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 spacial 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), "Pi" 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.