Il Nuovo Cimento B (1971-1996)

, Volume 25, Issue 2, pp 757–785 | Cite as

Strong and weak gravity: A class of generally covariant mixing models of spin-2 neutral fields: Linearization

  • W. C. Hammel


By linearizing, in an unorthodox way, the nonlinear partial differential equation system proposed to describe the interaction of a massless and massive graviton, it is discovered that the equations yield a variety of theories with different structure. The general system of equations which results from this linearization is a Lorentz-covariant system of coupled equations in two symmetric Lorentz tensor fields. We distinguish between situations where the system may be uncoupled by a linear transformation of the tensor fields, thus permitting the definition of diagonalizing fields, and those situations where this is not possible. The diagonalizable cases are examined for mass and spin structure, and are divided into perturbative and nonperturbative cases. One of the perturbative cases contains the theory of Aichelburg and Mansouri. Finally we discuss the procedure for solving the explicitly coupled equations.


Linearizzando, con procedimento non ortodosso, il sistema non lineare di equazioni differenziali parziali che è stato proposto per descrivere l’interazione fra un gravitone dotato di massa e uno privo di massa, si è scoperto che le equazioni producono una varietà di teorie con strutture diverse. Il sistema generale di equazioni che si ottiene da questa linearizzazione è un sistema covariante di Lorentz di equazioni accoppiate in due campi tensoriali simmetrici di Lorentz. Si fa distinzione fra casi in cui si può disaccoppiare il sistema per mezzo di una trasformazione, lineare dei campi tensoriali, permettendo così la definizione di campi diagonalizzanti, e casi in cui ciò non è possibile. Si esamina la massa e la struttura dello spin nei casi in cui si hanno campi diagonalizzabili e si distingue ulteriormente fra casi perturbativi e non perturbativi. Uno dei casi perturbativi comprende la teoria di Aichelburg e Mansouri. Infine si discute il procedimento per risolvere le equazioni accoppiate esplicitamente.


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  1. (1).
    E. Lubkin:Bull. Amer. Phys. Soc.,11, 397 (1966).Google Scholar
  2. (2).
    C. J. Isham, A. Salam andJ. Strathdee:Phys. Rev. D,3, 867 (1971).MathSciNetADSCrossRefGoogle Scholar
  3. (3).
    W. C. Hammel andE. Lubkin: University of Wisconsin-Milwaukee preprint, «Strong» and «weak» gravity: A class of generally covariant mixing models of spin-2neutral fields, UWM-4867-73-7, which supersedes UWM-4867-72-6.Google Scholar
  4. (4).
    V. I. Ogievetsky andI. V. Polubarinov:Ann. of Phys.,35, 167 (1965), see particularly p. 170 and Sect.5.ADSCrossRefGoogle Scholar
  5. (5).
    S. Deser: inNonpolynomial Lagrangians, Renormalization and Gravity, Vol.1, edited byA. Salam., ofLectures from the Coral Gables Conference on Fundamental Interactions at High Energies (New York, N. Y., 1971).Google Scholar
  6. (6).
    P. C. Aichelburg andR. Mansouri:Generally covariant massive gravitation, University of Vienna preprint (1971).Google Scholar
  7. (7).
    R. Mansouri:Allgemein.-kovariante Zwei-Tensor Theorie der Gravitation (Thesis), University of Vienna (1972);Acta Phys. Austriaca,37, 152 (1973).Google Scholar
  8. (8).
    J. K. Lawrence andE. T. Toton:Ann. of Phys.,72, 293 (1972).ADSCrossRefGoogle Scholar
  9. (9).
    J. L. Synge andA. Schild:Tensor Calculus (Toronto, 1962);E. Schrödinger:Space-Time Structure (Cambridge, 1963).Google Scholar
  10. (10).
    N. Rosen:Phys. Rev.,57, 147, 151, 154 (1940);Ann. of Phys.,22, 1 (1963);38, 170 (1966). See alsoA. Papapetrou:Proc. Roy. Irish Acad.,52 A, 11 (1948).MathSciNetADSCrossRefGoogle Scholar
  11. (11).
    S. N. Gupta:Rev. Mod. Phys.,29, 334 (1957);Phys. Rev.,96, 1683 (1954);Proc. Phys. Soc.,65 A, 161 (1952).MathSciNetADSCrossRefMATHGoogle Scholar
  12. (12).
    Y. Choquet-Bruhat:Comm. Math. Phys.,12, 16 (1969).MathSciNetADSCrossRefGoogle Scholar
  13. (13).
    See for exampleYa. B. Zeldovich andI. D. Novikov:Relativistic, Astrophysics (Chicago, Ill., 1971). See in particular Subsect.2.5.Google Scholar
  14. (14).
    See for exampleA. Peres:Invariant evolution of gravitational field, inRelativity and Gravitation, edited byC. G. Kuper andA. Peres (New York, N. Y., 1971).Google Scholar
  15. (15).
    B. S. De Witt:The quantum theory of gravity, publication No. 18, Institute of Field Physics, University of North Carolina (1966).Google Scholar
  16. (16).
    J. A. Wheeler:Geometrodynamics (New York, N. Y., 1961);Geometrodynamics and the issue of the final state, inRelativity, Groups and Topology, edited byC. De Witt andB. De Witt (New York, N. Y., 1964).Google Scholar
  17. (17).
    To be published inJourn. Math. Phys. Google Scholar
  18. (18).
    R. Adler, M. Bazin andM. Schiffer:Introduction to General Relativity (New York, N. Y., 1965).Google Scholar
  19. (19).
    R. U. Sexl:Fortschr. Phys.,15, 269 (1967).CrossRefGoogle Scholar
  20. (20).
    G. Wentzel:Quantum Theory of Fields (New York, N. Y., 1949), p. 205.Google Scholar
  21. (21).
    J. L. Synge andA. Schild:Tensor Calculus (Toronto, 1962).Google Scholar
  22. (22).
    E.g.,J. A. Wolf:Spaces of Constant Curvature (New York, N. Y., 1967).Google Scholar
  23. (23).
    W. E. Thirring:Ann. of Phys.,16, 96 (1961).MathSciNetADSCrossRefMATHGoogle Scholar
  24. (24).
    A. Bers, R. Fox, C. G. Kuper andS. G. Lipson:The impossibility of free tachyons, inRelativity and Gravitation, edited byC. G. Kuper andA. Peres (New York, N. Y., 1971).Google Scholar

Copyright information

© Società Italiana di Fisica 1975

Authors and Affiliations

  • W. C. Hammel
    • 1
  1. 1.Department of PhysicsUniversity of Wisconsin-MilwaukeeMilwaukee

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