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

where stands for the three-dimensional Euclidean rotations, is the relative velocity determining Galilean boosts, a stands for spacial 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 .[3]

Extended Galilei Algebra

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), 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 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

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]

See also

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.

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