Electromagnetic coupling of a sine-Gordon topological current ($j^{\mu }=({\theta }_x,-{\theta }_t),Q=\int {\theta }_{x}dx$)

The identically conserved topological current $j$ couples to an external electromagnetic gauge potential through $-A_{\mu }{j}^{\mu }=-A_0{\theta }_x+A_1{\theta }_t$. A gauge transformation changes this coupling only by a divergence. The Euler-Lagrange field equation becomes ${\theta }_{tt}-{\theta }_{xx}+\sin \theta +E=0$, with $E={\partial }_t{A}_1-{\partial }_x{A}_0$. A unit Sine-Gordon kink has unnormalized charge $Q=2\pi$ and mass $8$. For spatially uniform forcing, its field momentum balance is $\dot{P}=QE$ when the boundary stress contributions cancel.