1. Time dilation, length contraction, simultaneity
Special Relativity
Time dilation, length contraction, simultaneity

2. Electron-electron Interaction
Center-of mass rest frame S' - -
Coulombic repulsion only
Electron acceleration is a'

3. Electron-electron Interaction
Frame S in which electrons move v - v -
Coulombic repulsion plus Lorentz attraction
Electron acceleration is a = a' (1– b2) (b  v/c)

4. c in Inertial Frames To simplify we used 1/c2 = m0e0.
Is that valid in all inertial frames?
Is that true for all directions (isotropy)?
If you move against the light, is it faster?
if you move with the light, is it slower?
Is c isotropic only in one absolute rest frame (the ether frame)?

5. Einstein’s Postulate
The speed of light is isotropically c in all inertal frames.

6. Time Dilation

7. Times in Different Frames
Transverse light flash in a moving car
mirror v
source/ detector
Flash emitted, reflected, received
Reference frames: S for track, S' for car

8. Detection time in S' Light signal travels distance 2h
Detection time t' = 2h/c

9. Detection time in S h
Light signal travels distance 2h vertically and vt horizontally
Total distance is 2d h d vt
Solve for detection time t, find t/t'

10. Time Dilation
If two events at the same place in frame S' are separated by time t', in frame S they are separated by time
t = t' (1 – b2)–1/2

11. Length Contraction

12. Lengths in Different Frames
Longitudinal light flash in a moving car
mirror v
source/ detector l0
Flash emitted, reflected, received
Reference frames: S for track, S' for car

13. Back to our Electrons a = a' (1 – b2)1/2- -
What does time dilation tell us?

14. Doppler Shift
Of light

15. Moving Source Equations of Motion: Source xS = –vt1 2 3 D
Equations of Motion:
Source xS = –vt
ith wavefront xi = –v (i/fs) + c (t – i/fs) 15

16. Classical Formula 1 fD = fS 1 + b But recall Ts = gTs' So fs = fs'/g16

17. Doppler Shift of Light Relativistic Doppler shift: 1 + b fD = fS'
where
b = vS/c
Detector frame of reference (vD = 0) 17

18. Redshift z Dl l' l – l' l' l l' – 1 z = = = = c/fD c/fS' – 1 fD – 1

20. Relativistic Mechanics
Momentum and energy

21. Momentum
p = gmv

22. Total Energy
E = gmc2

23. Rest Energy E = gmc2 If v = 0 then g = 1, E = mc2 Rest energy is mc2
Kinetic energy is (g–1)mc2

24. Correspondence At small b: Momentum gmv  mv
Energy gmc2 = (1–b2)–1/2 mc2
Binomal approximation (1+x)n  1+nx for small x
So (1–b2)–1/2  1 + (–1/2)(–b2) = 1 + b2/2
gmc2  mc2 + 1/2 mv2
Is this true? Let’s check:

26. Convenient Formula E2 = (mc2)2 + (pc)2
Derivation: show R side = (gmc2)2

27. A massless photon p = h/l E = hf = hc/l
h = 6.62610–36 J·s (Planck constant)
gmc2 incalculable: g =  and m = 0
But E2 = (mc2)2 + (pc)2 works:
E2 = 0 + (hc/l)2 = (hf)2
E = hf