Potential energy in Special Relativity

Let's start with Newtonian mechanics. Of the fundamental forces of nature, the only one that can be handled at all by Newtonian mechanics is gravity. Newtonian mechanics can't handle electromagnetism. Electromagnetism is inherently relativistic (i.e., Maxwell's equations only make sense in the context of SR, not Galilean relativity).

Now let's pass from the Newtonian approximation to SR. We lose the ability to model gravity, since that would require GR. We gain the ability to model electromagnetism. In electromagnetism, we don't really have a useful concept of a scalar potential energy $q\Phi$, where $\Phi$ is the electric potential. The reason for this is that although the charge $q$ is a relativistic scalar, the electrical potential $\Phi$ is not a relativistic scalar, it's the timelike component of a four-vector. The conserved energy in Maxwell's equations is not really the energy of a point particle in some external field, it's the energy of the electromagnetic field itself, which depends on energy densities proportional to $E^2$ and $B^2$.

I'll expose what I've understood.

In classical mechanics, $E=T+U$. Since for a free particle in SR, $E=\sqrt{p^2+m^2}$ (here $c=1$). We could try to introduce potential energy as: $E-U=\sqrt{p^2+m^2}$. But this would not be a covariant equation.

So we have to use the 4-vector $Q^\mu=(U, \textbf{Q})$, which is the potential four-momentum while $\textbf{Q}$ is called just the potential momentum.

If we subtract $Q^\mu$ to $p^\mu=(E,\textbf{p})$, we get:


The potential momentum is closely related to the Aharonov-Bohm effect.

This way to introduce potential energy the one used in gauge theories. There are two more possible ways: for gravity, using space-time curvature or supposing that the potential energy is a scalar field (Higgs field).