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.
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References
Staruszkiewicz, A.: Gravity theory in three dimensional space. Acta Phys. Pol. 24, 734 (1963)
Aichelburg, P.C., Sexl, R.U.: On the gravitational field of a massless particle. Gen. Relativ. Gravit. 2, 303 (1971)
Deser, S., Jackiw, R., ’t Hooft, G.: Three dimensional Einstein gravity: dynamics of flat space. Ann. Phys. 152, 220 (1984)
’t Hooft, G.: Cosmology in 2+1 dimensions. Nucl. Phys. B 30, 200 (1993)
’t Hooft, G.: The evolution of gravitating point particles in 2+1 dimensions. Class. Quantum Gravity 10, 1023 (1993)
Gott, J.R.: Phys. Rev. Lett. 66, 1126 (1991)
Ori, A.: Phys. Rev. D 44, R2214 (1991)
Gödel, K.: An example of a new type of cosmological solution of Einstein’s field equations of gravitation. Rev. Mod. Phys. 21, 447 (1949)
Deser, S., Jackiw, R., ’t Hooft, G.: Physical cosmic strings do not generate closed timelike curves. Phys. Rev. Lett. 68, 267 (1992)
Kadar, Z.: Polygon model from first order gravity. Class. Quantum Gravity 22, 809 (2005). e-Print: gr-qc/0410012
Witten, E.: (2+1)-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46 (1988)
Carlip, S.: Six ways to quantize (2+1)-dimensional gravity. Can. Gen. Rel. 0215-234, (1993) (QC6:C25:1993), gr-qc/9305020
’t Hooft, G.: Quantization of point particles in (2+1) dimensional gravity and spacetime discreteness. Class. Quantum Gravity 13, 1023 (1996), gr-qc/9607022
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
Brandenberger, R., Firouzjahi, H., Karouby, J.: Lensing and CMB anisotropies by cosmic strings at a junction. arXiv:0710.1636 (gr-qc)
’t Hooft, G.: A mathematical theory for deterministic quantum mechanics. J. Phys. Conf. Ser. 67, 012015 (2007), quant-ph/0604008
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’t Hooft, G. A Locally Finite Model for Gravity. Found Phys 38, 733–757 (2008). https://doi.org/10.1007/s10701-008-9231-3
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DOI: https://doi.org/10.1007/s10701-008-9231-3