Geodesic
Three-dimensional volume of a closed universe as a canonical time parameter
Teoretičeskaâ i matematičeskaâ fizika, Tome 95 (1993) no. 3, pp. 541-548 
N. N. Gorobey; A. S. Lukyanenko. Three-dimensional volume of a closed universe as a canonical time parameter. Teoretičeskaâ i matematičeskaâ fizika, Tome 95 (1993) no. 3, pp. 541-548. http://geodesic.mathdoc.fr/item/TMF_1993_95_3_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The dynamics of a spatially closed universe is studied in gravitation theory by means of the eigenfunctions and eigenvalues of the $3D$ projection of the Dirac operator. In a gauge that we call the Ashtekar gauge and construct by using an eigenfunction of this operator, the $3D$ volume of a spatial section is a canonical parameter, and the energy is a positive-definite functional of the dynamical variables (on the region of the phase space in which $\dot V>0$) proportional to the corresponding eigenvalue. There is a discrete set of frames of reference distinguished in this manner.

[1] Ashtekar A., Physica, 124A (1984), 51–60 | DOI | MR | Zbl

[2] Mizner Ch., Torn K., Uiller Dzh., Gravitatsiya, t. 2, Mir, M., 1977 | MR

[3] Witten E., Commun. Math. Phys., 80 (1981), 381 | DOI | MR | Zbl

[4] Lukyanenko A. S., DAN SSSR, 289:3 (1986), 579–583 | MR | Zbl

[5] Ashtekar A., Phys. Rev., D36 (1987), 1587 | MR

[6] Jacobson T., Smolin L., Phys. Lett., 196B (1987), 39 | DOI | MR

[7] Jacobson T., Class. Quantum Grav., 5 (1988), 923–935 | DOI | MR | Zbl