Lifting the Eisenhart-Duval Lift to a Minimal Brane

Abstract

The motion of a dynamical system on an $n$-dimensional configuration space may be regarded as the lightlike shadow of null geodsics moving in an $(n+2)$ dimensional spacetime known as its Einsenhart-Duval lift. In this paper it is shown that if the configuration space is $n$-dimensional Euclidean space, and in the absence of magnetic type forces, the Eisenhart-Duval lift may be regarded as an $(n+1)$-brane moving in a flat $(n+4)$ -dimensional space with two times. If the Eisenhart-Duval lift is Ricci flat, then the $(n+1)$-brane moves in such a way as to extremise its spacetime volume. A striking example is provided by the motion of $N$ point particles moving in three-dimensional Euclidean space under the influence of their mutual gravitational attraction. Embeddings with curved configuration space metrics and velocity dependent forces are also be constructed. Some of the issues arising from the two times are addressed.

References in corpus (6)

• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• The Memory Effect for Plane Gravitational Waves
• Hidden Symmetries of Dynamics in Classical and Quantum Physics
• Soft Gravitons & the Memory Effect for Plane Gravitational Waves
• Eisenhart lift for higher derivative systems
• Causality of spacetimes admitting a parallel null vector and weak KAM theory