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.^[6][7]

Galilean manifold

The Galilei transformations are

${\displaystyle {\begin{aligned}\mathbf {x} '&=R\mathbf {x} -\mathbf {v} t+\mathbf {a} \\t'&=t+\mathbf {b} .\end{aligned))}$

where ${\displaystyle R}$ stands for the three-dimensional Euclidean rotations, ${\displaystyle \mathbf {v} }$ is the relative velocity determining Galilean boosts, a stands for spatial translations and b, for time translations. Consider a free mass particle ${\displaystyle m}$; the mass shell relation is given by ${\displaystyle p^{2}-2mE=0}$.

We can then define a 5-vector,

${\displaystyle p^{\mu }=(p_{x},p_{y},p_{z},m,E)=(p_{i},m,E)}$,

with ${\displaystyle i=1,2,3}$.

Thus, we can define a scalar product of the type

${\displaystyle p_{\mu }p_{\nu }g^{\mu \nu }=p_{i}p_{i}-p_{5}p_{4}-p_{4}p_{5}=p^{2}-2mE=k,}$

where

${\displaystyle g^{\mu \nu }=\pm {\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&-1\\0&0&0&-1&0\end{pmatrix)),}$

is the metric of the space-time, and ${\displaystyle p_{\nu }g^{\mu \nu }=p^{\mu ))$.^[3]

Extended Galilei algebra

A five dimensional Poincaré algebra leaves the metric ${\displaystyle g^{\mu \nu ))$ invariant,

${\displaystyle {\begin{aligned}[][P_{\mu },P_{\nu }]&=0,\\{\frac {1}{i))~[M_{\mu \nu },P_{\rho }]&=g_{\mu \rho }P_{\nu }-g_{\nu \rho }P_{\mu },\\{\frac {1}{i))~[M_{\mu \nu },M_{\rho \sigma }]&=g_{\mu \rho }M_{\nu \sigma }-g_{\mu \sigma }M_{\nu \rho }-g_{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho },\end{aligned))}$

We can write the generators as

${\displaystyle {\begin{aligned}J_{i}&={\frac {1}{2))\epsilon _{ijk}M_{jk},\\K_{i}&=M_{5i},\\C_{i}&=M_{4i},\\D&=M_{54}.\end{aligned))}$

The non-vanishing commutation relations will then be rewritten as

${\displaystyle {\begin{aligned}\left[J_{i},J_{j}\right]&=i\epsilon _{ijk}J_{k},\\\left[J_{i},C_{j}\right]&=i\epsilon _{ijk}C_{k},\\\left[D,K_{i}\right]&=iK_{i},\\\left[P_{4},D\right]&=iP_{4},\\\left[P_{i},K_{j}\right]&=i\delta _{ij}P_{5},\\\left[P_{4},K_{i}\right]&=iP_{i},\\\left[P_{5},D\right]&=-iP_{5},\\[4pt]\left[J_{i},K_{j}\right]&=i\epsilon _{ijk}K_{k},\\\left[K_{i},C_{j}\right]&=i\delta _{ij}D+i\epsilon _{ijk}J_{k},\\\left[C_{i},D\right]&=iC_{i},\\\left[J_{i},P_{j}\right]&=i\epsilon _{ijk}P_{k},\\\left[P_{i},C_{j}\right]&=i\delta _{ij}P_{4},\\\left[P_{5},C_{i}\right]&=iP_{i}.\end{aligned))}$

An important Lie subalgebra is

${\displaystyle {\begin{aligned}[][P_{4},P_{i}]&=0\\[][P_{i},P_{j}]&=0\\[][J_{i},P_{4}]&=0\\[][K_{i},K_{j}]&=0\\\left[J_{i},J_{j}\right]&=i\epsilon _{ijk}J_{k},\\\left[J_{i},P_{j}\right]&=i\epsilon _{ijk}P_{k},\\\left[J_{i},K_{j}\right]&=i\epsilon _{ijk}K_{k},\\\left[P_{4},K_{i}\right]&=iP_{i},\\\left[P_{i},K_{j}\right]&=i\delta _{ij}P_{5},\end{aligned))}$

${\displaystyle P_{4))$ is the generator of time translations (Hamiltonian), P_i is the generator of spatial translations (momentum operator), ${\displaystyle K_{i))$ is the generator of Galilean boosts, and ${\displaystyle J_{i))$ stands for a generator of rotations (angular momentum operator). The generator ${\displaystyle P_{5))$ is a Casimir invariant and ${\displaystyle P^{2}-2P_{4}P_{5))$ is an additional Casimir invariant. This algebra is isomorphic to the extended Galilean Algebra in (3+1) dimensions with ${\displaystyle P_{5}=-M}$, The central charge, interpreted as mass, and ${\displaystyle P_{4}=-H}$.^{[citation needed]}

The third Casimir invariant is given by ${\displaystyle W_{\mu \,5}W^{\mu }{}_{5))$, where ${\displaystyle W_{\mu \nu }=\epsilon _{\mu \alpha \beta \rho \nu }P^{\alpha }M^{\beta \rho ))$ is a 5-dimensional analog of the Pauli–Lubanski pseudovector.^[4]

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

${\displaystyle g^{\mu \nu }={\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\end{pmatrix)).}$

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.^[9]