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8/24/00 NE 409

8/24/00 NE 409. Relativity Propagation of waves in various media was well understood by classical physicists. E.g., the speed of sound depends on the density of air. How did light propagate in space? Theory: light must propagate in a propagate in a low density and ridged medium.

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8/24/00 NE 409

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  1. 8/24/00 NE 409 • Relativity • Propagation of waves in various media was well understood by classical physicists. • E.g., the speed of sound depends on the density of air. • How did light propagate in space? • Theory: light must propagate in a propagate in a low density and ridged medium.

  2. Michelson and Morley Experiment 1881 • If an observer moves towards source of a wave, the speed of the wave appears to be: • ws,a = v + u • where v is the speed of wave and u is the speed of observer towards source of wave. • If an observer moves away from source of a wave, the speed of the wave appears to be: • ws,b = v - u

  3. Experimental Set Up • Experiment used motion of earth and light from stars • Conclusion: speed of light does not change with respect to inertial frames.

  4. Galilean Transformation(a transform where Newtons 2nd law is invariant) • Consider two reference frames S and S’ S S’ u p ut

  5. Location of Point p • S frame: p(x,y,z) • S’ frame: p(x’,y’,z’) • therefore: • x’= x – ut • y’ = y • z’ = z

  6. Newton’s Law is invariant in S and S’ • dx’/dt = (dx/dt) – u • d2x’/dt2 = d2x/dt2 • dy’/dt = dy/dt • d2y’/dt2 = d2y/dt2 • dz’/dt = dz/dt • d2z’/dt2 = d2z/dt2

  7. c’ =c for all inertial framesNewtonian mechanics must change • As uc, Newtonian mechanics does not work • In Newtonian mechanics: • We assume that rulers do not shrink or expand • We assume that clocks do not slow down or speed up

  8. x - ut x’ = 1 – u2/c2 Lorentz Transformation y’ = y z’ = z t – ux/c2 t’ = 1 – u2/c2

  9. Dx’ x2’ – x1’ v’ = = Dt’ t2’ – t1’ v – u v’ = 1 – vu/c2 Velocity Transformation • Velocity can be expressed as a finite difference:

  10. v’ + u v = 1 + v’u/c2 Inverse Transformation

  11. 1 g = 1 – u2/c2 Definition of b and g (u/c)2 = b

  12. fs = fs = fs’ fs’ 1 + b 1 - b 1 - b 1 + b Relativistic Doppler Shift Approaching wave Receding wave

  13. Relativistic Momentum p = g m v

  14. Force and Motion dgmv dp S F = = dt dt

  15. Work and Energy E = g mc2 Kinetic Energy is found by subtracting mc2 Ek = g mc2 – mc2 E2 = (mc2)2 + (pc)2 E = mc2 1 + (p/mc)2

  16. Massless Particle • m = 0 and v = c E2 = (mc2)2 + (pc)2 E = pc A massless particle has momentum. What massless particles have been shown to exist? Photons!

  17. Invarients • Conservation of Energy • E2 – p2c2 = constant • For a system of i particles having energies Ei and momenta pi, the quantity m is defined as: • m = (SEi)2 – (SPi)2

  18. Relationship between Mass and Energy • E = mc2

  19. Questions?

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