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phys documents (dragged) 20 - Chapter 3 Relativity 3.1...

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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 x 2 + 2 y 2 + 2 z 2 - 1 c 2 2 t 2 = 2 x 2 + 2 y 2 + 2 z 2 - 1 c 2 2 t 2 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 + ( γ - 1)( x · v ) v | v | 2 - γ vt , t = γ t - x · v c 2 where γ = 1 1 - v 2 c 2 The velocity difference v between two observers transforms according to: v = γ 1 - v 1 · v 2 c 2 - 1 v 2 + ( γ - 1) v 1 · v 2 v 2 1 v 1 - γ v 1 If the velocity is parallel to the x -axis, this becomes y = y , z = z and: x = γ ( x - vt ) , x = γ ( x + vt ) t = γ t - xv c 2 , t = γ t + x v c 2 , v = v 2 - v 1 1 - v 1 v 2 c 2 If v = ve x holds: p x = γ p x - β W c , W = γ ( W - vp x ) With β = v/c the electric field of a moving charge is given by:
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