Eisenhart-Duval lift ($g=d{\mathbf{x}}^2+2dt(du-2\mathbf{A}\cdot d\mathbf{x}-Udt)$)

A mechanical Lagrangian quadratic in velocities can be represented by geodesics of a Lorentzian metric with an extra null coordinate. For $L=\frac{1}{2}|\dot{\mathbf{x}}{|}^2-2\mathbf{A}\cdot \dot{\mathbf{x}}-U$, use $g=d{\mathbf{x}}^2+2dt(du-2\mathbf{A}\cdot d\mathbf{x}-Udt)$. The null Killing vector field ${\partial }_u$ gives the conserved momentum $p_u=t\prime$. Fixing $p_u=1$ projects the geodesic equation to the mechanical Euler-Lagrange equations. This is useful for relating mechanical forces to spacetime geometry.

A time-independent gauge transformation $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ in an Eisenhart-Duval lift is the coordinate change $u\mapsto u+2\chi$. It preserves the metric one-form $du-2\mathbf{A}\cdot d\mathbf{x}-Udt$ and adds only $-2d\chi /dt$ to the reduced Lagrangian.

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 $\dot{u}=U+2\mathbf{A}\cdot \dot{\mathbf{x}}-\frac{1}{2}|\dot{\mathbf{x}}{|}^2$.