, Volume 61, Issue 3, pp 370–374 | Cite as

Note on the Interpretation of Proper Mass as a Constant Lagrange Multiplier

  • R. A. KrikorianEmail author

The timelike world line C0 of a free particle is a maximizing curve for the integral I = ∫ds in the class Γ of neighboring admissible timelike curves joining the events A, B, and satisfying the side-condition imposed on the 4-velocity \( \upvarphi ={g}_{ij}\ {\overset{\cdot }{x}}^i{\overset{\cdot }{x}}^j=1\left({\overset{\cdot }{x}}^i={dx}^i/ ds\right) \). Considering the problem of extremizing integral I as a time optimal problem, we show that the multiplier λ (s) associated with the equation φ = 1 is constant along C0 and may be identified with the proper mass m of the free particle. The constancy of m can thus be regarded as a consequence of the path dependence of proper time.


proper mass: Lagrange multiplier 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. Fock, The theory of Space, Time and Gravitation, 2nd revised Edition (Pergamon Press) 1966.Google Scholar
  2. 2.
    Yu. V. Novozhilov and Yu. A. Yappa, Electrodynamics, Mir, Moscow, 1981.Google Scholar
  3. 3.
    A. Peres and N. Rosen, Nuovo Cimento, XVIII, N.4, 644, 1960.Google Scholar
  4. 4.
    G. A. Bliss, Lectures on the calculus of variations, Univ. Chicago Press, Chicago, 1946; Am. J. Math., 52, 673, 1930.CrossRefGoogle Scholar
  5. 5.
    M. R. Hestenes, Am. J. Math., 58, 391, 1936.CrossRefGoogle Scholar
  6. 6.
    M. Morse, The Calculus of Variations in the Large, American Mathematical Society, New York, 1934.CrossRefzbMATHGoogle Scholar
  7. 7.
    L. Infeld, Bull. Acad. Pol. Sci., 5, 491, 1951.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Collège de France - Sorbonne Université, CNRS, Institut d’Astrophysique de ParisParisFrance

Personalised recommendations