Galilean Covariance
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Galilean Covariance

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][8]

## Galilean Manifold

The Galilei transformations are

${\displaystyle {\textbf {x}}'=R{\textbf {x}}-{\textbf {v}}t+{\textbf {a}}}$
${\displaystyle t'=t+{\textbf {b}}.}$

where ${\displaystyle R}$ stands for the three-dimensional Euclidean rotations, ${\displaystyle {\textbf {v}}}$ is the relative velocity determining Galilean boosts, a stands for spacial 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,

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

We can write the generators as

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

The non-vanishing commutation relations will then be rewritten as

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

An important Lie subalgebra is

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

${\displaystyle P_{4}}$ is the generator of time translations (Hamiltonian), Pi 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 ${\displaystyle P_{4}=H}$.[]

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.[]

## 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.[9] Gravitational models in this framework have shown to precisely calculate the mercury precession.[10]

## References

1. ^ Takahashi, Yasushi (1988). "Towards the Many-Body Theory with the Galilei Invariance as a Guide: Part I". Fortschritte der Physik/Progress of Physics. 36 (1): 63-81. Bibcode:1988ForPh..36...63T. doi:10.1002/prop.2190360105. eISSN 1521-3978.
2. ^ Takahashi, Yasushi (1988). "Towards the Many-Body Theory with the Galilei invariance as a Gluide Part II". Fortschritte der Physik/Progress of Physics. 36 (1): 83-96. Bibcode:1988ForPh..36...83T. doi:10.1002/prop.2190360106. eISSN 1521-3978.
3. ^ a b Omote, M.; Kamefuchi, S.; Takahashi, Y.; Ohnuki, Y. (1989). "Galilean Covariance and the Schrödinger Equation". Fortschritte der Physik/Progress of Physics (in German). 37 (12): 933-950. Bibcode:1989ForPh..37..933O. doi:10.1002/prop.2190371203. eISSN 1521-3978.
4. ^ Santana, A. E.; Khanna, F. C.; Takahashi, Y. (1998-03-01). "Galilei Covariance and (4,1)-de Sitter Space". Progress of Theoretical Physics. 99 (3): 327-336. arXiv:hep-th/9812223. Bibcode:1998PThPh..99..327S. doi:10.1143/PTP.99.327. ISSN 0033-068X. S2CID 17091575.
5. ^ Duval, C.; Burdet, G.; Künzle, H. P.; Perrin, M. (1985). "Bargmann structures and Newton-Cartan theory". Physical Review D. 31 (8): 1841-1853. Bibcode:1985PhRvD..31.1841D. doi:10.1103/PhysRevD.31.1841. PMID 9955910.
6. ^ Pinski, G. (1968-11-01). "Galilean Tensor Calculus". Journal of Mathematical Physics. 9 (11): 1927-1930. Bibcode:1968JMP.....9.1927P. doi:10.1063/1.1664527. ISSN 0022-2488.
7. ^ Kapu?cik, Edward. (1985). On the relation between Galilean, Poincaré and Euclidean field equations. IFJ. OCLC 835885918.
8. ^ Horzela, Andrzej; Kapu?cik, Edward; Kempczy?ski, Jaroslaw (December 1993). "The Relativistic Invariant and the Galilean Mass of Bodies". Physics Essays. 6 (4): 536-539. Bibcode:1993PhyEs...6..536H. doi:10.4006/1.3029090. ISSN 0836-1398.
9. ^ Goldberger, Walter D. (2009). "AdS/CFT duality for non-relativistic field theory". Journal of High Energy Physics. 2009 (3): 069. arXiv:0806.2867. Bibcode:2009JHEP...03..069G. doi:10.1088/1126-6708/2009/03/069. S2CID 118553009.
10. ^ Ulhoa, Sérgio C.; Khanna, Faqir C.; Santana, Ademir E. (2009-11-20). "Galilean covariance and the gravitational field". International Journal of Modern Physics A. 24 (28n29): 5287-5297. arXiv:0902.2023. Bibcode:2009IJMPA..24.5287U. doi:10.1142/S0217751X09046333. ISSN 0217-751X. S2CID 119195397.