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Chapter.1.Appendix.Covariant.Form

# Chapter.1.Appendix.Covariant.Form - A − ∂ ∂ = R5 Note...

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Appendix B: Relativistic four vector notation and covariant form for Maxwell’s Equations Under a Lorentz transformation, L , the space-time position four vector transforms as follows: x = L x with the following choice of metric tensor g  = [ L T L ]  = −  for , = 1,2,3; = 0 = 0 otherwise. R1 R1b The x , and are given by x = ( ct , r ) ; x = ( ct , r ) ...... = 0,1,2,3 = ( 1 c t , ∇) ; = ( 1 c t , −∇) = 1 c 2 2 t 2 − ∇ 2 R2a R2b R2c The electrostatic potential, , and the vector potential, A , form a four vector (transform under Lorentz transformation as a four vector), A = ( , A ) ; A ≡ ( , A ) ; R3 A = A − ∂ = (( ( r , t ) − 1 c t ) , A + ∇ ) = ( ( r , t ) , A ) A = A − ∂ = (( ( r , t ) − 1 c t ) , A − ∇ ) = ( ( r , t ) , A ) R4 Eq. 31 (the Lorentz gauge condition) becomes A = ∂ A − ∂ = 1 c t ( r , t ) + ∇
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Unformatted text preview: A − ∂ ∂ = R5 Note that the general condition on which ensures the Lorentz gauge is ∂ ∂ = −∇ 2 ( r , t ) + 1 c 2 ∂ 2 ∂ t 2 ( r , t ) = ∇ A ( r , t ) + 1 c ∂ ∂ t ( r , t ) , Finally, the Maxwell’s equations become ∂ ∂ A = 4 c J where J = ( c , J ) . R6a R6b Conservation of charge is given by ∂ J = ∂ ∂ t + ∇ J = R7 In summary, the wave equations, the Lorentz condition and the conservation of charge condition take on manifestly covariant forms. That is, they have the same form in all Lorentz frames: ∂ ∂ A = 4 c J ∂ A = ∂ J = R8 R9 R10...
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