Remarks on the “Non-canonicity puzzle”: Lagrangian symmetries of the Einstein-Hilbert action. (English) Zbl 1255.83020
Summary: Given the non-canonical relationship between variables used in the Hamiltonian formulations of the Einstein-Hilbert action (due to Pirani, Schild, Skinner (PSS) and Dirac) and the Arnowitt-Deser-Misner (ADM) action, and the consequent difference in the gauge transformations generated by the first-class constraints of these two formulations, the assumption that the Lagrangians from which they were derived are equivalent leads to an apparent contradiction that has been called “the non-canonicity puzzle”. In this work we shall investigate the group properties of two symmetries derived for the Einstein-Hilbert action: diffeomorphism, which follows from the PSS and Dirac formulations, and the one that arises from the ADM formulation. We demonstrate that unlike the diffeomorphism transformations, the ADM transformations (as well as others, which can be constructed for the Einstein-Hilbert Lagrangian using Noether’s identities) do not form a group. This makes diffeomorphism transformations unique (the term “canonical” symmetry might be suggested). If the two Lagrangians are to be called equivalent, canonical symmetry must be preserved. The interplay between general covariance and the canonicity of the variables used is discussed.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C40 | Gravitational energy and conservation laws; groups of motions |
References:
| [1] | Pirani, F.A.E., Schild, A., Skinner, R.: Phys. Rev. 87, 452 (1952) · Zbl 0047.21102 · doi:10.1103/PhysRev.87.452 |
| [2] | Dirac, P.A.M.: Proc. R. Soc. A 246, 333 (1958) · Zbl 0080.41403 · doi:10.1098/rspa.1958.0142 |
| [3] | Kiriushcheva, N., Kuzmin, S.V., Racknor, C., Valluri, S.R.: Phys. Lett. A 372, 5101 (2008) · Zbl 1221.83011 · doi:10.1016/j.physleta.2008.05.081 |
| [4] | Kiriushcheva, N., Kuzmin, S.V.: Cent. Eur. J. Phys. 9, 576 (2011) · doi:10.2478/s11534-010-0072-2 |
| [5] | Castellani, L.: Ann. Phys. 143, 357 (1982) · doi:10.1016/0003-4916(82)90031-8 |
| [6] | Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004) · Zbl 1091.83001 |
| [7] | Arnowitt, R., Deser, S., Misner, C.W.: In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, p. 227. Wiley, New York (1962). arXiv:gr-qc/0405109 |
| [8] | Arnowitt, R., Deser, S., Misner, C.W.: Gen. Relativ. Gravit. 40, 1997 (2008) · Zbl 1152.83320 · doi:10.1007/s10714-008-0661-1 |
| [9] | Frolov, A.M., Kiriushcheva, N., Kuzmin, S.V.: Gravit. Cosmol. 17, 314 (2011) · Zbl 1470.83012 · doi:10.1134/S0202289311040049 |
| [10] | Cianfrani, F., Lulli, M., Montani, G.: arXiv:1104.0140 [gr-qc] |
| [11] | Shestakova, T.P.: Class. Quantum Gravity 28, 055009 (2011) · Zbl 1210.83006 · doi:10.1088/0264-9381/28/5/055009 |
| [12] | Shestakova, T.P.: Gravit. Cosmol. 17, 67 (2011) · Zbl 1232.83016 · doi:10.1134/S0202289311010191 |
| [13] | Kiriushcheva, N., Komorowski, P.G., Kuzmin, S.V.: arXiv:1107.2981 [gr-qc] |
| [14] | Noether, E.: Nachrichten von der König. Gesellsch. der Wiss. zu Göttingen. Math. Phys. Kl. 2, 235 (1918) |
| [15] | Noether, E. (M.A. Tavel’s English translation): arXiv:physics/0503066 |
| [16] | Schwinger, J.: Phys. Rev. 130, 1253 (1963) · Zbl 0111.42401 · doi:10.1103/PhysRev.130.1253 |
| [17] | Kiriushcheva, N., Kuzmin, S.V.: Gen. Relativ. Gravit. 42, 2613 (2010) · Zbl 1204.83070 · doi:10.1007/s10714-010-1003-7 |
| [18] | Banerjee, R., Gangopadhyay, S., Mukherjee, P., Roy, D.: J. High Energy Phys. 1002, 075 (2010) |
| [19] | Shestakova, T.P.: Proceedings of Russian summer school–seminar on Gravitation and Cosmology GRACOS–2007, Kazan, p. 179 (2007). arXiv:0801.4854 [gr-qc] |
| [20] | Bergmann, P.G., Komar, A.: Int. J. Theor. Phys. 1, 15 (1972) · doi:10.1007/BF00671650 |
| [21] | Hořava, P., Melby-Thompson, C.M.: Phys. Rev. D 82, 064027 (2010) · doi:10.1103/PhysRevD.82.064027 |
| [22] | Kiriushcheva, N., Kuzmin, S.V.: Eur. Phys. J. C 70, 389 (2010) · doi:10.1140/epjc/s10052-010-1458-4 |
| [23] | Bianchi, L.: Lezioni di Geometria Differenziale, 2nd edn., vol. 1. Spoerri, Pisa (1902) · JFM 33.0633.01 |
| [24] | Hilbert, D.: Nachrichten von der König. Gesellsch. der Wiss. zu Göttingen. Math. Phys. Kl. 8, 395 (1916) |
| [25] | Hilbert, D.: In: Renn, J. (ed.) The Genesis of General Relativity, vol. 4, p. 1003 (2007) |
| [26] | Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, 4th edn. Pergamon Press, Oxford (1975) · Zbl 0178.28704 |
| [27] | Carmeli, M.: Classical Fields, General Relativity and Gauge Theory. World Scientific, Singapore (2001) · Zbl 0992.83001 |
| [28] | Rosenfeld, L.: Ann. Phys. 397, 113 (1930) · JFM 56.1303.01 · doi:10.1002/andp.19303970107 |
| [29] | Salisbury, D.: Preprint 381, Max Planck Institute for the History of Science (2009) |
| [30] | Samanta, S.: Int. J. Theor. Phys. 48, 1436 (2009) · Zbl 1171.83306 · doi:10.1007/s10773-008-9914-8 |
| [31] | Torre, C.G., Anderson, I.M.: Phys. Rev. Lett. 70, 3525 (1993) · Zbl 1051.83504 · doi:10.1103/PhysRevLett.70.3525 |
| [32] | Henneaux, M., Kleinschmidt, A., Gómez, G.L.: arXiv:1004.3769 [hep-th]. To appear in the proceedings of the conference ”Gauge Fields: Yesterday, Today, Tomorrow”, dedicated to the 70th anniversary of Professor A.A. Slavnov |
| [33] | Pons, J.M.: Class. Quantum Gravity 20, 3279 (2003) · Zbl 1056.83024 · doi:10.1088/0264-9381/20/15/301 |
| [34] | Mukherjee, P., Saha, A.: Int. J. Mod. Phys. A 24, 4305 (2009) · Zbl 1175.83033 · doi:10.1142/S0217751X09044759 |
| [35] | Pons, J.M., Salisbury, D.C., Shepley, L.C.: Phys. Rev. D 55, 658 (1997) · doi:10.1103/PhysRevD.55.658 |
| [36] | Kiriushcheva, N., Komorowski, P.G., Kuzmin, S.V.: arXiv:1108.6105 [gr-qc] |
| [37] | Isham, C.J., Kuchar, K.V.: Ann. Phys. 164, 316 (1985) · doi:10.1016/0003-4916(85)90019-3 |
| [38] | Norton, J.D., Brading, K., Castellani, E. (eds.): Symmetries in Physics: Philosophical Reflections, p. 110. Cambridge University Press, Cambridge (2003) · Zbl 1206.00016 |
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