×

Hamiltonian dynamics for Einstein’s action in \(G\rightarrow 0\) limit. (English) Zbl 1176.83015

Summary: The Hamiltonian analysis for the Einstein’s action in \(G\rightarrow 0\) limit is performed. Considering the original configuration space without involve the usual \(ADM\) variables we show that the version \(G\rightarrow 0\) for Einstein’s action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C40 Gravitational energy and conservation laws; groups of motions
83A05 Special relativity
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arnowitt, R., Deser, R., Misner, C.: In: Witten, L. (ed.) Gravitation: An Introduction to Current Research. Wiley, New York (1962)
[2] Ashtekar, A.: Phys. Rev. Lett. 77, 3288 (1986)
[3] Ashtekar, A.: Phys. Rev. D 36, 1587 (1987)
[4] Ashtekar, A.: Lectures on Non-Perturbative Canonical Gravity. World Scientific, Singapore (1991) · Zbl 0948.83500
[5] Ashtekar, A., Romano, J.D., Tale, R.S.: Phys. Rev. D 40, 2572 (1989)
[6] Morales-Tecotl, H.A., Urrutia, L.F., Vergara, J.D.: Class. Quantum Gravity 13, 2933–2940 (1996) · Zbl 0863.53064
[7] Samuel, J.: Pramana J. Phys. 28, L429 (1987)
[8] Jacobson, T., Smolin, L.: Class. Quantum Gravity 5, 583 (1988) · Zbl 0645.53054
[9] Capovilla, R., Dell, J., Jacobson, T., Manson, L.: Class. Quantum Gravity 8, 41 (1991) · Zbl 0716.53066
[10] Capovilla, R., Dell, J., Jacobson, T.: Class. Quantum Gravity 8, 59 (1991) · Zbl 0716.53067
[11] Barbero, J.F.: Phys. Rev. D 51, 5507 (1995)
[12] Holts, S.: Phys. Rev. D 53, 5966 (1996)
[13] Capovilla, R., Jacobson, T., Dell, J.: Phys. Rev. Lett. 63, 2325 (1989)
[14] Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004) · Zbl 1091.83001
[15] Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2007) · Zbl 1129.83004
[16] Smolin, L.: Class. Quantum Gravity 9, 883 (1992) · Zbl 0747.58064
[17] Barros e Sa, N., Bengtsson, I.: Phys. Rev. D 59, 107502 (1999)
[18] Newman, E.T., Rovelli, C.: Phys. Rev. Lett. 69, 1300 (1992)
[19] Newman, E.T., Rovelli, C.: In: Colomo, F., Lusanna, L., Marmo, G. (eds.) Constraint Theory and Quantization Methods, p. 401. World Scientific, Singapore (1994)
[20] Gitman, D.M., Tyutin, I.V.: Quantization of Fields with Constraints. Springer Series in Nuclear and Particle Physics. Springer, Berlin (1990) · Zbl 1113.81300
[21] Hanson, A., Regge, T., Teitelboim, C.: Constrained Hamiltonian Systems. Accademia Nazionale dei Lincei, Roma (1978)
[22] Mondragon, M., Montesinos, M.: J. Math. Phys. 47, 022301 (2006) · Zbl 1111.81136
[23] Escalante, A.: The Chern-Simons state for topological invariants. Phys. Lett. B (2008). DOI: 10.1016/j.physletb.2009.04.052
[24] Blagojevic, M.: Gravitation and Gauge Symmetries. IOP, Bristol (2002)
[25] Cartas-Fuentevilla, R., Escalante, A.: Topological terms and the global symplectic geometry of the phase space in string theory. In: Benton, C.V. (ed.) Trends in Mathematical Physics Research. Nova Science, Hauppauge (2004) · Zbl 1068.81586
[26] Escalante, A.: The symmetries for Einstein’s action in 2+1 dimensions without resort to ADM variables. In preparation (2008)
[27] Ashtekar, A.: Lectures on Non-Perturbative Canonical Gravity. World Scientific, Singapore (1991). (Notes prepared in collaboration with R.S. Tate) · Zbl 0948.83500
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.