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Erratum: Lattices of effectively nonintegral dimensionality

On the separation of Einsteinian substructures

J. Math. Phys. 18, 2511 (1977); doi:10.1063/1.523215

Issue Date: December 1977

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Jerzy F. Plebański
Centro de Investigación y de Estudios Avanzados del I.P.N., México D.F., 14−740, México
Within the spinorial version of the Cartan structure formulas with the built-in (complex) Einstein vacuum equations some closed semi-Einsteinian substructures are isolated and discussed. Then the idea of semigraviton is introduced, and its relationship to Penrose's nonlinear graviton is described. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
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PACS

  • 04.60.+n
    Relativity and gravitation Quantum theory of gravitation
  • YEAR: 1977

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (32)
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  1. J. F. Plebański and A. Schild, Nuovo Cimento 35B, 35–53 (1976).
  2. J. F. Plebański, J. Math. Phys. 16, 2395 (1975).
  3. J. F. Plebański and S. Hacyan, J. Math. Phys. 16, 2403 (1975).
  4. J. D. Finley, III and J. F. Plebański, J. Math. Phys. 17, 585 (1976).
  5. S. Hacyan and J. F. Plebański, J. Math. Phys. 17, 2203 (1976).
  6. C. P. Boyer and J. F. Plebański, J. Math. Phys. 18, 1022 (1977).
  7. E. T. Newman, in Proceedings of GR7 Conference, Tel Aviv, 1974.
  8. C. W. Fette, Allen I. Janis, and E. T. Newman, J. Math. Phys. 17, 660 (1976).
  9. E. T. Newman, In Proceedings of ASST Conf., Cincinnati, June 1976, edited by L. Witten (Plenum, New York, 1977), p. 229.
  10. This is of course, not the most general gauge group: The most general gauge transformations, [script G], contain in addition to (1.17): (i) the antilinear spinorial transformations which map the dotted spinors into undotted and vice versa; they correspond to the improper null tetrad transformations (g<sub>AB-dot</sub>)-->(g<sub>AB-dot</sub>)T with the determinant equal to minus one; these transformations are allowed if the orientation of M remains undetermined, and (ii) the dilatations of undotted spinors by say lambda[not-equal]0, compensated by the dilatations of the dotted spinors by lambda−1. This group component, however, does not act effectively on the fundamental objects gAB.
  11. “6−1 = 5” is to be understood on the basis of the fact that (16c) can be interpreted as 3SAB[logical and]SCD = rhodelta<sub>(C</sub><sup>A</sup>delta<sub>D)</sub><sup>B</sup>, with rho to be determined. Then, SAB[logical and]SCD = SCD[logical and]SAB implies that we end up with 6−1 = 5 of the effective conditions on SAB.
  12. A. Tomimatsu and H. Sato, Progr. Theor. Phys. 50, 95 (1973).
  13. M. Yamazaki, Kanazawa Univ. Preprint, 1976.
  14. F. Ernst, J. Math. Phys. 17, 1091 (1976).
  15. R. Penrose, a talk at the “Riddle of Gravity” symposium in honor of P. G. Bergmann's 60th birthday, in Syracuse, March 1975.
  16. R. Penrose, “Non-Linear Gravitation,” First Award Gravity Foundation Essay (1975) and Gen. Rel. Grav. 7, 31 (1976).
  17. I. Robinson, King's College Lectures, unpublished (1956).
  18. J. F. Plebański and I. Robinson, Phys. Rev. Lett. 37, 493 (1976).
  19. J. F. Plebański and I. Robinson, in Proceedings of Symposium on Asymptotic Structures of Space—Time, Cincinnati, June 1977, Edited by F. Eposito and L. Witten (Plenum, New York, 1977), p. 361
    20. and in Proceedings of V-e Colloque Int. de Theorie de Groups, Montreal, July 1976, edited by R. Sharp (Academic, New York, 1977).
  20. A. García, J. F. Plebański, and I. Robinson, “Null Strings and Complex Einstein—Maxwell Fields with lambda,” to be published in Gen. Rel. Grav.
  21. J. D. Finley, III and J. F. Plebański, J. Math. Phys. 17, 2207 (1976).
  22. A. García and J. F. Plebański, “Seven Parameter Type Metric in Canonical Coordinates,” to be published in Nuovo Cimento.
  23. J. D. Finley, III and J. F. Plebański, “Spinorial structures and electromagnetic hyperheavens,” J. Math. Phys. 18, 1662 (1977).
  24. For the general theory of the real slices of the complex V4[prime]s see thesis of K. Rózga, IFT, Hoźa 69, University of Warsaw, Real Slices of Complex Riemannian Spaces.”
  25. With M real,

    [dformula g[subformula -[over c]][sub AB-dot]]

    and

    [dformula g[subformula -[over i]][sub AB-dot]]

    can be chosen Hermitian; thus, the concept of a real “time” is meaningful—although uncorrelated—in [script S]<sub>[script H]</sub> and [script S]E in the case of a real structure.

  26. F. J. Ernst, Phys. Rev. 168, 1415 (1968),
    27. J. Math. Phys. 15, 1409 (1974).
  27. F. Ernst and J. F. Plebański, “Killing Structures and Complex [script E]-Potentials” to be published in Ann. Phys. (Oct., 1977).
  28. W. Kinnersley, J. Math. Phys. 14, 651 (1973).
  29. W. Kinnersley, J. Math. Phys. 18, 1529 (1977).
  30. M. Walker and R. Penrose, Commun. Math. Phys. 18, 265 (1970).
  31. S. Hacyan and J. F. Plebański, “D(1,0) Killing structures and [script E] potentials,” J. Math. Phys. 18, 1517 (1977).
  32. A suggestion in this direction by Dr. Frederick J. Ernst is gratefully appreciated.

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