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 NewtonâÂÂCartan theory. Some other authors also have developed a similar Galilean tensor formalism.
The Galilei transformations are
where stands for the three-dimensional Euclidean rotations, is the relative velocity determining Galilean boosts, a stands for spatial translations and b, for time translations. Consider a free mass particle ; the mass shell relation is given by .
We can then define a 5-vector,
with .
Thus, we can define a scalar product of the type
where
is the metric of the space-time, and .
A five dimensional Poincaré algebra leaves the metric invariant,
We can write the generators as
The non-vanishing commutation relations will then be rewritten as
An important Lie subalgebra is
is the generator of time translations (Hamiltonian), P<sub>i</sub> is the generator of spatial translations (momentum operator), is the generator of Galilean boosts, and stands for a generator of rotations (angular momentum operator). The generator is a Casimir invariant and is an additional Casimir invariant. This algebra is isomorphic to the extended Galilean Algebra in (3+1) dimensions with , The central charge, interpreted as mass, and .
The third Casimir invariant is given by , where is a 5-dimensional analog of the PauliâÂÂLubanski pseudovector.
In 1985 Duval, Burdet and Kunzle showed that four-dimensional NewtonâÂÂCartan theory of gravitation can be reformulated as KaluzaâÂÂKlein 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.