Charge-current four-vector

This reflects the fundamental symmetry between space and time coordinates in special relativity and thus also in electromagnetism. The three space coordinates and time are collected in a so-called four-vector in special relativity. In the same way is the charge density p(f the fourth, i.e. time, component of the current-charge-density four-vector and the scalar potential (r, f) is the fourth, i.e. time, component of the vector-scalar-potential four-vector. 

It is to be expected that the equations relating electromagnetic fields and potentials to the charge current, should bear some resemblance to the Lorentz transformation. Stating that the equations for A and (j> are Lorentz invariant, means that they should have the same form for any observer, irrespective of relative velocity, as long as it s constant. This will be the case if the quantity (Ax, Ay, Az, i/c) = V is a Minkowski four-vector. Easiest would be to show that the dot product of V with another four-vector, e.g. the four-gradient, is Lorentz invariant, i.e. to show that... 

By defining a four-vector with the components of the current-density vector J and charge density, i.e. 

The space-charge current density in vacuo expressed by Eqs. (3) and (4) constitutes the essential part of the present extended theory. To specify the thus far undetermined velocity C, we follow the classical method of recasting Maxwell s equations into a four-dimensional representation. The divergence of Eq. (1) can, in combination with Eq. (4), be expressed in terms of a fourdimensional operator, where (j, 7 p) thus becomes a 4-vector. The potentials A and are derived from the sources j and p, which yield... 

These four vector relations compactly summarize the experimental laws describing all known electrical and magnetic phenomena. In these expressions, p is the electric charge density, J, the current density, E, the electric field and B, the magnetic induction. Maxwell s equations in free space (in the absence of dielectric or magnetic media) can be written... 

So far we have just defined another four-component quantity Af, but by now it is not clear whether it properly transforms under Lorentz transformations in order to justify the phrase 4-vector. In order to prove the transformation property of the gauge field, we re-express the inhomogeneous Maxwell equations in Lorenz gauge as given by Eq. (2.138) in explicitly covariant form by employment of the charge-current density and the gauge field A, ... 

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