Abstract
We are concerned with the precise modalities by which mathematical constructions related to energy tensors can be adapted to a tetrad-affine setting. We show that, for fairly general gauge field theories formulated in that setting, two notions of energy tensor (the canonical tensor and the stress-energy tensor) exactly coincide with no need for tweaking. Moreover, we show how both notions of energy tensor can be naturally extended to include the gravitational field itself, represented by a couple constituted by the tetrad and the spinor connection. Then we examine the on-shell divergences of these tensors in relation to the issue of local energy conservation in the presence of torsion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Goldsmith and S. Sternberg, “The Hamilton-Cartan formalism in the calculus of variations,” Ann. Inst. Fourier, Grenoble 23, 203–267 (1973).
M. J. Gotay and J. E. Marsden, “Stress-energymomentum tensors and the Belinfante-Rosenfeld formula,” Contemp. Math. 132, 367–392 (1992).
B. Kuperschmidt, “Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms,” Lecture Notes in Math. 775, 162–218 (1979).
L. Landau and E. Lifchitz, Théorie du champ (EditionsMir, Moscou, 1968).
L. Mangiarotti and M. Modugno, “Some results on the calculus of variations on jet spaces,” Ann. Inst. H. Poinc. 39, 29–43 (1983).
A. Trautman, “Noether equations and conservation laws,” Commun. Math. Phys. 6, 248–261 (1967).
M. Ferraris and M. Francaviglia, “Conservation laws in general relativity,” Class. Quantum Grav. 9, S79 (1992).
L. Fatibene, M. Ferraris, and M. Francaviglia, “Noether formalism for conserved quantities in classical gauge field theories,” J. Math. Phys. 35 1644 (1994).
M. Forger and H. Römer, “Currents and the energymomentum tensor in classical field theory, a fresh look at an old problem,” Annals Phys. 309, 306–389 (2004).
F. J. Belinfante, “On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,” Physica VII, 449–474 (1940).
L. Rosenfeld, “Sur le tenseur d’impulsion-énergie,” Mem. Acad. Roy. Belg. Sci. 18, 1–30 (1940).
K. Bamba and K. Shimizu, “Construction of energymomentum tensor of gravitation,” Int. J. Geom. MethodsMod. Phys. 13, 1650001 (2016).
C. M. Chen and J. M. Nester, “A symplectic Hamiltonian derivation of quasilocal energy-momentum for GR,” Grav. Cosmol. 6, 257–270 (2000).
J. Kijowski, “A simple derivation of canonical structure and quasi-local Hamiltonians in General Relativity,” Gen. Rel. Grav. 29, 307–343 (1997).
A. Komar, “Covariant conservation laws in general relativity,” Phys. Rev. 113, 934–936 (1959).
Lau Loi So, “Gravitational energy froma combination of a tetrad expression and Einstein’s pseudotensor,” Class. Quantum Grav. 25, 175012 (2008).
M. Leclerc, “Canonical and gravitational stressenergy tensors,” Int. J.Mod. Phys. D 15, 959–990 (2006).
T. Padmanabhan, “General Relativity from a thermodynamic perspective,” Gen. Rel. Grav. 46, 1673 (2014).
D. Canarutto, “Overconnections and the energytensors of gauge and gravitational fields,” J. Geom. Phys. 106, 192–204 (2016).
É. Cartan, “Sur une généralisation de la notion de courbure de Riemann et les espaces átorsion,” C. R. Acad. Sci. (Paris) 174, 593–595 (1922).
É. Cartan, “Sur les variétés áconnexion affine et la théorie de la relativitégénéralisée,” Part I: Ann. Éc. Norm. 40, 325–412 (1923) and 41 (1924), 1–25; Part II: Ann. Éc. Norm. 42 (1925), 17–88 (1925).
F. W. Hehl, “Spin and torsion in general relativity: I. Foundations,” Gen. Rel. Grav. 4, 333–349 (1973).
F. W. Hehl, “Spin and torsion in General Relativity: II. Geometry and field equations,” Gen. Rel. Grav. 5, 491–516 (1974).
F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J.M. Nester, “General relativity with spin and torsion: Foundations and prospects,” Rev. Mod. Phys. 48, 393–416 (1976).
F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, “Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance,” Phys. Rep. 258, 1–171 (1995).
M. Henneaux, “On geometrodynamics with tetrad fields,” Gen. Rel. Grav. 9, 1031–1045 (1978).
T. W. B. Kibble, “Lorentz invariance and the gravitational field,” J.Math. Phys. 2, 212 (1961).
N. J. Popławsky, “Geometrization of electromagnetism in tetrad-spin-connection gravity,” Mod. Phys. Lett. A 24 (6), 431–442 (2009).
D. W. Sciama, “On a non-symmetric theory of the pure gravitational field,” Proc. Cambridge Philos. Soc. 54, 72–80 (1958).
A. Trautman, “Einstein-Cartan theory,” in Encyclopedia of Mathematical Physics, Vol. 2, Ed. J.-P. Franзoise, G. L. Naber, and S. T. Tsou (Elsevier, Oxford, 2006), pp. 189–195.
R. Penrose and W. Rindler, Spinors and Space-Time. I: Two-Spinor Calculus and Relativistic Fields (Cambridge University Press, Cambridge, 1984).
R. Penrose and W. Rindler, Spinors and Space-Time. II: Spinor and Twistor Methods in Space-Time Geometry (Cambridge University Press, Cambridge, 1988).
D. Canarutto, “Possibly degenerate tetrad gravity and Maxwell-Dirac fields,” J. Math. Phys. 39 (9), 4814–4823 (1998).
D. Canarutto, “Two-spinors, field theories and geometric optics in curved spacetime,” Acta Appl. Math. 62 (2), 187–224 (2000).
D. Canarutto, “Minimal geometric data” approach to Dirac algebra, spinor groups and field theories,” Int. J. Geom.Met.Mod. Phys., 4 (6), 1005–1040 (2007).
D. Canarutto, “Tetrad gravity, electroweak geometry and conformal symmetry,” Int. J. Geom. Met. Mod. Phys., 8 (4), 797–819 (2011).
D. Canarutto, “Natural extensions of electroweak geometry and Higgs interactions,” Ann. Inst. H. Poincaré 16 (11), 2695–2711 (2015).
A. Frölicher and A. Nijenhuis, “Theory of vector–valued differential forms, Part I,” Indagationes Mathematicae 18, 338–360 (1956).
L. Mangiarotti and M. Modugno, “Graded Lie algebras and connections on a fibred space,” Journ.Math. Pur. et Applic. 63, 111–120 (1984).
I. Kolář, P. Michor, and J. Slovák, Natural Operations in Differential Geometry (Springer-Verlag, 1993).
D. Canarutto, “Covariant-differential formulation of Lagrangian field theory,” arXiv: 1607.03864.
P. L. Garcнa, “The Poincarй-Cartan invariant in the calculus of variations,” Symposia Mathematica 14, 219–246 (1974).
D. Krupka, Introduction to Global Variational Geometry (Springer, 2015).
D. J. Saunders, The Geometry of Jet Bundles (Cambridge University Press, 1989).
V. Bargmann and E. P. Wigner, “Group theoretical discussion of relativistic wave equations,” Proc. Nat. Acad. Sci. of the USA 34 (5), 211–223 (1948).
N. H. Barth and S. M. Christensen, “Arbitrary spin field equations on curved manifolds with torsion,” J. Phys. A 16, 543–563 (1983).
D. Canarutto, “A first-order Lagrangian theory of fields with arbitrary spin,” Int. J. Geom. Met. Mod. Phys. (to appear), arXiv: 1704.01110.
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, (Cambridge Univ. Press, Cambridge, 1973).
R. Hermann, “Gauge fields and Cartan-Ehresmann connections, Part A,” in Interdisciplinary Mathematics, Vol. X, (Math. Sci. Press, Brooklyn, 1975).
D. Canarutto and M. Modugno, “Ehresmann’s connections and the geometry of energy-tensors in Lagrangian field theories,” Tensor 42, 112–120 (1985).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Canarutto, D. On the Notions of Energy Tensors in Tetrad-Affine Gravity. Gravit. Cosmol. 24, 122–128 (2018). https://doi.org/10.1134/S0202289318020056
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0202289318020056