Relativistic Fractal Cosmologies

  • Marcelo B. Ribeiro
Part of the NATO ASI Series book series (NSSB, volume 332)


This article presents a review of an approach for constructing a simple relativistic fractal cosmology, whose main aim is to model the observed inhomogeneities of the distribution of galaxies by means of the Tolman solution of Einstein’s field equations for spherically symmetric dust in comoving coordinates. Such model is based on earlier works developed by L. Pietronero and J. R. Wertz on Newtonian cosmology, and the main points of these models are also discussed. Observational relations in Tolman’s spacetime are presented, together with a strategy for finding numerical solutions which approximate an averaged and smoothed out single fractal structure in the past light cone. Such fractal solutions are actually obtained and one of them is found to be in agreement with basic observational constraints, namely the linearity of the redshift-distance relation for z < 1, the decay of the average density with the distance as a power law (the de Vaucouleurs’ density power law), the fractal dimension within the range 1 ≤ D ≤ 2, and the present range of uncertainty for the Hubble constant. The spatially homogeneous Friedmann model is discussed as a special case of the Tolman solution, and it is found that once we apply the observational relations developed for the fractal model we find that all Friedmann models look inhomogeneous along the backward null cone, with a departure from the observable homogeneous region at relatively close ranges. It is also shown that with these same observational relations the Einstein-de Sitter model can have an interpretation where it has zero global density, a result consistent with the “zero global density postulate” advanced by Wertz for hierarchical cosmologies and conjectured by Pietronero for fractal cosmological models. The article ends with a brief discussion on the possible link between this model and nonlinear and chaotic dynamics.


Fractal Dimension Hubble Constant Fractal Solution Luminosity Distance Null Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Mandelbrot, B. B., 1983, The Fractal Geometry of Nature, (New York: Freeman).Google Scholar
  2. [2]
    Pietronero, L., 1987, Physica A, 144, 257.ADSzbMATHCrossRefGoogle Scholar
  3. [3]
    Wertz, J. R., 1970, Newtonian Hierarchical Cosmology, (PhD thesis), University of Texas at Austin.Google Scholar
  4. [4]
    Ellis, G. F. R. and Stoeger, W., 1987, Class. Quantum Grav., 4, 1697.MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. [5]
    de Lapparent, V., Geller, M. J. and Huchra, J. P., 1986, Astrophys. J. Lett., 302, L1.ADSCrossRefGoogle Scholar
  6. [6]
    Kopylov, A. I. et al., 1988, Large Scale Structures of the Universe, 130th IAU Symp., ed J Audouze et al., (Dordrecht: Kluwer), p 129.CrossRefGoogle Scholar
  7. [7]
    Geller, M., 1989, Astronomy, Cosmology and Fundamental Physics, ed M Caffo et al., (Dordrecht: Kluwer), p 83.CrossRefGoogle Scholar
  8. [8]
    Geller, M. J. and Huchra, J. P., 1989, Science, 246, 897.ADSCrossRefGoogle Scholar
  9. [9]
    Saunders, W. et al., 1991, Nature, 349, 32.ADSCrossRefGoogle Scholar
  10. [10]
    Ramella, M., Geller, M. J. and Huchra, J. P., 1992, Astrophys. J., 384, 396.ADSCrossRefGoogle Scholar
  11. [11]
    Charlier, C. V. L., 1908, Ark. Mat. Astron. Fys., 4, 1.Google Scholar
  12. [12]
    Charlier, C. V. L., 1922, Ark. Mat. Astron. Fys., 16, 1.Google Scholar
  13. [13]
    Feder, J., 1988, Fractals, (New York: Plenum).zbMATHGoogle Scholar
  14. [14]
    Fournier D’Albe, E. E., 1907, Two New Worlds: I The Infra World; II The Supra World, (London: Longmans Green).Google Scholar
  15. [15]
    Selety, F., 1922, Ann. Phys., 68, 281.CrossRefGoogle Scholar
  16. [16]
    Einstein, A., 1922, Ann. Phys., 69, 436.zbMATHCrossRefGoogle Scholar
  17. [17]
    Selety, F., 1923, Ann. Phys., 72, 58.zbMATHCrossRefGoogle Scholar
  18. [18]
    Selety, F., 1924, Ann. Phys., 73, 290.Google Scholar
  19. [19]
    de Vaucouleurs, G., 1970, Science, 167, 1203.ADSCrossRefGoogle Scholar
  20. [20]
    de Vaucouleurs, G., 1970, Science, 168, 917.ADSCrossRefGoogle Scholar
  21. [21]
    Wertz, J. R., 1971, Astrophys. J., 164, 277.ADSCrossRefGoogle Scholar
  22. [22]
    Haggerty, M. J., 1971, Astrophys. J., 166, 257.ADSCrossRefGoogle Scholar
  23. [23]
    Haggerty, M. J. and Wertz, J. R., 1972, M. N. R. A. S., 155, 495.ADSGoogle Scholar
  24. [24]
    de Vaucouleurs, G. and Wertz, J. R., 1971, Nature, 231, 109.ADSCrossRefGoogle Scholar
  25. [25]
    Sandage, A., Tamman, G. A. and Hardy, E., 1972, Astrophys. J., 172, 253.ADSCrossRefGoogle Scholar
  26. [26]
    de Vaucouleurs, G., 1986, Gamow Cosmology, ed F Melchiorri and R Ruffini, (Amsterdam: North-Holland), p 1.Google Scholar
  27. [27]
    Feng, L. L., Mo, H. J. and Ruffini, R., 1991, Astron. Astrophys., 243, 283.ADSGoogle Scholar
  28. [28]
    Bonnor, W. B., 1972, M. N. R. A. S., 159, 261.ADSGoogle Scholar
  29. [29]
    Wesson, P. S., 1978, Astrophys. Space Sci., 54, 489.MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    Wesson, P. S., 1979, Astrophys. J., 228, 647.ADSCrossRefGoogle Scholar
  31. [31]
    Coleman, P. H., Pietronero, L. and Sanders, R. H., 1988, Astron. Astrophys., 200, L32.ADSGoogle Scholar
  32. [32]
    Davis, M. et al., 1988, Astrophys. J. Lett., 333, L9.ADSCrossRefGoogle Scholar
  33. [33]
    Peebles, P. J. E., 1989, Physica D, 38, 273.ADSCrossRefGoogle Scholar
  34. [34]
    Calzetti, D. and Giavalisco, M., 1991, Applying Fractals in Astronomy, ed A Heck and JM Perdang, (Berlin: Springer-Verlag), p 119.CrossRefGoogle Scholar
  35. [35]
    Coleman, P. H. and Pietronero, L., 1992, Phys. Reports, 213, 311.ADSCrossRefGoogle Scholar
  36. [36]
    Maurogordato, S., Schaeffer, R. and da Costa, L. N., 1992, Astrophys. J., 390, 17.ADSCrossRefGoogle Scholar
  37. [37]
    Ruffini, R., Song, D. J. and Taraglio, S., 1988, Astron. Astrophys., 190, 1.ADSGoogle Scholar
  38. [38]
    Calzetti, D., Giavalisco, M. and Ruffini, R., 1988, Astron. Astrophys., 198, 1.MathSciNetADSzbMATHGoogle Scholar
  39. [39]
    Pietronero, L., 1988, Order and Chaos in Nonlinear Physical Systems, ed S Lundqvist et al., (New York: Plenum Press), p 277.Google Scholar
  40. [40]
    Tolman, R. C., 1934, Proc. Nat. Acad. Sci. (Wash.), 20, 169.ADSCrossRefGoogle Scholar
  41. [41]
    Ribeiro, M. B., 1992, Astrophys. J., 388, 1.ADSCrossRefGoogle Scholar
  42. [42]
    Ribeiro, M. B., 1993, Astrophys. J., 415, 469.ADSCrossRefGoogle Scholar
  43. [43]
    Ellis, G. F. R., 1971, Relativistic Cosmology, Proc. of the Int. School of Physics “Enrico Fermi”, General Relativity and Cosmology, ed RK Sachs, (New York: Academic Press), p 104.Google Scholar
  44. [44]
    McVittie, G. C., 1974, Q. Journal R. A. S., 15, 246.ADSGoogle Scholar
  45. [45]
    Saunders, W. et al., 1990, M. N. R. A. S., 242, 318.ADSGoogle Scholar
  46. [46]
    Ribeiro, M. B., 1992, Astrophys. J., 395, 29.ADSCrossRefGoogle Scholar
  47. [47]
    Ribeiro, M. B., 1992, On Modelling a Relativistic Hierarchical (Fractal) Cosmology by Tolman’s Spacetime, (PhD thesis), Queen Mary & Westfield College, University of London.Google Scholar
  48. [48]
    Peacock, J., 1991, Nature, 352, 378.ADSCrossRefGoogle Scholar
  49. [49]
    Bondi, H., 1947, M. N. R. A. S., 107, 410.MathSciNetADSzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Marcelo B. Ribeiro
    • 1
  1. 1.Departamento de AstrofísicaObservatório Nacional-CNPqRio de JaneiroBrazil

Personalised recommendations