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A new Lie algebra expansion method: Galilei expansions to Poincare and Newton-Hooke

Francisco J. Herranz·Mariano Santander
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Abstract

We modify a Lie algebra expansion method recently introduced for the (2+1)-dimensional kinematical algebras so as to work for higher dimensions. This new improved and geometrical procedure is applied to expanding the (3+1)-dimensional Galilei algebra and leads to its physically meaningful `expanded' neighbours. One expansion gives rise to the Poincare algebra, introducing a curvature $-1/c^2$ in the flat Galilean space of worldlines, while keeping a flat spacetime which changes from absolute to relative time in the process. This formally reverses, at a Lie algebra level, the well known non-relativistic contraction $c\to \infty$ that goes from the Poincare group to the Galilei one; this expansion is done in an explicit constructive way. The other possible expansion leads to the Newton-Hooke algebras, endowing with a non-zero spacetime curvature $\pm 1/τ^2$ the spacetime, while keeping a flat space of worldlines.

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Francisco J. Herranz, Mariano Santander. 2000-10-09. A new Lie algebra expansion method: Galilei expansions to Poincare and Newton-Hooke. https://arxiv.org/abs/math-ph/9909005

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