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]
The Galilei transformations are
\begin{align} x'&=Rx-vt+a\\ t'&=t+b. \end{align}
where
R
v
m
p2-2mE=0
We can then define a 5-vector,
p\mu=(px,py,pz,m,E)=(pi,m,E)
i=1,2,3
Thus, we can define a scalar product of the type
p\mup\nug\mu\nu=pipi-p5p4-p4p5=p2-2mE=k,
where
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
p\nug\mu\nu=p\mu
A five dimensional Poincaré algebra leaves the metric
g\mu\nu
\begin{align}[] [P\mu,P\nu]&=0,\\
| 1 | |
| i |
~[M\mu\nu,P\rho]&=g\mu\rhoP\nu-g\nu\rhoP\mu,\\
| 1 | |
| i |
~[M\mu\nu,M\rho\sigma]&=g\mu\rhoM\nu\sigma-g\mu\sigmaM\nu\rho-g\nu\rhoM\mu\sigma+η\nu\sigmaM\mu\rho, \end{align}
We can write the generators as
\begin{align} Ji&=
| 1 | |
| 2 |
\epsilonijkMjk,\\ Ki&=M5i,\\ Ci&=M4i,\\ D&=M54. \end{align}
The non-vanishing commutation relations will then be rewritten as
\begin{align} \left[Ji,Jj\right]&=i\epsilonijkJk,\\ \left[Ji,Cj\right]&=i\epsilonijkCk,\\ \left[D,Ki\right]&=iKi,\\ \left[P4,D\right]&=iP4,\\ \left[Pi,Kj\right]&=i\deltaijP5,\\ \left[P4,Ki\right]&=iPi,\\ \left[P5,D\right]&=-iP5,\\[4pt] \left[Ji,Kj\right]&=i\epsilonijkKk,\\ \left[Ki,Cj\right]&=i\deltaijD+i\epsilonijkJk,\\ \left[Ci,D\right]&=iCi,\\ \left[Ji,Pj\right]&=i\epsilonijkPk,\\ \left[Pi,Cj\right]&=i\deltaijP4,\\ \left[P5,Ci\right]&=iPi. \end{align}
An important Lie subalgebra is
\begin{align}[] [P4,Pi]&=0\\[] [Pi,Pj]&=0\\[] [Ji,P4]&=0\\[] [Ki,Kj]&=0\\ \left[Ji,Jj\right]&=i\epsilonijkJk,\\ \left[Ji,Pj\right]&=i\epsilonijkPk,\\ \left[Ji,Kj\right]&=i\epsilonijkKk,\\ \left[P4,Ki\right]&=iPi,\\ \left[Pi,Kj\right]&=i\deltaijP5, \end{align}
P4
Ki
Ji
P5
| 2-2P | |
| P | |
| 4P |
5
P5=-M
P4=-H
The third Casimir invariant is given by
W\mu5
| \mu{} | |
| W | |
| 5 |
W\mu\nu=\epsilon\mu\alpha\beta\rho\nuP\alphaM\beta\rho
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
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]