On the differential geometry of time-like curves in Minkowski spacetime

American Journal of Physics

Volumn 74, Issue 11, 2006, Pages 1012-1016

  Formiga, J B ^a   Romero, C ^a  

^a FEDERAL UNIVERSITY OF PARAÍBA   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33751202720     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.2232644     Document Type: Article

References (28)

1. A readable account of the geometry of Minkowski spacetime may be found in G. L. Naber, The Geometry of Minkowski Spacetime (Dover, New York, 2003).
2. See, for instance, M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, Englewood Cliffs, NJ, 1976), Chap. 1.
3. The reader interested in the geometry of null curves is referred to W. B. Bonnor, "Null curves in a Minkowski spacetime," Tensor 20, 229-242 (1969);
4. J. L. Synge, "An action principle for photons," Nature (London) 317, 675-675 (1985);
5. H. Urbantke, "Local differential geometry of null curves in conformally flat spacetime," J. Math. Phys. 30, 2238-2245 (1989).
6. See, for example, J. L. Synge and A. Schild, Tensor Calculus (Dover, New York, 1978), Chap. 2.
7. C. Lanczos, Space through the Ages (Academic, London, 1970).
8. 1, to be non-negative.
9. The index lowered form of Σ is antisymmetric, which means that the original matrix lies in the Lie algebra of the Lorentz group.
10. It is usual to consider a Poincaré transformation as a coordinate transformation in the passive sense. Here, we regard it in the active sense, that is, as motion in Minkowski spacetime.
11. See, for instance, the appendix to Chap. 4 of Ref. 2.
12. In developing the Serret-Frenet formalism in Minkowski spacetime we are ultimately interested in applications to the motion of physical particles. Therefore we shall restrict ourselves to time-like curves, although the formalism may also work for space-like curves. The consideration of null curves should introduce some difficulties.
13. A matrix Λ is said to be a Lorentz matrix if it is pseudo-orthogonal, that is, if it satisfies the condition Λ′η Λ= η, where Λ′ is the transpose of Λ and η denotes the Minkowski metric. If det Λ=1. then Λ is called a proper Lorentz matrix.
14. The Serret-Frenet equations describe a curve of Lorentz transformation matrices generated by the curve of Lorentz Lie algebra matrices, which describe how the Serret-Frenet frame changes with respect to a constant orthonormal frame.
15. S. Lang, Analysis 1 (Addison-Wesley, Reading, MA, 1968), pp. 383-386.
16. E. Kreyszig, Differential Geometry (Dover, New York, 1991), Chap. 2.
17. M. Carmeli, "Motion of a charge in a gravitational field," Phys. Rev. B 138, 1003-1007 (1965).
18. J. L. Synge, "Time-like helices in flat spacetime," Proc. R. Ir. Acad., Sect. A 65, 27-41 (1967).
19. E. Pina, "Lorentz transformation and the motion of a charge in a constant electromagnetic field," Rev. Mex. Fis. 16, 233-236 (1967).
20. E. Honig, E. Schucking, and C. Vishveshwara, "Motion of charged particles in homogeneous electromagnetic fields," J. Math. Phys. 15, 774-781 (1974).
21. H. Ringermacher, "Intrinsic geometry of curves and the Minkowski force," Phys. Lett. A 74, 381-383 (1979).
22. J. H. Caltenco, R. Linares, and J. L. López-Bonilla, "Intrinsic geometry of curves and the Lorentz equation," Czech. J. Phys. 52, 839-842 (2002).
23. See, for instance, L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Addison-Wesley, Reading, MA, 1962), Chap. 3.
24. See, for instance, H. Stephani, General Relativity (Cambridge U.P., Cambridge, 1994), pp. 187-188.
25. W. Rindler, Essential Relativity (Springer, New York, 1977), Chap. 2.
26. G. Muñoz, "Charged particle in a constant electromagnetic field: Covariant solution," Am. J. Phys. 65, 429-433 (1997).
27. AB.
28. 1, as non-negative scalars (Ref. 18).