Galilei-covariant tensor formulation
The Galilei-covariant tensor formulation is a method for treating non-relativistic physics using the extended Galilei group as the representation group of the theory. It is constructed in the light cone of a five dimensional manifold. Takahashi et al., in 1988, began a study of Galilean symmetry, where an explicitly covariant non-relativistic field theory could be developed. The theory is constructed in the light cone of a (4,1) Minkowski space. Previously, in 1985, Duval et al. constructed a similar tensor formulation in the context of Newton–Cartan theory. Some other authors also have developed  a similar Galilean tensor formalism. Galilean Manifold. The Galilei transformations are where formula_3 stands for the three-dimensional Euclidean rotations, formula_4 is the relative velocity determining Galilean boosts, a stands for spatial translations and b, for time translations. Consider a free mass particle formula_5; the mass shell relation is given by formula_6. We can then define a 5-vector, formula_7, with formula_8. Thus, we can define a scalar product of the type where is the metric of the space-time, and formula_11. Extended Galilei algebra. A five dimensional Poincaré algebra leaves the metric formula_12 invariant, We can write the generators  as The non-vanishing commutation relations will then be rewritten as An important Lie subalgebra is formula_42 is the generator of time translations (Hamiltonian), "P" is the generator of spatial translations (momentum operator), formula_43 is the generator of Galilean boosts, and formula_44 stands for a generator of rotations (angular momentum operator). The generator formula_45 is a Casimir invariant and formula_46 is an additional Casimir invariant. This algebra is isomorphic to the extended Galilean Algebra in (3+1) dimensions with formula_47, The central charge, interpreted as mass, and formula_48. The third Casimir invariant is given by formula_49, where formula_50 is  a 5-dimensional analog of the Pauli–Lubanski pseudovector. Bargmann structures. In 1985 Duval, Burdet and Kunzle showed that four-dimensional Newton–Cartan theory of gravitation can be reformulated as Kaluza–Klein reduction of five-dimensional Einstein gravity along a null-like direction. The metric used is the same as the Galilean metric but with all positive entries This lifting is considered to be useful for non-relativistic holographic models. Gravitational models in this framework have been shown to precisely calculate the mercury precession.