Gravitation and electromagnetism as geometrical objects of a Riemann-Cartan spacetime structure. (English) Zbl 1260.83028
Summary: In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature \(R\) of a particular kind of Riemann-Cartan spacetime structure. In our model the gravitational and electromagnetic fields are identified as geometrical objects of the structure.We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term \(A\wedge dA\) that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic field plus the gravitational field, together with a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.
MSC:
| 83C22 | Einstein-Maxwell equations |
| 83E05 | Geometrodynamics and the holographic principle |
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 78A25 | Electromagnetic theory (general) |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
Keywords:
Riemann-Cartan connection; electromagnetism; gravitation; Maxwell equations; Chern-Simons term; spin density; Lorentz forceReferences:
| [1] | R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity. Chapter 13, McGraw-Hill, New York, 1965. · Zbl 0144.47604 |
| [2] | Andrade V. C., Pereira J. G.: Torsion and the Electromagnetic Field. Int. J. Mod. Phys. D 8, 141–151 (1999) · Zbl 0964.83015 · doi:10.1142/S0218271899000122 |
| [3] | Appelquist T., Chodos A., Freund P. G. O.: Modern Kaluza-Klein Theories. Addison-Wesley, New York (1987) · Zbl 0984.83501 |
| [4] | S. V. Babak and L. P. Grishckuk, The Energy Momentum Tensor for the Gravitational Field. Phys. Rev. D 61 (1999), 024038, 1–18. [arXiv: gr-qc/9907027 v2]. |
| [5] | Barut A. O.: Electrodynamics and Classical Theory of Fields and Particles. Dover, New York (1980) |
| [6] | Beil R. G.: Electrodynamics from a Metric. Int. J. Theor. Phys. 26, 189–197 (1986) · Zbl 0615.53072 · doi:10.1007/BF00669600 |
| [7] | Borchsenius K.: An extension of the Nonsymmetric Unified Field Theory. Gen. Rel. Grav. 7, 527–534 (1976) · doi:10.1007/BF00766412 |
| [8] | Capozzielo S., Lambiase G., Stornailo C.: Geometric Classification of Torsion Tensor of Space-Time. Ann. Phys. 10, 713–727 (2001) [arXiv: gr-qc 0101038 v1]. · Zbl 0974.83035 · doi:10.1002/1521-3889(200108)10:8<713::AID-ANDP713>3.0.CO;2-2 |
| [9] | Cartan E.: On Manifolds with an Affine Connection and the Theory of General Relativity. Bibliopolis, Napoli (1986) · Zbl 0657.53001 |
| [10] | Chyba C. F.: Kaluza-Klein Unified Field Theory and Apparent Four- Dimensional Space-Time. Am. J. Phys. 53, 863–872 (1985) · doi:10.1119/1.14353 |
| [11] | Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Physics. Revised edition, North-Holland, Amsterdam, 1982. · Zbl 0385.58001 |
| [12] | R. Coquereaux and A. Jadczyk, Riemannian Geometry, Fiber Bundles, Kaluza- Klein Theories And All That. Lectures Notes in Physics 16, World Scientific, Singapore, 1988. · Zbl 0734.53002 |
| [13] | W. N. Cottingham and D. A. Greenwood, An Introduction to the Standard Model of Particle Physics. Second edition, Cambridge Univ. Press, Cambridge, 2007. · Zbl 1126.81002 |
| [14] | R. da Rocha and W. A. Rodrigues, Jr., Pair and Impar, Even and Odd Differential Forms and Electromagnetism. Ann. Phys. (Berlin) 19 (2010), 6–34. · Zbl 1192.78008 |
| [15] | A. S. Eddington, The Mathematical Theory of Relativity. Third edition, Chelsea, New York, 1975. |
| [16] | A. Einstein, The Meaning of Relativity. Appendix II. Princeton University Press, Princeton, 1956. · Zbl 0071.41205 |
| [17] | V. V. Fernández and W. A. Rodrigues, Jr., Gravitation as a Plastic Distortion of the Lorentz Vacuum. Fundamental Theories of Physics 168, Springer, Heidelberg, 2010. |
| [18] | M. Fontaine and P. Amiot, Geometrical Aspects of Electromagnetism, I. Ann. Phys. 147 (1983), 269–280. |
| [19] | E. S. Fradkin and A. A. Tseytlin, Effective Field Theory from Quantized Strings. Phys. Lett. B 158 (1985), 316–322. · Zbl 0967.81516 |
| [20] | L. C. G. de Andrade and M. Lopes, Detecting Torsion from Massive Electrodynamics. Gen. Rel. Grav. 25 (1993), 1101–1106. |
| [21] | H. de Vries, On the Electromagnetic Chern Simons Spin Density as Hidden Variable and EPR Correlations. http://physicsquest.org/ChernSimonsSpinDensity.pdf , 2012. |
| [22] | J. Garecki, Is Torsion Needed in Theory of Gravity?. Rel. Grav. Cosmol. 1 (2004), 43–59, [arXiv:gr-qc/0103029 v1]. |
| [23] | M. Göckeler and T. Schücker, Differential Geometry, Gauge, Gauge Theories, and Gravity. Cambridge University Press, Cambridge, 1987. · Zbl 0649.53001 |
| [24] | R. T. Hammond, Gravitation, Torsion, and Electromagnetism. Gen. Rel. Grav. 20 (1988), 813–827. |
| [25] | R. T. Hammond, Spin, Torsion, Forces. Gen. Rel. Grav. 26 (1994), 247–263. · Zbl 0798.53085 |
| [26] | R. T. Hammond, New Fields in General Relativity. Cont. Phys. 36 (1995), 103–114. |
| [27] | R. T. Hammond, Upper Limit on the Torsion Coupling Constant. Phys. Rev. D 52 (1995), 6918–6921. |
| [28] | F. W. Hehl, P. von der Heyde, and G. Kerlick, General Relativity with Spin and Torsion: Foundations and Prospects. Rev. Mod. Phys. 48 (1976), 393–416. · Zbl 1371.83017 |
| [29] | F. W. Hehl, Spin and Torsion in General Relativity: I. Foundations. Gen. Rel. Grav. 4 (1973), 333–349. |
| [30] | F. W. Hehl and Y. N. Obukhov, How Does the Electromagnetic Field Couple to Gravity, in Particular to Metric, Nonmetricity, Torsion and Curvature?. Lecture Notes in Physics 562, Springer, Heidelberg, 2001, 479–504 [arXiv: grqe/ 0001010 v2]. · Zbl 0982.83033 |
| [31] | F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics, Charge, Flux and Metric. Birkhäuser, Boston, 2003. · Zbl 1032.78001 |
| [32] | A. Jakubiec and J. Kijowski, On Interaction of the Unified Maxwell-Einstein Field with Spinorial Matter. Lett. Math. Phys. 9 (1985), 1–11. |
| [33] | Th. Kaluza, Zum Unitätsproblem in der Physik. Sitzungsber Preuss. Akad. Wiss. Berlin, (1921), 966–972. · JFM 48.1032.03 |
| [34] | T. Kibble, Lorentz Invariance and the Gravitational Field. J. Math. Phys. 2 (1961), 212–221. · Zbl 0095.22903 |
| [35] | R. M. Kiehn, Non-Equilibrium and Irreversible Thermodynamics from a Topological Perspective. Adventures in Applied Topology, Vol. 1 (2008), p. 38. http://www.lulu.com/spotlight/kiehn , 2012. |
| [36] | O. Klein, Quantentheorie und Fünfdimensionale Relativitätstheorie. Z. Phys. 37 (1926), 895–906. · JFM 52.0970.09 |
| [37] | H. Kleinert, Gauge Fields in Condensed Matter. World Scientific, Singapore, 1989. · Zbl 0785.53061 |
| [38] | L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields. Pergamon, Oxford, 1971. · Zbl 0178.28704 |
| [39] | J. W. Maluf, Hamiltonian Formulation of the Teleparallel Description of General Relativity. J. Math. Phys. 35 (1994), 335–343. · Zbl 0797.53067 |
| [40] | J. W. Moffat, Space-time Structure in a Generalization of Gravitation Theory. Phys. Rev. D 15 (1977), 3520–3529. |
| [41] | Moffat, J. W., Reinventing Gravity, Smithsonian Books, Harper Coolins Publ, New York, 2008. |
| [42] | R. J. McKellar, Asymmetric Connection Treatment of the Einstein-Maxwell Field Equations. Phys. Rev. D 20 (1979), 356–361. · Zbl 0413.53021 |
| [43] | M. Nakahara, Geometry, Topology and Physics, IOP Publ. Ltd, Bristol and Philadelphia, 1990. |
| [44] | E. A. Note-Cuello, R. da Rocha, and W. A. Rodrigues, Jr., Some Thoughts on Geometries and on the Nature of the Gravitational Field. J. Phys. Math. 2 (2010), 20–40. |
| [45] | W. Pauli, Theory of Relativity. Dover Publ. Inc., New York, 1981. · Zbl 0518.73075 |
| [46] | N. J. Popławski, Torsion as Electromagnetism and Spin. Int. J. Theor. Phys. 49 (2010), 1481–1488. · Zbl 1197.83098 |
| [47] | H. Ringermacher, An Electrodynamic Connection. Class. Quant. Grav. 11 (1997), 2383–2394. · Zbl 0821.53069 |
| [48] | W. A. Rodrigues, Jr. and E. Capelas de Oliveira, The Many Faces of Maxwell, Dirac and Einstein Equations. Lecture Notes in Physics 722, Springer, Heidelberg, 2007. · Zbl 1124.83002 |
| [49] | W. A. Rodrigues, Jr., Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan Structures and Some Applications to Physics. Ann. Fond. L. de Broglie 32 (special issue dedicated to torsion) (2007), 425–478, [arXiv:0712.3067v6 [math-ph]]. · Zbl 1329.53112 |
| [50] | W. A. Rodrigues, Jr., On the Nature of the Gravitational Field and its Legitimate Energy-Momentum Tensor. To appear in Rep. Math. Phys. (2012) [arXiv:1109.5272math-ph]. · Zbl 1252.83017 |
| [51] | H. Weyl, Gravitation und Elektrizität. Sitzungsber. Preuss. Akad. Wiss. 26 (1918), 465–478. · JFM 46.1300.01 |
| [52] | H. Weyl, Space, Time, Matter. Fourth edition, Dover Publications, New York, 1950. |
| [53] | T. Sauer, Field Equations in Teleparallel Spacetime: Einstein’s Fernparallelismus Approach Towards Unified Field Theory. Historia Math. 33 (2006), 399– 439, [arXiv:physics/0405142v1]. · Zbl 1108.01013 |
| [54] | R. K. Sachs and H. Wu, General Relativity for Mathematicians. Springer- Verlag, New York, 1997. · Zbl 0373.53001 |
| [55] | D. Sciama, The Physical Structure of General Relativity. Rev. Mod. Phys. 36 (1964), 463–469. |
| [56] | E. Schrödinger, The General Unitary Theory of the Physical Fields. Proc. R. Irish Acad. A 49 (1943), 43–58. · Zbl 0060.44201 |
| [57] | C. Sivaram and L. C. G. de Andrade, Torsion Gravity Effects on Carged- Particle and Neutron Interferometers. [arXiv:gr-qc/0111009 v1]. |
| [58] | Q. A. G. Souza and W. A. Rodrigues, Jr, The Dirac Operator and the Structure of Riemann-Cartan-Weyl Spaces. In Gravitation: The Spacetime Structure, P. Letelier and W. A. Rodrigues, Jr., Eds., World Scientific, Singapore, 1994, 179–212. · Zbl 0972.58507 |
| [59] | D. Utiyama, Invariant Theoretical Interpretation of Interaction. Phys. Rev. 101 (1956), 1597–1607. · Zbl 0070.22102 |
| [60] | J. G. Vargas, Geometrization of the Physics with Teleparallelism I. The Classical Interactions. Found. Phys. 22 (1992), 507–526. |
| [61] | J. G. Vargas, D. G. Torr, and A. Lecompte, Geometrization of the Physics with Teleparallelism. II. Towards a Fully Geometric Dirac Equation. Found. Phys. 22 (1992), 527–547. |
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.
