1. Special Relativity Jeffrey Eldred1 1
Special Relativity
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 2 2
Relativity 2
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3. Maxwell’s Equations are Universal3
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
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4. 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
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5. 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
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6. Particle Beams are Relativistic Trains6
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
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7. Lorentz Transformations7
Lorentz Transformations
Lorentz Shift:
Hyperbolic Rotation:
Invariant Interval: 7
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8. 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
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9. Example: Undulator Wavelength9
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 = m
X-ray light source from macroscopic undulator period. 9
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10. Example: Doppler Off-Angle10
Example: Doppler Off-Angle
Chavanne ESRF 10
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11. 11
Velocity Addition
We can calculate the Lorentz shift on a differential length:
Then we can calculate the Lorentz shift of a velocity: 11
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12. Relativistic Kinematics12
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
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13. Kinematics Example: Pion/Kaon Decay13
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
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14. 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
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15. Kinematics Example: Pion/Kaon Decay (cont.)15
Kinematics Example: Pion/Kaon Decay (cont.)
For the neutrino particle β’ = 1. For the muon β’ =
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
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16. Twin Paradox (Just for Fun!)16
Twin Paradox (Just for Fun!) 16
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17. 17
Twin Paradox 17
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18. 18
Twin Paradox 18
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19. Minkowski Diagram It’s not just a rotation, it’s a shear 19 19 19 19
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