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.
[1][2][3][4] Previously, in 1985, Duval et al. constructed a similar tensor formulation in the context of Newton–Cartan theory.
[5] Some other authors also have developed a similar Galilean tensor formalism.
stands for the three-dimensional Euclidean rotations,
is the relative velocity determining Galilean boosts, a stands for spatial translations and b, for time translations.
Thus, we can define a scalar product of the type where is the metric of the space-time, and
[3] 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), Pi is the generator of spatial translations (momentum operator),
stands for a generator of rotations (angular momentum operator).
μ α β ρ ν
[4] 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.
[8] Gravitational models in this framework have been shown to precisely calculate the Mercury precession.