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Unformatted text preview: Lecture 4 The spacetime interval  a Lorentz invariant The Doppler effect The twin paradox The pole & barn paradox Summary of Lecture 3 Spacetime diagrams Lorentz contraction: an object in motion contracts in the direction relative motion Time dilation: A moving clock slows down Relative velocity transformation: u ' = u v 1 c u L ' = L PROPER T ' = T PROPER Lorentz Invariance: Spacetime interval s ( ) 2 = c t ( ) 2 x ( ) 2 + y ( ) 2 + z ( ) 2 { } So far, weve seen that both time intervals and distances are relative. However, we can define certain quantities that are absolute and do not change from one inertial frame to another. These are known as Lorentz invariants. First example: the spacetime interval _ s Prove for yourself that this is invariant under the Lorentz transformation Lorentz Invariance: Spacetime interval s ( ) 2 = c t ( ) 2 x ( ) 2 ds = 0: lightlike (ds) 2 > 0: timelike (ds) 2 < 0: spacelike (Harmonic) Waves A reminder about waves: Phase velocity v Frequency f Angular frequency Period T Wavelength _ Wavenumber k v = f f = 2 = 1 T k = 2 y = y m sin kx t ( ) The Classical Doppler Effect f ' = f 1 v Observer v sound Observer moves with respect to medium: f ' = f 1 m v Source v sound...
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This note was uploaded on 05/08/2008 for the course PHYS 237 taught by Professor Stephonalexander during the Spring '08 term at Pennsylvania State University, University Park.
 Spring '08
 STEPHONALEXANDER

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