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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 γ= (γ − 1)(x · v )v x·v − γ vt , t = γ t − 2 |v |2 c 1 2 1 − v2 c The velocity difference v between two observers transforms according to: v= γ 1− v1 · v2 c2 −1 v2 + (γ − 1) v1 · v2 2 v1 − γ v1 v1 If the velocity is parallel to the x-axis, this becomes y = y , z = z and: x = γ (x − vt) , x = γ (x + vt ) xv xv t =γ t− 2 , t=γ t + 2 c c If v = v ex holds: px = γ px − , v= v2 − v1 v1 v2 1− 2 c βW c , W = γ (W − vpx ) With β = v/c the electric field of a moving charge is given by: E= Q (1 − β 2 )er 4πε0 r2 (1 − β 2 sin2 (θ))3/2 The electromagnetic field transforms according to: E = γ (E + v × B ) , B = γ B− v×E c2 Length, mass and time transform according to: ∆t r = γ ∆t0 , mr = γ m0 , lr = l0 /γ , 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 τ is defined as: dτ 2 = ds2 /c2 , so ∆τ = ∆t/γ . For energy and momentum holds: W = m r c2 = γ W0 , ...
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