= Null Kaluza-Klein reduction of a stationary force
{title2=$p_u=t\prime\ne0$}
For an <Eisenhart-Duval lift>, reduction along the null isometry fixes a nonzero conserved $p_u=t\prime$ and projects <geodesics> to $(t,\mathbf x)$. One never divides by $g_{uu}=0$. When $\mathbf E=-\nabla U$ and $\nabla\times\mathbf A=\mathbf B$, the reduced equation is $\ddot{\mathbf x}=\mathbf E+2\mathbf B\times\dot{\mathbf x}$. Every trajectory admits a null lift by choosing $\dot u=U+2\mathbf A\cdot\dot{\mathbf x}-\tfrac12|\dot{\mathbf x}|^2$.
Back to article page
