= Eisenhart-Duval lift
{c}
{title2=$g=d\mathbf x^2+2dt(du-2\mathbf A\cdot d\mathbf x-Udt)$}
= Eisenhart lift
{c}
{synonym}
= Eisenhart-Duval metric lift
{c}
{synonym}
A mechanical Lagrangian quadratic in velocities can be represented by <geodesics> of a <Lorentzian metric> with an extra null coordinate. For $L=\tfrac12|\dot{\mathbf x}|^2-2\mathbf A\cdot\dot{\mathbf x}-U$, 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.
Back to article page
