Foundations of Physics

, 38:733 | Cite as

A Locally Finite Model for Gravity

Open Access
Article

Abstract

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate matter of a form that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Globally, however, the model is not finite, because solutions tend to generate infinite fractals. The model is not (yet) quantized, but could serve as an interesting setting for analytical approaches to classical general relativity, as well as a possible stepping stone for quantum models. Details of its properties are explained, but some problems remain unsolved, such as a complete description of the most violent interactions, which can become quite complex.

Keywords

Locally flat Straight strings Holonomy Classical general relativity Exact solutions 

References

  1. 1.
    Staruszkiewicz, A.: Gravity theory in three dimensional space. Acta Phys. Pol. 24, 734 (1963) MathSciNetGoogle Scholar
  2. 2.
    Aichelburg, P.C., Sexl, R.U.: On the gravitational field of a massless particle. Gen. Relativ. Gravit. 2, 303 (1971) CrossRefADSGoogle Scholar
  3. 3.
    Deser, S., Jackiw, R., ’t Hooft, G.: Three dimensional Einstein gravity: dynamics of flat space. Ann. Phys. 152, 220 (1984) CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    ’t Hooft, G.: Cosmology in 2+1 dimensions. Nucl. Phys. B 30, 200 (1993) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    ’t Hooft, G.: The evolution of gravitating point particles in 2+1 dimensions. Class. Quantum Gravity 10, 1023 (1993) CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Gott, J.R.: Phys. Rev. Lett. 66, 1126 (1991) MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Ori, A.: Phys. Rev. D 44, R2214 (1991) CrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Gödel, K.: An example of a new type of cosmological solution of Einstein’s field equations of gravitation. Rev. Mod. Phys. 21, 447 (1949) MATHCrossRefADSGoogle Scholar
  9. 9.
    Deser, S., Jackiw, R., ’t Hooft, G.: Physical cosmic strings do not generate closed timelike curves. Phys. Rev. Lett. 68, 267 (1992) MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Kadar, Z.: Polygon model from first order gravity. Class. Quantum Gravity 22, 809 (2005). e-Print: gr-qc/0410012 MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Witten, E.: (2+1)-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46 (1988) CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Carlip, S.: Six ways to quantize (2+1)-dimensional gravity. Can. Gen. Rel. 0215-234, (1993) (QC6:C25:1993), gr-qc/9305020
  13. 13.
    ’t Hooft, G.: Quantization of point particles in (2+1) dimensional gravity and spacetime discreteness. Class. Quantum Gravity 13, 1023 (1996), gr-qc/9607022 MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Barret, J.W., et al.: A parallelizable implicit evolution scheme for Regge calculus. Int. J. Theor. Phys. 36, 815 (1997), gr-qc/9411008, and references therein CrossRefGoogle Scholar
  15. 15.
    Brandenberger, R., Firouzjahi, H., Karouby, J.: Lensing and CMB anisotropies by cosmic strings at a junction. arXiv:0710.1636 (gr-qc)
  16. 16.
    ’t Hooft, G.: A mathematical theory for deterministic quantum mechanics. J. Phys. Conf. Ser. 67, 012015 (2007), quant-ph/0604008 CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Spinoza InstituteUtrechtThe Netherlands

Personalised recommendations