Chapter.1.Appendix.Covariant.Form

Chapter.1.Appendix.Covariant.Form - A = R5 Note that the...

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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 Maxwells 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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This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

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