A categorical approach for relativity theory

  • Marcelo CarvalhoEmail author
Original Paper


We give a mathematical interpretation for a previously investigated model, which is based on a set of axioms incorporating fundamental aspects from both Galilei and special relativity (SR). The model considers two time variables, one of them identified with absolute time [characteristic of Galilei relativity (GR)] and the other identified with the local time of SR. Its main characteristics rely on two classes of transformations. One is what we call generalized Lorentz transformation, which includes the standard Lorentz transformations as a particular case. The other is the h-map, which essentially relates absolute and local times. The h-map also determines the basic kinematics of SR from the corresponding kinematics of GR. This allows us to express a Lorentz transformation in terms of a Galilei transformation. It suggests us to see the h-map as a natural transformation between two functors \(\overline{G}\) and \(\overline{L}\), representing the notions of Galilei and Lorentz transformations in functorial language. The categorical framework we develop not only elucidates the structure of Galilei and Lorentz transformations but also provides a unified model for Galilei and special relativity, where the former acquires a distinguished role in itself, not being a mere low speed limit of the latter.


Special relativity Galilei relativity Absolute time Category Natural transformation 


03.30.+p 01.55.+b 02.90.+p 



The author thanks Aurelina Carvalho, José Evaristo Carvalho, Teodora Pereira, Ying Chen, Rina Chen, Aureliana Cabral Raposo, Alexandre Lyra and Américo Cruz for the constant support. The author also thanks Sebastião Alves Dias for improving the final form of the text. This work is done in honor of \(\overline{\text{ IC }}\;\overline{\text{ XC }}\), \(\overline{\text{ MP }}\;\overline{\Theta \Upsilon }\).


  1. [1]
    M Carvalho and A L Oliveira ISRN Mathematical Physics  2013 Article ID 156857 (2013)Google Scholar
  2. [2]
    L P Horwitz, R I Arshansky and A C Elitzur Found. Phys.  18 1159 (1988)ADSCrossRefGoogle Scholar
  3. [3]
    V A Fock The Theory of Space, Time and Gravitation (Oxford: Pergamon Press) (1964)zbMATHGoogle Scholar
  4. [4]
    C H Moller Theory of Relativity (Delhi: Oxford University Press) (1976)Google Scholar
  5. [5]
    J Cushing Am. J. Phys.  35 858 (1967)ADSCrossRefGoogle Scholar
  6. [6]
    H Schubert Categories (New York: Springer Verlag) (1992)Google Scholar
  7. [7]
    S Mac Lane Categories for the Working Mathematician (New York: Springer Verlag) (1998)zbMATHGoogle Scholar
  8. [8]
    Z Oziewicz Int. J. Geom. Methods Mod. Phys  4(5) 739 (2007)MathSciNetCrossRefGoogle Scholar
  9. [9]
    A Chand and I Weiss arXiv:1511.00746 [math.CT] (2015)
  10. [10]
    C W Misner, K S Thorne, J A Wheeler Gravitation (Princeton: Princeton University Press) (2017)zbMATHGoogle Scholar
  11. [11]
    J M Levy-Leblond, M Le Bellac Nuovo Cimento  14 217 (1973)CrossRefGoogle Scholar

Copyright information

© Indian Association for the Cultivation of Science 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de Santa CatarinaFlorianópolisBrazil

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