ABSTRACT
In relation to our study of the potential description of electrodynamics in curved-space time (Sec. 8.4) I promised to show that the potential in Minkowski space satisfies the covariant field equation in (8.116). In preparation for this we first note that the relations between (E,B) and (A, φ = cA0) given in Eqs. (8.103) and (8.104) imply that the Maxwell-Lorentz equations [Eqs. (8.49) and (8.50)] automatically are met. Expressed in terms of the fourpotential, the inhomogeneous field equation given in Eq. (8.52) may be written in the form
A−∇ ( ∇ ·A+ 1
c
∂
∂t A0 ) = −µ0J (12.1)
because∇×(∇×A) =∇∇·A−∇2A. With J0 = cρ the inhomogeneous Maxwell-Lorentz equation in Eq. (8.51) becomes in the potential version
∇2A0 + 1 c
∂
∂t ∇ ·A = −µ0J0 (12.2)
remembering that ǫ0µ0 = c −2. By adding and subtracting a term c−2∂2A0/∂t2, Eq. (12.2)
can be given a form closely resembling the one in Eq. (12.1), viz.,
A0 + 1
c
∂
∂t
( ∇ ·A+ 1
c
∂
∂t A0 ) = −µ0J0. (12.3)
In general, Eqs. (12.1) and (12.3) constitute a set of coupled equations for A and A0. However, the equations decouple in the Lorenz gauge, where the potentials satisfy the constraint in Eq. (10.11) [with K = 0]. Thus, in this gauge we have
A(r, t) = −µ0J(r, t), (12.4) A0(r, t) = −µ0J0(r, t). (12.5)
Using covariant notation [{∂µ} = ( 1 c ∂ ∂t ,∇
) , {∂µ} = (− 1c ∂∂t ,∇), {Aµ} = (A0,A), {Jµ} =(
J0,J ) , ∂µ∂
µ = and ∂µA µ =∇ ·A+ c−1 ∂∂tA0], Eqs. (12.1) and (12.3) can be written in
the compact form
(∂ν∂ ν)Aµ(x) − ∂µ (∂νAν(x)) = −µ0Jµ(x), µ = 0− 3, (12.6)
with x = {xµ} = (ct, r). the result in Eq. (12.6) is just the contravariant form of the field equation given in covariant form in Eq. (8.116).
