Special Relativity Jeffrey Eldred 1 1 Special Relativity Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU 1 1 1 1 1 1

2 2 Relativity 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 2 2 2 2 2 2

Maxwell’s Equations are Universal 3 Maxwell’s Equations are Universal Maxwell’s Equations in a vacuum: Plane wave solution: Implies the speed of light is a constant c. But if Maxwell’s equations work the same in every moving reference frame, than the speed of light must always be c. 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 3 3 3

Clock on a Train (Time Dilation) 4 Clock on a Train (Time Dilation) Time Dilation: Intervals of time are longer (slower) for objects in motion. 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 4 4 4

Train passing by a Marker (Length Contraction) 5 Train passing by a Marker (Length Contraction) Length Contraction: Intervals of length are shorter for objects in motion. 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 5 5 5

Particle Beams are Relativistic Trains 6 Particle Beams are Relativistic Trains Clear example of relativity in particle beams: Muons. The Muon mean decay lifetime is τ 0 = 2.2 μs at rest. The muon rest mass is m0 =107 MeV/c2, so if E = γ m0 c2, a 6 GeV muon has a value of γ = 57. A 6 GeV muon beam has a decay lifetime of τ = γ τ0 = 235 μs. In that time the muon travels L0 = β c τ = 70 m. But from the muon’s perspective, it’s stationary and the lab is traveling relativistically around it. It decays rapidly in 2.2 μs, but the relativistic lab around it is compressed by a factor of γ. 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 6 6 6

Lorentz Transformations 7 Lorentz Transformations Lorentz Shift: Hyperbolic Rotation: Invariant Interval: 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 7 7 7

Doppler Shift For an electromagnetic plane wave is invariant: 8 Doppler Shift For an electromagnetic plane wave is invariant: Just like the invariant interval: The frequency shifts are given by: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 8 8 8

Example: Undulator Wavelength 9 Example: Undulator Wavelength λ λ0 Time for beam to travel undulator period: Propagation of light in that time interval: Distance between particle and light: 6 GeV electron, γ=11820, λ0 = 2.8 mm λ = 10-10 m X-ray light source from macroscopic undulator period. 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 9 9 9

Example: Doppler Off-Angle 10 Example: Doppler Off-Angle Chavanne ESRF 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 10 10 10

11 Velocity Addition We can calculate the Lorentz shift on a differential length: Then we can calculate the Lorentz shift of a velocity: 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 11 11 11

Relativistic Kinematics 12 Relativistic Kinematics Velocity addition informs relativistic momentum: The Lorentz transformation is: The rest mass of a particle is invariant: The relativistic energy and momentum are: 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 12 12 12

Kinematics Example: Pion/Kaon Decay 13 Kinematics Example: Pion/Kaon Decay A charged Pions will decay into a muon and mu-neutrino: Energy and momentum must be conserved. In the rest-frame of the pion the total energy is just the rest mass of the pion. The momentum of the decay particles is equal and opposite. 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 13 13 13

Kinematics Example: Pion/Kaon Decay (cont.) 14 Kinematics Example: Pion/Kaon Decay (cont.) Shifting-back to lab-frame: We can calculate a relation for the decay particle: 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 14 14 14

Kinematics Example: Pion/Kaon Decay (cont.) 15 Kinematics Example: Pion/Kaon Decay (cont.) For the neutrino particle β’ = 1. For the muon β’ = 0.28 . The parameter θ’ is uniformly distributed, which allows one to calculate or simulate the real angle θ. For neutrinos from relativistic parents β = 1 we can use: And calculate the energy of the neutrino from its angle: S. Kopp Thesis 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 15 15 15

Twin Paradox (Just for Fun!) 16 Twin Paradox (Just for Fun!) 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 16 16 16

17 Twin Paradox 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 17 17 17

18 Twin Paradox 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 18 18 18

Minkowski Diagram It’s not just a rotation, it’s a shear 19 19 19 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/28/2018 19 19 19