Skip to main content
SpringerLink
Log in

The proper rigid frame and the principle of objectivity. A relativistic approach

Собственная недеформируемая система отсчета и принцип обБективности. Релятивистский подход

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

A new principle of objectivity, appropriate for relativistic constitutive equations, is proposed here. This principle is the combination of the concepts of proper observer and rigid frame of reference. A specific frame is introduced and its properties are studied. It is concluded that such a frame is appropriate for derivation of objective constitutive relations.

Riassunto

Qui si propone un nuovo principio di obiettività, appropriato alle equazioni constitutive relativistiche. Questo principio è una combinazione dei concetti di osservatore proprio e di sistema di riferimento rigido. Si introduce un sistema specifico e se ne studiano le proprietà Si conclude che tale sistema è adatto a dedurre relazioni costitutive obiettive.

Резюме

В этой работе предлагается новый принцип обБективности, соответствующий релятвистским конститутивным уравнениям. Этот принцип представляет комбинацию концепций соответствующего наблюдателя и недеформируемой системы отсчета. Вводится специальная система отсчета и исследуются ее свойства. Утверждается, что такая система отсчета удобна для вывода обБективных конститутивных соотношений.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Lianis:Nuovo Cimento,14 B, 57 (1973).

    Article  ADS  MathSciNet  Google Scholar 

  2. R. S. Rivlin andJ. L. Ericksen:Journ. Rat. Mech. Anal.,4, 323 (1955).

    Google Scholar 

  3. A. E. Green andR. S. Rivlin:Arch. Rat. Anal.,4, 487 (1949/60).

    Google Scholar 

  4. W. Noll: Carnegie Institute of Technical, Department of Mathematics, Report No. 17 (June 1957).

  5. W. Noll:Arc. Rat. Mech. Anal.,4, 197 (1958/59).

    Article  Google Scholar 

  6. I. Muller:Arch. Rat. Mech. Anal.,45, 241 (1972).

    Article  Google Scholar 

  7. D. Edelen andJ. McLennan:Int. Journ. Eng. Sci.,11, 813 (1973).

    Article  Google Scholar 

  8. R. J. Atkin andN. Fox:Zeits. Ang. Math. Phys.,24, 853 (1973).

    Article  Google Scholar 

  9. C. C. Wang:Arch. Rat. Mech. Anal.,58, 381 (1976).

    Article  Google Scholar 

  10. A. Bressan:Rend. Sem. Mat. Univ. di Padova,34, 74 (1964).

    Google Scholar 

  11. A. Bressan:Nuovo Cimento,48 B, 201 (1967).

    Article  ADS  Google Scholar 

  12. A. C. Pipkin andR. S. Rivlin:Arch. Rat. Mech. Anat.,4, 129 (1959/60).

    Article  Google Scholar 

  13. R. S. Rivlin:Arch. Rat. Mech. Anal.,4, 262 (1959/60).

    Article  Google Scholar 

  14. R. A. Grot andA. C. Eringen:Int. Journ. Engng. Sci.,4, 611 (1966).

    Article  Google Scholar 

  15. G. Lianis:Nuovo Cimento,66 B, 239 (1970).

    Article  ADS  MathSciNet  Google Scholar 

  16. This is an inertial frame which, at a given instant of time, is at rest with respect to the particle. The use of the local proper Lorentz frame is the basis of theasynchronous formulation of the relativistic statics and dynamics of extended bodies (or of continua). See Subsect.2 .2,G. Cavalleri andG. Salgarelli:Nuovo Cimento,62 A, 722 (1969);S. Aranoff:Nuovo Cimento,10 B, 155 (1972);S. Pahor andJ. Strnd:Nuovo Cimento,20 B, 105 (1974);Ø. Grøn:Lett. Nuovo Cimento,13, 447 (1975).

    Article  ADS  Google Scholar 

  17. L. E. Bragg:Arch. Rat. Mech. Anal.,18, 127 (1965).

    Article  Google Scholar 

  18. L. Söderholm:Arch. Rat. Mech. Anal.,39, 89 (1970).

    Article  Google Scholar 

  19. J. L. Synge:Relativity: The General Theory (Amsterdam, 1960), p. 83.

  20. A rotating frame becomes especially controversial in relativity, due to the famous «rotating-disc paradox», when the disc changes its angular velocity; seeG. Cavalleri:Nuovo Cimento,53 B, 415 (1968);Lett. Nuovo Cimento,3, 608 (1972);A. Brotas:Lett. Nuovo Cimento,1, 217 (1969).

    Article  ADS  Google Scholar 

  21. Greek indices correspond to 1, 2, 3 and 4, while Latin indices correspond to 1, 2 and 3. The latter signify space co-ordinates or space components of tensors, while the index 4 signifies the temporal co-ordinates or components.

  22. F. Fermi:Att. R. Accad. Lincei Rend. Cl. Sci. Fis. Mat. Nat.,31, 21 (1922).

    Google Scholar 

  23. The second index (capital) of the Fermi matrix {itΛ}{it{suά}}{it{inΣ}} is raised or lowered through the Minkowski metric (3.6).

  24. A tensorA ……ϱ is called spatial with respect to the index ϱ, ifA ……ϱ u ϱ=0.

  25. Lower-case Greek and Latin indices will be used for the arbitrary co-ordinate systemx, while capital Greek and Latin indices will correspond to Fermi co-ordinates. We will also use capital indices together with capital letters to denote the material labels,e.g. X R.

  26. In the sequel, unless necessary, we will omit the dependence on\(\bar s(P)\),i.e. \(\Lambda ^\alpha _\Delta = \Lambda ^\alpha _\Delta (\bar s),\Gamma ^\alpha _{\beta \gamma ,\delta } = (\partial \Gamma ^\alpha _{\beta \gamma } /\partial x^\delta )|\bar s,\), etc.

  27. Round brackets () denote symmetrization, while square brackets [] denote skew symmetrization.

  28. M. Born:Ann. der Phys.,30, 1 (1909).

    Article  ADS  Google Scholar 

  29. G. Herglotz:Ann. der Phys.,31, 393 (1910).

    Article  ADS  Google Scholar 

  30. F. Noether:Ann. der Phys.,31, 919 (1910).

    Article  ADS  Google Scholar 

  31. R. H. Boyer:Proc. Roy. Soc.,283 A, 343 (1965).

    Article  ADS  Google Scholar 

  32. N. Rosen:Phys. Rev.,249, 54 (1947).

    Article  ADS  Google Scholar 

  33. G. Saltzman andA. H. Taub:Phys. Rev.,95, 1659 (1954).

    Article  ADS  Google Scholar 

  34. F. A. F. Pirani andG. Williams:Seminaire Janet (Paris, 1962).

  35. K. Gödel:Proceedings of the VI International Congress of Mathematicians, Vol.1 (Providence, R. I., 1950), p. 175.

    Google Scholar 

  36. The word «local» here should be understood to mean that this frame is defined only in an arbitrary small spatial neighborhood of Q.

  37. To avoid proliferation of symbols, we will denote the above frame, the corresponding congruence and the fictitious continuous body by the same symbol.

  38. In what follows, quantities with capital indices will be understood as components in F.C., while those with lower-case indices will be the components in the arbitrary systemx.

  39. It is obvious from (3.2), (3.6) and (3.7) that\(u_\varrho \Lambda ^\varrho _Q = \Lambda _{\varrho 4} \Lambda ^\varrho _Q = n_{4Q} = 0\) and, therefore.

  40. Here the rotation is understood with respect to the compass of inertia or, equivalently, with respect to the Fermi triad.

Download references

Author information

Author notes

  1. G. Lianis (Chair)

    Present address: Technical Mechanics, University of Thessaloniki, Greece

Authors and Affiliations

  1. School of Aeronautics and Astronautics, Purdue University, West Lafayette, Ind.

    G. Lianis (Chair) & J. G. Papastavridis

Authors
  1. G. Lianis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. G. Papastavridis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Traduzione a cura della Redazione.

Перевебено ребакцией.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lianis, G., Papastavridis, J.G. The proper rigid frame and the principle of objectivity. A relativistic approach. Nuovo Cim B 38, 37–60 (1977). https://doi.org/10.1007/BF02726210

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02726210

Search

Navigation