "slow down" when moving? Other clocks?
Phenomena • Analyze in detail • lnvoke Principle of Relativity Discrepancy not allowed • Study laws of physics (e.g. EM) rather than phenomena
Transformation of velocity • Galilean transformation • Relativistic transformation • Using Lorentz transformation directly • Using addition of "angles"
V P x x' Vt Transformation of velocity 1. Galilean Same t !! "Addition of velocities"
A. Using Lorentz transformation 2. Relativistic Note +
Cannot add to more than c • If v'or V << c, the reduce to Galilean
Example "0.01 + 0.01" "0.9 + 0.9"
S S' P • Obvious that resultant bsatisfies B. Using addition of angles • Easy to do multiple additions
V frame v, v' particle Four velocity • Velocity transforms in a complicated nonlinear manner
Displacement is 4-vector Simple case:
because we divide by , and • is not an invariant, • Velocity does not transform simply
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