General Relativity and Gravitation

, Volume 42, Issue 3, pp 601–622 | Cite as

Gauge-invariant localization of infinitely many gravitational energies from all possible auxiliary structures

  • J. Brian Pitts
Research Article


The problem of finding a covariant expression for the distribution and conservation of gravitational energy–momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann’s realization that there are infinitely many conserved gravitational energy–momenta. Initially use is made of a flat background metric (or rather, all of them) or connection, because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of pseudotensors of a given type, such as the Einstein pseudotensor, in every coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression (Landau–Lifshitz, Goldberg, Papapetrou or the like) or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether’s theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many.


Conservation laws Localization Gauge invariance Infinite-component Gravitational energy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Einstein A., Grossmann M.: Outline of a generalized theory of relativity and of a theory of gravitation. In: Beck, A., Howard, D. (eds) The Collected Papers of Albert Einstein, vol. 4. The Swiss Years: Writings, 1912–1914, English Translation. The Hebrew University of Jerusalem and Princeton University, Princeton, 1996. Translated from Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, Teubner, Leipzig (1913)Google Scholar
  2. 2.
    Norton J.: How Einstein found his field equations, 1912–1915. In: Howard, D., Stachel, J. (eds) Einstein and the History of General Relativity, Einstein Studies, vol. 1, pp. 101. Birkhäuser, Boston (1989)Google Scholar
  3. 3.
    Janssen M.: Of pots and holes: Einstein’s bumpy road to general relativity. Annalen der Physik 14(Suppl), 58 (2005)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Janssen M., Renn J.: Untying the knot: how Einstein found his way back to field equations discarded in the Zurich notebook. In: Renn, J. (eds) The Genesis of General Relativity, vol. 2: Einstein’s Zurich Notebook: Commentary and Essays, pp. 839–925. Springer, Dordrecht (2007)Google Scholar
  5. 5.
    Fletcher J.G.: Local conservation laws in generally covariant theories. Rev. Mod. Phys. 32, 65 (1960)zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Trautman A.: Conservation laws in general relativity. In: Witten, L. (eds) Gravitation: an Introduction to Current Research, pp. 169. Wiley, New York (1962)Google Scholar
  7. 7.
    Goldberg J.: Invariant transformations, conservation laws, and energy-momentum. In: Held, A. (eds) General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein, vol. 1, pp. 469–489. Plenum Press, New York (1980)Google Scholar
  8. 8.
    Szabados, L.B.: Quasi-local energy-momentum and angular momentum in general relativity. Living Rev. Relat. 12(4), (2009). cited 22 June 2009Google Scholar
  9. 9.
    Graves J.C.: The Conceptual Foundations of Contemporary Relativity Theory. MIT, Cambridge (1971)Google Scholar
  10. 10.
    Einstein, A.: The foundation of the general theory of relativity. In: Lorentz, H.A., Einstein, A., Minkowski, H., Weyl, H., Sommerfeld, A., Perrett, W., Jeffery, G.B. (eds.) The Principle of Relativity. Dover reprint, New York, 1952, 1923. Translated from “Die Grundlage der allgemeinen Relativitätstheorie,” Annalen der Physik 49, 769–822 (1916)Google Scholar
  11. 11.
    Schrödinger E.: Die Energiekomponenten des Gravitationsfeldes. Physikalische Zeitschrift 19, 4 (1918)Google Scholar
  12. 12.
    Bauer H.: Über die Energiekomponenten des Gravitationsfeldes. Physikalische Zeitschrift 19, 163 (1918)Google Scholar
  13. 13.
    Pauli, W.: Theory of Relativity. Pergamon, New York (1921). English translation 1958 by Field, G.; republished by Dover, New York (1981)Google Scholar
  14. 14.
    Cattani, C., Maria, M.D.: Conservation laws and gravitational waves in general relativity (1915–1918). In: Earman, J., Janssen, M., Norton, J.D. The Attraction of Gravitation: New Studies in the History of General Relativity, Einstein Studies, vol. 5, p. 63. Birkhäuser, Boston (1993)Google Scholar
  15. 15.
    Rosen N.: General Relativity and flat space. I., II. Phys. Rev. 57, 147, 150 (1940)ADSGoogle Scholar
  16. 16.
    Rosen N.: Flat-space metric in general relativity theory. Ann. Phys. 22, 1 (1963)zbMATHCrossRefADSGoogle Scholar
  17. 17.
    Cornish F.H.J.: Energy and momentum in general relativity. I. The 4-momentum expressed in terms of four invariants when space-time is asymptotically flat. Proc. R. Soc. A 282, 358 (1964)CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Møller C.: Further remarks on the localization of the energy in the general theory of relativity. Ann. Phys. 12, 118 (1961)CrossRefADSGoogle Scholar
  19. 19.
    Møller C.: Momentum and energy in general relativity and gravitational radiation. Kongelige Danske Videnskabernes Selskab, Matematisk-Fysiske Meddelelser. 34(3) (1964)Google Scholar
  20. 20.
    Sorkin R.D.: The gravitational-electromagnetic Noether operator and the second-order energy flux. Proc. R. Soc. Lond. A Math. Phys. Sci. 435, 635 (1991)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Fatibene L., Francaviglia M.: Natural and Gauge Natural Formalism for Classical Field Theories: a Geometric Perspective including Spinors and Gauge Theories. Kluwer, Dordrecht (2003)zbMATHGoogle Scholar
  22. 22.
    Misner C., Thorne K., Wheeler J.A.: Gravitation. Freeman, New York (1973)Google Scholar
  23. 23.
    Brading K.: A note on general relativity, energy conservation, and Noether’s theorems. In: Kox, A.J., Eisenstaedt, J. (eds) The Universe of General Relativity, Einstein Studies, vol. 11, pp. 125. Birkhäuser, Boston (2005)CrossRefGoogle Scholar
  24. 24.
    Kiriushcheva N., Kuzmin S., Racknor C., Valluri S.: Diffeomorphism invariance in the Hamiltonian formulation of General Relativity. Phys. Lett. A 372, 5101 (2008) arXiv:0808.2623v1 [gr-qc]CrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Kiriushcheva, Kiriushcheva, N., Kuzmin, S.: The Hamiltonian Formulation of General Relativity: Myths and Reality. (2008). arXiv:0809.0097v1 [gr-qc]Google Scholar
  26. 26.
    Brown H.R.: Physical Relativity: Space-time Structure from a Dynamical Perspective. Oxford University Press, New York (2005)zbMATHGoogle Scholar
  27. 27.
    Trautman, A.: Foundations and current problems of General Relativity. In: Deser, S., Ford, K.W. (eds.) Lectures on General Relativity, pp. 1–248. Prentice Hall, Englewood Cliffs (1965). Brandeis Summer Institute in Theoretical PhysicsGoogle Scholar
  28. 28.
    Nijenhuis, A.: Natural bundles and their general properties: geometric objects revisited. In: Kobayashi, S., Obata, M., Takahashi, T. Differential Geometry: In Honor of Kentaro Yano, p. 317. Kinokuniya Book-store Co, Tokyo (1972)Google Scholar
  29. 29.
    Faddeev L.D.: The energy problem in Einstein’s theory of gravitation (dedicated to the memory of V. A. Fock). Sov. Phys. Uspekhi 25, 130 (1982)CrossRefADSGoogle Scholar
  30. 30.
    Bergmann P.G.: Conservation laws in general relativity as the generators of coordinate transformations. Phys. Rev. 112, 287 (1958)zbMATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Komar A.: Covariant conservation laws in general relativity. Phys. Rev. 113, 934 (1959)zbMATHCrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Petrov, A.N., Katz J.: Relativistic conservation laws on curved backgrounds and the theory of cosmological perturbations. In: Proceedings of the Royal Society (London) A, vol. 458, p. 319 (2002). gr-qc/9911025v3Google Scholar
  33. 33.
    Komar A.: Asymptotic covariant conservation laws for gravitational radiation. Phys. Rev. 127, 1411 (1962)zbMATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Komar A.: Positive-definite energy density and global consequences for General Relativity. Phys. Rev. 129, 1873 (1963)zbMATHCrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Anderson J.L.: Principles of Relativity Physics. Academic, New York (1967)Google Scholar
  36. 36.
    Nijenhuis, A.: Theory of the Geometric Object. PhD thesis, University of Amsterdam (1952). Supervised by Jan A. SchoutenGoogle Scholar
  37. 37.
    Grishchuk L.P., Petrov A.N., Popova A.D.: Exact theory of the (Einstein) gravitational field in an arbitrary background space-time. Commun. Math. Phys. 94, 379 (1984)CrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Petrov, A.N.: Nonlinear perturbations and conservation laws on curved backgrounds in GR and other metric theories. In: Christiansen, M.N., Rasmussen, T.K. (eds.) Classical and Quantum Gravity Research, p. 79. Nova Science, Hauppauge (2008). arXiv:0705.0019Google Scholar
  39. 39.
    Katz J., Bičák J., Lynden-Bell D.: Relativistic conservation laws and integral constraints for large cosmological perturbations. Phys. Rev. D 55, 5957 (1997) gr-qc/0504041CrossRefMathSciNetADSGoogle Scholar
  40. 40.
    Katz J.: Gravitational energy. Class. Quantum Gravity 22, 5169 (2005) gr-qc/0510092zbMATHCrossRefADSGoogle Scholar
  41. 41.
    Chang C.-C., Nester J.M., Chen C.-M.: Pseudotensors and quasilocal energy-momentum. Phys. Rev. Lett. 83, 1897 (1999) gr-qc/9809040zbMATHCrossRefMathSciNetADSGoogle Scholar
  42. 42.
    Nester J.M.: General pseudotensors and quasilocal quantities. Class. Quantum Gravity 21, S261 (2004)zbMATHCrossRefMathSciNetADSGoogle Scholar
  43. 43.
    Zel’dovich Y.B., Grishchuk L.P.: Gravitation, the general theory of relativity, and alternative theories. Sov. Phys. Uspekhi 29, 780 (1986)CrossRefMathSciNetADSGoogle Scholar
  44. 44.
    Grishchuk L.P.: The general theory of relativity: familiar and unfamiliar. Sov. Phys. Uspekhi 33, 669 (1990)CrossRefMathSciNetADSGoogle Scholar
  45. 45.
    Pinto-Neto N., Trajtenberg P.I.: On the localization of the gravitational energy. Brazil. J. Phys. 30, 181 (2000)Google Scholar
  46. 46.
    Yano K.: The Theory of Lie Derivatives and Its Applications. North-Holland, Amsterdam (1957)zbMATHGoogle Scholar
  47. 47.
    Pitts J.B., Schieve W.C.: Null cones and Einstein’s equations in Minkowski spacetime. Found. Phys. 34, 211 (2004) gr-qc/0406102zbMATHCrossRefMathSciNetADSGoogle Scholar
  48. 48.
    Pitts J.B., Schieve W.C.: Slightly bimetric gravitation. Gen. Relat. Gravit. 33, 1319 (2001) gr-qc/0101058v3zbMATHCrossRefMathSciNetADSGoogle Scholar
  49. 49.
    Marolf, D.: Group averaging and refined algebraic quantization: where are we now? In: Jantzen, R.T., Ruffini, R., Gurzadyan, V.G. (eds.) Proceedings of the Ninth Marcel Grossmann Meeting (held at the University of Rome “La Sapienza”, 2–8 July 2000). World Scientific, River Edge (2002). gr-qc/0011112Google Scholar
  50. 50.
    Hilbert, D.: The foundations of physics (second communication). In: Renn, J., Schemmel, M. (eds.) The Genesis of General Relativity, vol. 4: Gravitation in the Twilight of Classical Physics: The Promise of Mathematics, vol. 4, p. 1017. Springer, Dordrecht (2007). Translated from “Die Grundlagen der Physik. (Zweite Mitteilung),” Nachrichten von der Königliche Gesellschaft der Wissenschaft zu Göttingen. Mathematisch-Physikalische Klasse (1917), p. 53Google Scholar
  51. 51.
    Møller C.: The Theory of Relativity. Clarendon, Oxford (1952)Google Scholar
  52. 52.
    Hawking S.W., Horowitz G.T.: The gravitational Hamiltonian, action, entropy, and surface terms. Class. Quantum Gravity 13, 148 (1996) gr-qc/9501014MathSciNetGoogle Scholar
  53. 53.
    Choquet-Bruhat Y., DeWitt-Morette C.: Analysis, Manifolds, and Physics. Part II: 92 Applications. North-Holland, Amsterdam (1989)zbMATHGoogle Scholar
  54. 54.
    Gates S.J. Jr, Grisaru M.T., Roček M., Siegel W.: Superspace, or One Thousand and One Lessons in Supersymmetry. Benjamin/Cummings, Reading (1983)Google Scholar
  55. 55.
    Ogievetskĭ V.I., Polubarinov I.V.: Spinors in gravitation theory. Sov. Phys. JETP 21, 1093 (1965)ADSGoogle Scholar
  56. 56.
    Ogievetsky V.I., Polubarinov I.V.: Interacting field of spin 2 and the Einstein equations. Ann. Phys. 35, 167 (1965)CrossRefADSGoogle Scholar
  57. 57.
    Bilyalov R.F.: Spinors on Riemannian manifolds. Russ. Math. (Iz. VUZ) 46(11), 6 (2002)MathSciNetGoogle Scholar
  58. 58.
    Bilyalov R.F.: Conservation laws for spinor fields on a Riemannian space-time manifold. Theor. Math. Phys. 90, 252 (1992)CrossRefMathSciNetGoogle Scholar
  59. 59.
    Szybiak A.: On the Lie derivative of geometric objects from the point of view of functional equations. Prace Matematyczne=Schedae Mathematicae 11, 85 (1966)MathSciNetGoogle Scholar
  60. 60.
    Szybiak A.: Covariant derivative of geometric objects of the first class. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 11, 687 (1963)zbMATHMathSciNetGoogle Scholar
  61. 61.
    Isham C.J., Salam A., Strathdee J.: Nonlinear realizations of space-time symmetries. Scalar and tensor gravity. Ann. Phys. 62, 98 (1971)zbMATHCrossRefMathSciNetADSGoogle Scholar
  62. 62.
    Cho Y.M., Freund P.G.O.: Non-Abelian gauge fields as Nambu-Goldstone fields. Phys. Rev. D 12, 1711 (1975)CrossRefMathSciNetADSGoogle Scholar
  63. 63.
    Misner C.W.: Gravitational field energy and g 00. Phys. Rev. 130, 1590 (1963)zbMATHCrossRefMathSciNetADSGoogle Scholar
  64. 64.
    Thirring W., Wallner R.: The use of exterior forms Einstein’s gravitation theory. Revista Brasileira de Física 8, 686 (1978)Google Scholar
  65. 65.
    Sundermeyer, K.: Constrained Dynamics: With Applications to Yang–Mills Theory, General Relativity, Classical Spin, Dual String Model. Lecture Notes in Physics, vol. 169. Springer, Berlin (1982)Google Scholar
  66. 66.
    Papapetrou A.: Einstein’s theory of gravitation and flat space. Proc. R. Ir. Acad. A 52, 11 (1948)MathSciNetGoogle Scholar
  67. 67.
    Leclerc, M.: Noether’s theorem, the stress-energy tensor and Hamiltonian constraints (2006). arXiv:gr-qc/0608096v4Google Scholar
  68. 68.
    Goldberg J.N.: Conservation laws in general relativity. Phys. Rev. 111, 315 (1958)zbMATHCrossRefMathSciNetADSGoogle Scholar
  69. 69.
    Chang, C.-C., Nester, J.M., Chen, C.-M.: Energy-momentum (quasi-)localization for gravitating systems. In: Liu, L., Luo, J. Li, X.-Z., Hsu, J.-P. (eds.) The Proceedings of the Fourth International Workshop on Gravitation and Astrophysics: Beijing Normal University, China, 10–15 October 1999, p. 163. World Scientific, Singapore (2000). gr-qc/9912058v1Google Scholar
  70. 70.
    Kucharzewski M., Kuczma M.: Basic concepts of the theory of geometric objects. Rozprawy Matematyczne=Dissertationes Mathematicae 43, 1 (1964)MathSciNetGoogle Scholar
  71. 71.
    Wald R.M.: General Relativity. University of Chicago, Chicago (1984)zbMATHGoogle Scholar
  72. 72.
    Lee J.M.: Introduction to Smooth Manifolds. Springer, New York (2003)Google Scholar
  73. 73.
    Bergqvist G.: Positivity and definitions of mass. Class. Quantum Gravity 9, 1917 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  74. 74.
    Petrov A.N.: The Schwarzschild black hole as a point particle. Found. Phys. Lett. 18, 477 (2005) gr-qc/0503082v2zbMATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Rosen N., Virbhadra K.S.: Energy and momentum of cylindrical gravitational waves. Gen. Relat. Gravit. 25, 429 (1993)zbMATHCrossRefMathSciNetADSGoogle Scholar
  76. 76.
    Xulu S.S.: Total energy of the Bianchi type I universes. Int. J. Theor. Phys. 39, 115 (2000) gr-qc/9910015CrossRefMathSciNetGoogle Scholar
  77. 77.
    Regge T., Teitelboim C.: Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. 88, 286 (1974)zbMATHCrossRefMathSciNetADSGoogle Scholar
  78. 78.
    Katz J.: Note on Komar’s anomalous factor. Class. Quantum Gravity 2, 423 (1985)CrossRefADSGoogle Scholar
  79. 79.
    Katz J., Ori A.: Localisation of field energy. Class. Quantum Gravity 7, 787 (1990)CrossRefMathSciNetADSGoogle Scholar
  80. 80.
    Pavelle R.: Conserved vectors of the Komar type and compatibility identities in Lagrangian field theories. J. Math. Phys. 16, 696 (1975)zbMATHCrossRefMathSciNetADSGoogle Scholar
  81. 81.
    Pavelle R.: Conserved vector densities and their curl expressions. J. Math. Phys. 16, 1199 (1975)CrossRefMathSciNetADSGoogle Scholar
  82. 82.
    Schutz B.F., Sorkin R.: Variational aspects of relativistic field theories, with applications to perfect fluids. Ann. Phys. 107, 1 (1977)zbMATHCrossRefADSGoogle Scholar
  83. 83.
    Sorkin R.: On stress-energy tensors. Gen. Relat. Gravit. 8, 437 (1977)zbMATHCrossRefADSGoogle Scholar
  84. 84.
    Szabados L.B.: On canonical pseudotensors, Sparling’s form and Noether currents. Class. Quantum Gravity 9, 2521 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  85. 85.
    Trautman A.: The general theory of relativity. Sov. Phys. Uspekhi 89, 319 (1966)CrossRefADSGoogle Scholar
  86. 86.
    Tashiro Y.: Sur la dérivée de Lie de l’être géométrique et son groupe d’invariance. Tôhoku Math. J. 2, 166 (1950)zbMATHCrossRefMathSciNetGoogle Scholar
  87. 87.
    Tashiro Y.: Note sur la dérivée de Lie d’un être géométrique. Math. J. Okayama Univ. 1, 125 (1952)zbMATHMathSciNetGoogle Scholar
  88. 88.
    Gotay, M.J., Isenberg, J., Marsden, J.E., Montgomery, R., Śniatycki, J., Yasskin, P.B.: Momentum maps and classical fields. Part I: Covariant field theory, (2004). physics/9801019v2
  89. 89.
    Deser S.: Self-interaction and gauge invariance. Gen. Relat. Gravit. 1, 9 (1970) gr-qc/0411023v2CrossRefMathSciNetADSGoogle Scholar
  90. 90.
    Feynman, R.P., Morinigo, F.B., Wagner, W.G., Hatfield, B., Preskill, J., Thorne, K.S.: Feynman Lectures on Gravitation. Addison-Wesley, Reading (1995). Original by California Institute of Technology, 1963Google Scholar
  91. 91.
    Earman J., Norton J.: What price spacetime substantivalism? The hole story. Br. J. Philos. Sci. 38, 515 (1987)CrossRefMathSciNetGoogle Scholar
  92. 92.
    Weinberg S.: Gravitation and Cosmology. Wiley, New York (1972)Google Scholar
  93. 93.
    Hoefer C.: Energy conservation in GTR. Stud. Hist. Philos. Mod. Phys. 31, 187 (2000)CrossRefMathSciNetGoogle Scholar
  94. 94.
    Logunov A.A., Folomeshkin V.N.: The energy-momentum problem and the theory of gravitation. Theor. Math. Phys. 32, 749 (1977)CrossRefGoogle Scholar
  95. 95.
    Logunov A.A., Loskutov Y.M., Chugreev Y.V.: Does general relativity explain gravitational effects?. Theor. Math. Phys. 69, 1179 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  96. 96.
    Zel’dovich Y.B., Grishchuk L.P.: The general theory of relativity is correct!. Sov. Phys. Uspekhi 31, 666 (1988)CrossRefMathSciNetADSGoogle Scholar
  97. 97.
    Pitts J.B.: Has Robert Gentry refuted Big Bang cosmology? On energy conservation and cosmic expansion. Perspect. Sci. Christ. Faith 56(4), 260 (2004)Google Scholar
  98. 98.
    Pitts J.B.: Reply to Gentry on cosmological energy conservation and cosmic expansion. Perspect. Sci. Christ. Faith 56(4), 278 (2004)Google Scholar
  99. 99.
    Bunge M.: Energy: Between physics and metaphysics. Sci. Educ. 9, 457 (2000)CrossRefGoogle Scholar
  100. 100.
    Pitts J.B.: Nonexistence of Humphreys’ “volume cooling” for terrestrial heat disposal by cosmic expansion. Perspect. Sci. Christ. Faith 61(1), 23 (2009)Google Scholar
  101. 101.
    Collins R.: Modern physics and the energy-conservation objection to mind-body dualism. Am. Philos. Q. 45, 31 (2008)Google Scholar
  102. 102.
    Tryon E.P.: Is the universe a vacuum fluctuation?. Nature 246, 396 (1973)CrossRefADSGoogle Scholar
  103. 103.
    Thirring, W.E.: God’s traces in the laws of nature. In: The Cultural Values of Science, p. 362. The Pontifical Academy of Sciences, Vatican City (2003)Google Scholar
  104. 104.
    Laudan L.: Progress and Its Problems: Towards a Theory of Scientific Growth. University of California, Berkeley (1977)Google Scholar
  105. 105.
    Howard D.: Astride the divided line: Platonism, empiricism, and Einstein’s epistemological opportunism. In: Shanks, N. (eds) Idealization IX: Idealization in Contemporary Physics. Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 63, pp. 143–163. Rodopi, Amsterdam (1998)Google Scholar
  106. 106.
    Callan C.G. Jr, Coleman S., Jackiw R.: A new improved energy-momentum tensor. Ann. Phys. 59, 42 (1970)zbMATHCrossRefMathSciNetADSGoogle Scholar
  107. 107.
    Leclerc M.: Canonical and gravitational stress-energy tensors. Int. J. Mod. Phys. D 15, 959 (2006) gr-qc/0510044v6zbMATHCrossRefMathSciNetADSGoogle Scholar
  108. 108.
    Pons, J.M.: Noether Symmetries, Energy-Momentum Tensors and Conformal Invariance in Classical Field Theory (2009). arxiv:0902.4871Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.University of Notre DameNotre DameUSA

Personalised recommendations