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Deformations of three-dimensional metrics. (English) Zbl 1317.83063
Summary: We examine three-dimensional metric deformations based on a tetrad transformation through the action the matrices of scalar field. We describe by this approach to deformation the results obtained by B. Coll et al. [Gen. Relativ. Gravitation 34, No. 2, 269–282 (2002; Zbl 1008.53039)], where it is stated that any three-dimensional metric was locally obtained as a deformation of a constant curvature metric parameterized by a 2-form. To this aim, we construct the corresponding deforming matrices and provide their classification according to the properties of the scalar \(\sigma\) and of the vector \(\mathbf {s}\) used in Coll et al. [loc. cit.] to deform the initial metric. The resulting causal structure of the deformed geometries is examined, too. Finally, we apply our results to a spherically symmetric three geometry and to a space sector of Kerr metric.

83C80 Analogues of general relativity in lower dimensions
53Z05 Applications of differential geometry to physics
83C57 Black holes
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[1] Ellis, GFR; Stoeger, W, The ‘fitting problem’ in cosmology, Class. Quantum Gravity, 4, 1697, (1987) · Zbl 0626.53071
[2] Acena, A; Valiente Kroon, JA, Conformal extensions for stationary spacetimes, Class. Quantum Gravity, 28, 225023, (2011) · Zbl 1230.83035
[3] Friedrich, H; Frauendiener, J (ed.); Friedrich, H (ed.), The conformal structure of spacetime: geometry, analysis, numerics, (2002), Berlin · Zbl 1030.00021
[4] Lübbe, C; Valiente Kroon, JA, The extended conformal Einstein field equations with matter: the Einstein-Maxwell system, J. Geom. Phys., 62, 1548, (2012) · Zbl 1239.53018
[5] Lübbe, C; Valiente Kroon, JA, A class of conformal curves in the Reissner-Nordström spacetime, Annales Henri Poincare, 15, 1327, (2014) · Zbl 1295.83042
[6] Lübbe, C; Kroon, JAV, A conformal approach for the analysis of the non-linear stability of pure radiation cosmologies, Ann. Phys., 328, 1, (2013) · Zbl 1263.83188
[7] Valiente Kroon, J.A.: Conformal Methods in General Relativity. Cambridge University Press, Cambridge (2015, in preparation) · Zbl 1368.83004
[8] Capozziello, S; Laurentis, M, A review about invariance induced gravity: gravity and spin from local conformal-affine symmetry, Found. Phys., 40, 867, (2010) · Zbl 1200.83042
[9] Capozziello, S; Laurentis, M, Extended theories of gravity, Phys. Rep., 509, 167, (2011) · Zbl 1260.83030
[10] Ter-Kazarian, G, Two-step spacetime deformation induced dynamical torsion, Class. Quantum Gravity, 28, 055003, (2011) · Zbl 1210.83041
[11] Carfora, M; Marzuoli, A, Smoothing out spatially closed cosmologies, Phys. Rev. Lett., 53, 2445, (1984)
[12] Mukhanov, VF; Feldman, HA; Brandenberger, RH, Theory of cosmological perturbations. part 1. classical perturbations. part 2. quantum theory of perturbations. part 3. extensions, Phys. Rep., 215, 203, (1992)
[13] Newman, ET; Janis, AI, Note on the Kerr spinning particle metric, J. Math. Phys., 6, 915, (1965) · Zbl 0142.46305
[14] Carlip, S; Teitelboim, CT, Aspects of black hole quantum mechanics and thermodynamics in (2+1)-dimensions, Phys. Rev. D, 51, 622, (1995)
[15] Coll, B; Llosa, J; Soler, D, Three-dimensional metrics as deformations of a constant curvature metric, Gen. Relativ. Gravit., 34, 269, (2002) · Zbl 1008.53039
[16] Llosa, J; Soler, D, On the degrees of freedom of a semi-Riemannian metric, Class. Quantum Gravity, 22, 893, (2005) · Zbl 1069.53056
[17] Soler, D, Reference frames and rigid motions in relativity: applications, Found. Phys., 36, 1718, (2006) · Zbl 1117.83005
[18] Llosa, J; Carot, J, Flat deformation theorem and symmetries in spacetime, Class. Quantum Gravity, 26, 055013, (2009) · Zbl 1160.83006
[19] Llosa, J; Carot, J, Flat deformation of a spacetime admitting two commuting Killing fields, Class. Quantum Gravity, 27, 245006, (2010) · Zbl 1206.83044
[20] Coll, B; Hildebrandt, SR; Senovilla, JMM, Kerr-schild symmetries, Gen. Relativ. Gravit., 33, 649, (2001) · Zbl 0982.83015
[21] Coll, B.: A universal law of gravitational deformation for general relativity. In: Proceedings of the Spanish Relativistic Meeting, EREs, Salamanca Spain (1998) · Zbl 1210.83041
[22] Llosa, J; Soler, D, Reference frames and rigid motions in relativity, Class. Quantum Gravity, 21, 3067, (2004) · Zbl 1076.83007
[23] Capozziello, S; Stornaiolo, C, Space-time deformations as extended conformal transformations, Int. J. Geom. Meth. Mod. Phys., 5, 185, (2008) · Zbl 1157.83317
[24] Pugliese, D.: Deformazioni di metriche spaziotemporali, Thesis (unpublished) 2006-2007. Università degli studi di Napoli Federico II. Dipartimento di Scienze Fisiche, Biblioteca Roberto Stroffolini
[25] Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984) · Zbl 0549.53001
[26] Stephani, H., Kramer, D., MacCallum, M.: Exact Solutions of Einstein’s Field Equations, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009) · Zbl 1179.83005
[27] Banados, M; Teitelboim, C; Zanelli, J, The black hole in three-dimensional space-time, Phys. Rev. Lett., 69, 1849, (1992) · Zbl 0968.83514
[28] Banados, M; Henneaux, M; Teitelboim, C; Zanelli, J, Geometry of the (2+1) black hole, Phys. Rev. D, 48, 1506, (1993)
[29] Dain, S; Gabach-Clement, ME, Small deformations of extreme Kerr black hole initial data, Class. Quantum Gravity, 28, 075003, (2011) · Zbl 1213.83085
[30] Choquet-Bruhat, Y.: General Relativity and the Einstein equations. Oxford University Press, Oxford (2008) · Zbl 1157.83002
[31] Griffiths, J.B., Podolsk, J.: Exact Space-Times in Einstein’s General Relativity, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009)
[32] Pugliese, D., Stornaiolo, C., Capozziello, S.: Deformations of spacetime metrics. arXiv:0910.5738 [gr-qc] · Zbl 1317.83063
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