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5 Current density

The relativistic density $\rho$ of a collection of moving charges is defined as $\gamma\rho_0$ , where $\rho_0$ is the charge density in the rest frame of the moving charges, also referred to as proper density. A derived quantity is the current density 4-vector $J_{\mu}$ that is defined as:

$$J_{\mu}=\rho_0\frac{dx_{\mu}}{d\tau}$$ (18)

and is constructed quite similar to the energy-momentum 4-vector:

$$p_{\mu}=m_0\frac{dx_{\mu}}{d\tau}.$$ (19)

The current density 4-vector can be rewritten into Euclidean form in the same way as with the energy-momentum 4-vector:

$$(\rho_0c)^2 = (\gamma\rho_0c)^2-(\gamma\rho_0v)^2$$

$$(\gamma\rho_0c)^2 = (\rho_0c)^2+(\gamma\rho_0v)^2$$

$$(\gamma\rho_0c)^2 = (\gamma\rho_0\chi)^2+(\gamma\rho_0v)^2$$ (20)

saying that the 4-dimensional current density is the vector sum of the current density in space and the current density in the proper time dimension. A similar effect of the factor $\gamma$ is seen in the current density as in the energy-momentum 4-vector and this suggests that there will also be a justification that allows to leave out this factor in the equation for current density to reach an invariant result, equivalent to the relativistic Lagrangian for energy-momentum. That relativistic Lagrangian more or less represents the view on energy that a 'Hyperspacelander' with full 4D observational skills would have. Such an upgrade from 3D to 4D observational skills eliminates length contraction effects (length is invariant in 4D) and for the 4D observer this turns the relativistic charge density $\rho$ into the proper density $\rho_0$ . It is therefor justified to leave out $\gamma$ in Eq. (20).

With a single unit of charge the charge density will be the same for all inertial frames (charge is invariant). The equation for a single charge reads:

$$(\rho_0 c)^2=(\rho_0 \chi)^2+(\rho_0 v)^2$$ (21)

and is the only equation that is on equal footing with the Euclidean 4-vector for momentum, Eq. (14), when applied to a single elementary mass particle. It strikes that the factor $\gamma$ is eliminated automatically here. There is no need for an upgrade from 3D to 4D observational skills to justify any omission of $\gamma$ like was done for the energy-momentum 4-vector. This markedly distinguishes the properties charge and current in the electromagnetic field from the properties mass and energy-momentum in the gravity field and seems to suggests a dimensional hierarchy, i.e., the electromagnetic field seems to have one less dimension than gravity.

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