Greens tensors in the time domain

The time domain electromagnetic Green s tensors Gg, are introduced, as in frequency domain, as the fields of an elementary electric source (Felsen and Marcuvitz, 1994). They follow the Maxwell s equations 

Equation (8.143) suggests that in a homogeneous domain Ge also satisfies the equation 

The EM Green s tensors exhibit symmetry and can be shown, using the Lorentz lemma, to satisfy the following reciprocity relations (Felsen and Marcuvitz, 1994)  

The last conditions show that by replacing source and receiver (i.e. the points r and r) and by going simultaneously to the reverse time —t, (therefore, by retaining the causality, because the condition t t n ordinary time implies the condition —t —t in reverse time), we obtain the equivalent electromagnetic field, described by the Green s tensors Ge (r, t r,t) and G (r, t r,t). 

Following Morse and Feshbach (1953) and Felsen and Marcuvitz (1994), we can also introduce the adjoint Green s tensors  

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