Full-text resources of PSJD and other databases are now available in the new Library of Science.
Visit https://bibliotekanauki.pl

PL EN


Preferences help
enabled [disable] Abstract
Number of results

Journal

2011 | 9 | 3 | 576-615

Article title

The Hamiltonian formulation of general relativity: myths and reality

Content

Title variants

Languages of publication

EN

Abstracts

EN
A conventional wisdom often perpetuated in the literature states that: (i) a 3 + 1 decomposition of spacetime into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [21] and of ADM [22] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i–iii) have been shown to be incorrect in [45] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. By direct calculation we show that Dirac’s references to space-like surfaces are inessential and that such surfaces do not enter his calculations. In addition, we show that his assumption g
0k
= 0, used to simplify his calculation of different contributions to the secondary constraints, is unwarranted; yet, remarkably his total Hamiltonian is equivalent to the one computed without the assumption g
0k
= 0. The secondary constraints resulting from the conservation of the primary constraints of Dirac are in fact different from the original constraints that Dirac called secondary (also known as the “Hamiltonian” and “diffeomorphism” constraints). The Dirac constraints are instead particular combinations of the constraints which follow directly from the primary constraints. Taking this difference into account we found, using two standard methods, that the generator of the gauge transformation gives diffeomorphism invariance in four-dimensional space-time; and this shows that points (i–iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric g
μν to lapse and shift functions and the three-metric g
km, which is not canonical. This proves that point (iv) is incorrect. Points (i–iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein’s theory itself.

Publisher

Journal

Year

Volume

9

Issue

3

Pages

576-615

Physical description

Dates

published
1 - 6 - 2011
online
26 - 2 - 2011

References

  • [1] J.L. Lagrange, Mecanique Analytique (Académie des Sciences, Paris, 1788)
  • [2] J.L. Lagrange, Analytical Mechanics (Kluwer, Dordrecht, 1997)
  • [3] P.A.M. Dirac, Can. J. Math. 2, 129 (1950) http://dx.doi.org/10.4153/CJM-1950-012-1[Crossref]
  • [4] P.G. Bergmann, J.H.M. Brunings, Rev. Mod. Phys. 21, 480 (1949) http://dx.doi.org/10.1103/RevModPhys.21.480[Crossref]
  • [5] J.L. Anderson, P.G. Bergmann, Phys. Rev. 83, 1018 (1951) http://dx.doi.org/10.1103/PhysRev.83.1018[Crossref]
  • [6] L. Rosenfeld, Ann. Phys. 397, 113 (1930) http://dx.doi.org/10.1002/andp.19303970107[Crossref]
  • [7] P.G. Bergmann, I. Goldberg, Phys. Rev. 98, 531 (1955) http://dx.doi.org/10.1103/PhysRev.98.531[Crossref]
  • [8] B.S. DeWitt, Phys. Rev. 160, 1113 (1967) http://dx.doi.org/10.1103/PhysRev.160.1113[Crossref]
  • [9] B.S. DeWitt, Gen. Relat. Gravit. 1, 181 (1970) http://dx.doi.org/10.1007/BF00756896[Crossref]
  • [10] A. Hanson, T. Regge, C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale dei Lincei, Roma, 1976)
  • [11] P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Sciences (Yeshiva University, New York, 1964)
  • [12] K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics, vol. 169 (Springer, Berlin, 1982)
  • [13] D.M. Gitman, I.V. Tyutin, Quantization of Fields with Constraints (Springer, Berlin, 1990) http://dx.doi.org/10.1007/978-3-642-83938-2[Crossref]
  • [14] M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, New Jersey, 1992)
  • [15] H.J. Rothe, K.D. Rothe, Classical and Quantum Dynamics of Constrained Hamiltonian Systems, World Scientific Lecture Notes in Physics, vol. 81 (World Scientific, New Jersey, 2010)
  • [16] D. Salisbury, Preprint 381 (Max Planck Institute for the History of Science, Berlin, 2009)
  • [17] A. Einstein, Sitzber. Preuss. Akad. K1, 414 (1925)
  • [18] A. Einstein, In: I.E. Tamm, Ya.A. Smorodinskii, B.G. Kuznetsov (Eds.), The Complete Collection of Scientific Papers, vol. 2 (Nauka, Moscow, 1966) 171
  • [19] A. Unzicker, T. Case, arXiv:physics/0503046
  • [20] F.A.E. Pirani, A. Schild, R. Skinner, Phys. Rev. 87, 452 (1952) http://dx.doi.org/10.1103/PhysRev.87.452[Crossref]
  • [21] P.A.M. Dirac, P. R. Soc. A 246, 333 (1958) http://dx.doi.org/10.1098/rspa.1958.0142[Crossref]
  • [22] R. Arnowitt, S. Deser, C. W. Misner, In: L. Witten (Ed.), Gravitation: An Introduction to Current Research (Wiley, New York, 1962) 227
  • [23] E.C.G. Sudarshan, N. Mukunda, Classical Dynamics - A Modern Perspective (A Wiley-Interscience Publication, New York, 1974)
  • [24] N. Mukunda, Phys. Scripta 21, 783 (1980) http://dx.doi.org/10.1088/0031-8949/21/6/001[Crossref]
  • [25] L. Castellani, Ann. Phys. 143, 357 (1982) http://dx.doi.org/10.1016/0003-4916(82)90031-8[Crossref]
  • [26] M. Henneaux, C. Teitelboim, J. Zanelli, Nucl. Phys. B 332, 169 (1990) http://dx.doi.org/10.1016/0550-3213(90)90034-B[Crossref]
  • [27] R. Banerjee, H.J. Rothe, K.D. Rothe, Phys. Lett. B 479, 429 (2000) http://dx.doi.org/10.1016/S0370-2693(00)00323-3[Crossref]
  • [28] R. Banerjee, H.J. Rothe and K.D. Rothe, Phys. Lett. B 463, 248 (1999) http://dx.doi.org/10.1016/S0370-2693(99)00977-6[Crossref]
  • [29] N. Kiriushcheva, S.V. Kuzmin, arXiv:0912.3396[gr-qc]
  • [30] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W.H. Freeman and Company, San Francisco, 1973)
  • [31] R.M. Wald, General Relativity (The University of Chicago Press, Chicago, 1984)
  • [32] P. Mukherjee, A. Saha, Int. J. Mod. Phys. A 24, 4305 (2009) http://dx.doi.org/10.1142/S0217751X09044759[Crossref]
  • [33] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, fourth ed. (Pergamon Press, Oxford, 1975)
  • [34] P.G. Bergmann, A. Komar, Int. J. Theor. Phys. 5, 15 (1972) http://dx.doi.org/10.1007/BF00671650[Crossref]
  • [35] D.C. Salisbury, K. Sundermeyer, Phys. Rev. D 27, 740 (1983) http://dx.doi.org/10.1103/PhysRevD.27.740[Crossref]
  • [36] J.M. Pons, D.C. Salisbury, L.C. Shepley, Phys. Rev. D 62, 064026 (2000) http://dx.doi.org/10.1103/PhysRevD.62.064026[Crossref]
  • [37] J.M. Pons, Classical Quant. Grav. 20, 3279 (2003) http://dx.doi.org/10.1088/0264-9381/20/15/301[Crossref]
  • [38] J.M. Pons, D.C. Salisbury, Phys. Rev. D 71, 124012 (2005) http://dx.doi.org/10.1103/PhysRevD.71.124012[Crossref]
  • [39] J.M. Pons, D.C. Salisbury, L.C. Shepley, Phys. Rev. D 55, 658 (1997) http://dx.doi.org/10.1103/PhysRevD.55.658[Crossref]
  • [40] D.C. Salisbury, arXiv:physics/07012299
  • [41] S. Samanta, Int. J. Theor. Phys. 48, 1436 (2009) http://dx.doi.org/10.1007/s10773-008-9914-8[Crossref]
  • [42] S.W. Hawking, In: S.W. Hawking, W. Israel (Eds.), General Relativity. An Einstein Centenary Survey (Cambridge University Press, Cambridge, 1979) 746
  • [43] C. Rovelli, arXiv:gr-qc/0202079
  • [44] J. Pullin, arXiv:gr-qc/0209008
  • [45] N. Kiriushcheva, S.V. Kuzmin, C. Racknor, S.R. Valluri, Phys. Lett. A 372, 5101 (2008) http://dx.doi.org/10.1016/j.physleta.2008.05.081[Crossref]
  • [46] S.A. Hojman, K. Kuchar, C. Teitelboim, Ann. Phys. 96, 88 (1976) http://dx.doi.org/10.1016/0003-4916(76)90112-3[Crossref]
  • [47] K. Kuchar, In: W. Israel (Ed.), Relativity, Astrophysics and Cosmology (D. Reidel Publishing Company, Dordrecht, 1973) 237 http://dx.doi.org/10.1007/978-94-010-2639-0_5[Crossref]
  • [48] S.W. Hawking, R. Penrose, P. R. Soc. A 246, 529 (1970) http://dx.doi.org/10.1098/rspa.1970.0021[Crossref]
  • [49] F. Markopoulou, Commun. Math. Phys. 211, 559 (2000) http://dx.doi.org/10.1007/s002200050826[Crossref]
  • [50] T. Thiemann, arXiv:gr-qc/0110034
  • [51] L. Smolin, Three roads to Quantum Gravity (Basic Books, New York, 2001)
  • [52] K.R. Green, N. Kiriushcheva, S.V. Kuzmin, arXiv:0710.1430[gr-qc]
  • [53] M. Carmeli, Classical Fields: General Relativity and Gauge Theory (World Scientific, New Jersey, 2001)
  • [54] S. Chandrasekhar, The Mathematical Theory of Black Holes (Clarendon Press, Oxford, 2005)
  • [55] F. Antonsen, F. Markopoulou, arXiv:gr-qc/9702046
  • [56] C. Teitelboim, Phys. Rev. Lett. 38, 1106 (1977) http://dx.doi.org/10.1103/PhysRevLett.38.1106[Crossref]
  • [57] R. Arnowitt, S. Deser, C.W. Misner, Phys. Rev. 113, 745 (1959) http://dx.doi.org/10.1103/PhysRev.113.745[Crossref]
  • [58] C. Teitelboim, In: A. Held (Ed.), General Relativity and Gravitation, One Hundred Years After the Birth of Albert Einstein, v.1 (Plenum Press, New York and London, 1980) 195
  • [59] C. Teitelboim, Ann. Phys. 79, 542 (1973) http://dx.doi.org/10.1016/0003-4916(73)90096-1[Crossref]
  • [60] J.A. Wheeler, In: B.S. DeWitt, C.M. DeWitt (Eds.), Relativity, Groups and Topology (Gordon and Breach, New York - London, 1964) 346
  • [61] L.P. Faddeev, Usp. Fiz. Nauk+ 136, 435 (1982) http://dx.doi.org/10.3367/UFNr.0136.198203c.0435[Crossref]
  • [62] L.P. Faddeev, Sov. Phys. Uspekhi 25, 130 (1982) http://dx.doi.org/10.1070/PU1982v025n03ABEH004517[Crossref]
  • [63] C. Teitelboim, Phys. Rev. D 28, 297 (1983) http://dx.doi.org/10.1103/PhysRevD.28.297[Crossref]
  • [64] R.N. Ghalati, D.G.C. McKeon, arXiv:0712.2861v3[grqc]
  • [65] D.G.C. McKeon, Int. J. Mod. Phys. A 25, 3453 (2010) http://dx.doi.org/10.1142/S0217751X10050093[Crossref]
  • [66] A.M. Frolov, N. Kiriushcheva, S.V. Kuzmin, arXiv:0809.1198[gr-qc]
  • [67] N. Kiriushcheva, S.V. Kuzmin, Ann. Phys. 321, 958 (2006) http://dx.doi.org/10.1016/j.aop.2005.09.009[Crossref]
  • [68] D. Lovelock, J. Math. Phys. 12, 498 (1971) http://dx.doi.org/10.1063/1.1665613[Crossref]
  • [69] C.G. Torre, I.M. Anderson, Phys. Rev. Lett. 70, 3525 (1993) http://dx.doi.org/10.1103/PhysRevLett.70.3525[Crossref]
  • [70] L.P. Grishchuk, A.N. Petrov, A.D. Popova, Commun. Math. Phys. 94, 379 (1984) http://dx.doi.org/10.1007/BF01224832[Crossref]
  • [71] M. Ferraris, M. Francaviglia, C. Reina, Gen. Relat. Gravit. 14, 243 (1982) http://dx.doi.org/10.1007/BF00756060[Crossref]
  • [72] A. Palatini, Rend. Circolo Math. Palermo 43, 203 (1919) http://dx.doi.org/10.1007/BF03014670[Crossref]
  • [73] A. Palatini, In: P.G. Bergmann, V. De Sabbata (Eds.), Cosmology and Gravitation (Plenum Press, New York and London, 1979) 477
  • [74] W. Kummer, H. Schuetz, Eur. Phys. J. C 42, 277 (2005) http://dx.doi.org/10.1140/epjc/s2005-02244-7[Crossref]
  • [75] N. Kiriushcheva, S.V. Kuzmin, D.G.C. McKeon, Int. J. Mod. Phys. A 21, 3401 (2006) http://dx.doi.org/10.1142/S0217751X06029545[Crossref]
  • [76] R.N. Ghalati, D.G.C. McKeon, arXiv:0712.2861[grqc]
  • [77] R. Arnowitt, S. Deser, C.W. Misner, Phys. Rev. 116, 1322 (1959) http://dx.doi.org/10.1103/PhysRev.116.1322[Crossref]
  • [78] L.P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, New Jersey, 1997)
  • [79] C. Møller, The theory of Relativity (Clarendon Press, Oxford, 1952)
  • [80] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Relativity (Wiley, New York, 1972)
  • [81] S. Deser, R. Jackiw, S. Templeton, Ann. Phys. 140, 372 (1982) http://dx.doi.org/10.1016/0003-4916(82)90164-6[Crossref]
  • [82] S. Deser, Classical Quant. Grav. 23, 5773 (2006) http://dx.doi.org/10.1088/0264-9381/23/18/N01[Crossref]
  • [83] N. Kiriushcheva, S.V. Kuzmin, Classical Quant. Grav. 24, 1371 (2007) http://dx.doi.org/10.1088/0264-9381/24/5/N01[Crossref]
  • [84] R.N. Ghalati, N. Kiriushcheva, S.V. Kuzmin, D.G.C. McKeon, arXiv:hep-th/0609220
  • [85] E. Noether, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse (1918) 235
  • [86] E. Noether, arXiv:physics/0503066
  • [87] N. Kiriushcheva, S.V. Kuzmin, arXiv:0907.1999[gr-qc] (to appear in Gen. Relat. Gravit.)
  • [88] K. Kuchar, arXiv:gr-qc/9304012
  • [89] A. Einstein, Sitzber. K. Preuss Aka. 44, 778 (1915)
  • [90] A. Einstein, In: I.E. Tamm, Ya.A. Smorodinskii, B.G. Kuznetsov (Eds.), The Complete Collection of Scientific Papers, vol. 1 (Nauka, Moscow, 1965) 425
  • [91] T.P. Shestakova, arXiv:0801.4854[gr-qc]
  • [92] E. Anderson, J. Barbour, B.Z. Foster, B. Kelleher, N.O. Murchadha, Classical Quant. Grav. 22, 1795 (2005) http://dx.doi.org/10.1088/0264-9381/22/9/020[Crossref]
  • [93] P.A.M. Dirac, Phys. Rev. 114, 924 (1959) http://dx.doi.org/10.1103/PhysRev.114.924[Crossref]
  • [94] R. Arnowitt, S. Deser, C.W. Misner, Phys. Rev. 118, 1100 (1960) http://dx.doi.org/10.1103/PhysRev.118.1100[Crossref]
  • [95] R. Arnowitt, S. Deser, C.W. Misner, Phys. Rev. 121, 1556 (1961) http://dx.doi.org/10.1103/PhysRev.121.1556[Crossref]
  • [96] V.A. Franke, Theor. Math. Phys.+ 148, 995 (2006) http://dx.doi.org/10.1007/s11232-006-0096-3[Crossref]
  • [97] V.A. Franke, Theor. Math. Phys.+ 148, 143 (2006)
  • [98] R. Arnowitt, S. Deser, C.W. Misner, J. Math. Phys. 1, 434 (1960) http://dx.doi.org/10.1063/1.1703677[Crossref]
  • [99] H. Nicolai, K. Peeters, M. Zamaklar, Classical Quant. Grav. 22, R193 (2005) http://dx.doi.org/10.1088/0264-9381/22/19/R01[Crossref]
  • [100] B.S. DeWitt, Phys. Rev. 160, 1113 (1967) http://dx.doi.org/10.1103/PhysRev.160.1113[Crossref]
  • [101] C.J. Isham, K.V. Kuchar, Ann. Phys. 164, 316 (1985) http://dx.doi.org/10.1016/0003-4916(85)90019-3[Crossref]
  • [102] C. Lanczos, The variational princiles of mechanics, fourth ed. (Dover Publications, New York, 1970)
  • [103] E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, fourth ed. (Cambridge University Press, Cambridge, 1999)
  • [104] D. Dominici, J. Gomis, J. Math. Phys. 21, 2124 (1980) http://dx.doi.org/10.1063/1.524721[Crossref]
  • [105] D. Dominici, J. Gomis, J. Math. Phys. 23, 256 (1982) http://dx.doi.org/10.1063/1.525346[Crossref]
  • [106] J. Gomis, J. Liosa, N. Roman, J. Math. Phys. 25, 1348 (1984) http://dx.doi.org/10.1063/1.526303[Crossref]
  • [107] S. Deser, C.J. Isham, Phys. Rev. D 14, 2505 (1976) http://dx.doi.org/10.1103/PhysRevD.14.2505[Crossref]
  • [108] R. Di Stefano, R.T. Rauch, Phys. Rev. D 26, 1242 (1982) http://dx.doi.org/10.1103/PhysRevD.26.1242[Crossref]
  • [109] L. Castellani, P. van Nieuwenhuizen, M. Pilati, Phys. Rev. D 26, 352 (1982) http://dx.doi.org/10.1103/PhysRevD.26.352[Crossref]
  • [110] A.M. Frolov, N. Kiriushcheva, S.V. Kuzmin, Gravitation Cosmol. 16, 181 (2010) http://dx.doi.org/10.1134/S0202289310030011[Crossref]
  • [111] N. Kiriushcheva, S.V. Kuzmin, arXiv:0907.1553[gr-qc]
  • [112] R. Arnowitt, S. Deser, C.W. Misner, Phys. Rev. 116, 1322 (1959) http://dx.doi.org/10.1103/PhysRev.116.1322[Crossref]
  • [113] R. Banerjee, S. Gangopadhyay, P. Mukherjee, D. Roy, J. High Energy Phys. 1002, 075 (2010) http://dx.doi.org/10.1007/JHEP02(2010)075[Crossref]
  • [114] C. Battle, J. Gomis, J.M. Pons, N. Roman-Roy, J. Math. Phys. 27, 2953 (1986) http://dx.doi.org/10.1063/1.527274[Crossref]
  • [115] T. Regge, C. Teitelboim, Ann. Phys. 88, 286 (1974) http://dx.doi.org/10.1016/0003-4916(74)90404-7[Crossref]
  • [116] A.E. Fischer, J.E. Marsden, In: S.W. Hawking, W. Israel (Eds.), General Relativity. An Einstein Centenary Survey (Cambridge University Press, Cambridge, 1979) 138
  • [117] L.E. Kidder, M.A. Scheel, S.A. Teukolsky, Phys. Rev. D 64, 064017 (2001) http://dx.doi.org/10.1103/PhysRevD.64.064017[Crossref]
  • [118] O. Sarbach, G. Calabrese, J. Pullin, M. Tiglio, Phys. Rev. D 66, 064002 (2002) http://dx.doi.org/10.1103/PhysRevD.66.064002[Crossref]
  • [119] V. Paschalidis, A. Khokhlov, I. Novikov, Phys. Rev. D 75, 024026 (2007) http://dx.doi.org/10.1103/PhysRevD.75.024026[Crossref]
  • [120] M. Salgado, D. Martínez-del Río, M. Alcubierre, D. Núñez, Phys. Rev. D 77, 104010 (2008) http://dx.doi.org/10.1103/PhysRevD.77.104010[Crossref]
  • [121] B. Kelly et al., Phys. Rev. D 64, 084013 (2001) http://dx.doi.org/10.1103/PhysRevD.64.084013[Crossref]
  • [122] J. York, In: L. Smarr (Ed.), Sources of Gravitational Radiation (Cambridge University Press, Cambridge, 1979) 83
  • [123] G. Calabrese, J. Pullin, O. Sarbach, M. Tiglio, Phys. Rev. D 66, 064011 (2002) http://dx.doi.org/10.1103/PhysRevD.66.064011[Crossref]
  • [124] C. Bona, C. Palenzuela-Luque, Elements of Numerical Relativity, Lecture Notes in Physics v. 673 (Springer Verlag, Berlin, 2005)
  • [125] J.D. Brown, arXiv:0803.0334[gr-qc]

Document Type

Publication order reference

Identifiers

YADDA identifier

bwmeta1.element.-psjd-doi-10_2478_s11534-010-0072-2
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.