Poincare transformations of external fields

First we note that the following sixteen matrices, 

The behavior of the covariant potential K,ov under Poincare transformations immediately implies that the potential V (x) is transformed into 

It has to be assumed that V x) describes the external field at the space-time point X after a Poincare transformation. This requires a certain behavior of the coefficient functions Vj under Poincare transformations. Let me illustrate this with an example. 

It has been claimed already that the potential matrix 

In a similar way, we can give a meaning to the other coefficient functions. 

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