Il Nuovo Cimento B (1971-1996)

, Volume 69, Issue 2, pp 206–212 | Cite as

Some structural considerations on the Brans-Dicke theory of gravitation

  • S. Ikeda


Some interesting physical features of Brans-Dicke’s scalar ϕ are found in connection with the conformal scalar σ. First, by emphasizing the conformally Riemannian structure of Brans-Dicke’s field, it is shown that the main Brans-Dicke’s field equations are comparable with those derived from the conformal transformation of Einstein’s field equation, in which the two scalars ϕ and σ are found to play essentially the same role at the stage of field equations. Next, in order to obtain another new physical functions of ϕ, the spatial structure is extended to conformally non-Riemannian by introducing the torsion. By doing so, it is found that Brans-Dicke’s scalar ϕ also contributes to the torsion within the range of the Weyl-Dirac theory with torsion through the relation ϕ∼ exp ^-2σ]∼β2, β being Dirac’s conformal scalar.

Некоторые структурные рассмотрения теории гравитации бранса-дикка


Получены некоторые интересные физические особенности скаляра ф Бранса-Дикка, в связи с конформным скаляром σ. Сначала, обращая особое внимание на конформную риманову структуру поля Бранса-Дикка, показывается, что основные уравнения поля Бранса-Дикка сравнимы с уравнениями, полученными из конформного преобразования уравнения поля Эйнштейна, где два скаляра ϕ и σ играют, по существу, одинаковую роль на стадии уравнений поля. Затем, чтобы полуить другую новую физическую функцию ϕ, пространственная структура обобщается на конформно нериманову структуру, вводя кручения. Получено, что скаляр ϕ Бранса-Дикка также дает вклад в кручение, в рамках теории Вейля-Дирака с кручением, через соотношение ϕ∼ exp, ^-2σ]∼β2, где β конформный скаляр Дирака.


Si trovano alcune caratteristiche fisiche interessanti degli scalari ϕ di Brans-Dicke in relazione alla scalare conforme σ. Prima, enfatizzando la struttura conformemente riemanniana del campo di Brans-Dicke, si mostra che le equazioni di campo principali di Brans-Dicke sono confrontabili con quelle derivate dalla trasformazione conforme dell’equazione di campo di Einstein, dove si è trovato che i due scalari ϕ e σ svolgono essenzialmente lo stesso ruolo allo stadio di equazioni di campo. Quindi, allo scopo di ottenere un’altra nuova funzione fisica di ϕ, la struttura spaziale è ampliata a conformemente non riemanniana introducendo la torsione. Facendo ciò, si è trovato che lo scalare ϕ di Brans-Dicke contribuisce alla torsione nell’àmbito della teoria di Weyl-Dirac con torsione attraverso la relazione ϕ∼ exp ^-2σ]∼β2, dove β è lo scalare conforme di Dirac.


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  1. (1).
    C. Brans andR. H. Dicke:Phys. Rev.,124, 925 (1961).MathSciNetADSCrossRefMATHGoogle Scholar
  2. (2).
    R. H. Dicke:The Theoretical Significance of Experimental Relativity (New York, N. Y., London and Paris, 1965).Google Scholar
  3. (3).
    As has often been discussed with respect to Mach’s principle, the gravitational «constant» is influenced by the mass distribution (in a uniform expanding universe):GM/Rc 2∼1, whereM stands for the finite mass of the visible universe andR for the radius of the boundary of the visible universe (1,2).Google Scholar
  4. (4).
    J. Cohn:Gen. Rel. Grav.,6, 143 (1975).ADSCrossRefGoogle Scholar
  5. (5).
    D. Gregorash andG. Papini:Nuovo Cimento B,55, 37 (1980).MathSciNetADSCrossRefGoogle Scholar
  6. (6).
    Quite recently, eq. (10) has been regarded as a vacuum equation (under ηT μλ=0) byJ. Cervero andP. G. Estevezs:Lett. Nuovo Cimento,30, 323 (1981).CrossRefGoogle Scholar
  7. (7).
    P. A. M. Dirac:Proc. R. Soc. London Ser. A,165, 199 (1938);333, 403 (1973).ADSCrossRefGoogle Scholar
  8. (8).
    H. Weyl:Sitzungsber. Preuss. Acad. Wiss., 465 (1918).Google Scholar
  9. (9).
    F. W. Hehl, P. von der Heyde, G. D. Kerlick andJ. M. Nester:Rev. Mod. Phys.,48, 393 (1976).ADSCrossRefGoogle Scholar
  10. (10).
    From eq. (1), the relation ϕ∼exp[ε−1]σ is obtained. And another relation β∼exp[∼σ] is obtained by comparing the BD Lagrangian and the WD Lagrangian. Furthermore, the relation β∼exp[−σ] is obtained by comparing eqs. (16) and (17) (with α=σ). These three relations become compatible only when ε=−1. Fourtunately, these situations are ensured byCohn (4).ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica 1982

Authors and Affiliations

  • S. Ikeda
    • 1
  1. 1.Department of Mechanical Engineering, Faculty of Science and TechnologyScience University of TokyoNoda, ChibaJapan

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