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Unformatted text preview: Chapter 3 Relativity
3.1 Special relativity
3.1.1 The Lorentz transformation
The Lorentz transformation (x , t ) = (x (x, t), t (x, t)) leaves the wave equation invariant if c is invariant: 2 2 1 2 2 2 2 1 2 2 + 2+ 2- 2 2 = + + - 2 2 x2 y z c t x 2 y 2 z 2 c t This transformation can also be found when ds 2 = ds 2 is demanded. The general form of the Lorentz transformation is given by: x =x+ where = ( - 1)(x v )v xv - vt , t = t - 2 |v|2 c 1
2 1 - v2 c The velocity difference v between two observers transforms according to: v = 1- v1 v2 c2
-1 v2 + ( - 1) v1 v2 2 v1 - v1 v1 If the velocity is parallel to the x-axis, this becomes y = y, z = z and: x = (x - vt) , x = (x + vt ) xv xv t = t- 2 , t= t + 2 c c If v = vex holds: px = px - , v = v2 - v1 v1 v2 1- 2 c W c , W = (W - vpx ) With = v/c the electric field of a moving charge is given by: E= Q (1 - 2 )er 40 r2 (1 - 2 sin2 ())3/2 The electromagnetic field transforms according to: E = (E + v B ) , B = B- vE c2 Length, mass and time transform according to: t r = t0 , mr = m0 , lr = l0 /, with 0 the quantities in a co-moving reference frame and r the quantities in a frame moving with velocity v w.r.t. it. The proper time is defined as: d 2 = ds2 /c2 , so = t/. For energy and momentum holds: W = m r c2 = W0 , ...
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- Spring '10