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Year 2019, , 141 - 147, 30.09.2019
https://doi.org/10.32323/ujma.553017

Abstract

References

  • [1] A. Einstein, Sitzungsber, On the general theory of relativity, Preus. Akad. Wiss. Berlin (Math. Phys.) 778 (1915).
  • [2] L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, (Addison-Wesley Press, New York), (1962).
  • [3] C. Møller, On the localization of the energy of a physical system in the general theory of relativity, Ann. Phys., 4 (1958), 347.
  • [4] P. G. Bergmann, R. Thomson, Conservation laws in general relativity as the generators of coordinate transformations, Phys. Rev., 89 (1958), 400.
  • [5] S. Weinberg, Anthropic bound on the cosmological constant, Phys. Rev. Lett., 59 (1987), 2607.
  • [6] A. Papapetrou, Einstein’s theory of gravitation and flat space, Proc. R. Irish Acad., A52 (1948), 11.
  • [7] R. C. Tolman, Relativity, Thermodynamics and Cosmology (Oxford University Press, Oxford, 1934).
  • [8] C. W. Misner, D. H. Sharp, Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse, Phys. Rev. B, 136 (1964), 571.
  • [9] F. I. Cooperstock, R. S. Sarracino, The localisation of energy in general relativity, J. Phys. A: Math. Gen., 11 (1978), 877.
  • [10] K. S. Virbhadra, J. C. Parikh, Gravitational energy of a stringy charged black hole, Phys. Lett., B317 (1993), 312.
  • [11] K. S. Virbhadra, Naked singularities and Seifert?s conjecture, Phys. Rev., D60(1999), 104041
  • [12] K. P. Tod, Some examples of Penrose?s quasi-local mass construction, Proc. Roy. Soc., London A388(1983),1795.
  • [13] R. Penrose, Penrose?s quasi-local mass, Proc. Roy. Soc. London, A381 (1982), 53.
  • [14] J. W. Maluf, Hamiltonian formulation of the teleparallel description of general relativity, J. Math. Phys., 35 (1994), 335.
  • [15] M. Blagojevic, I. A. Nikolic, Hamiltonian structure of the teleparallel formulation of general relativity, Phys. Rev., D62 (2000), 024021.
  • [16] J. Schwinger, Quantized gravitational field, Phys. Rev., 130 (1963), 1253.
  • [17] F. I. Mikhail, M. I. Wanas, A. Hindawi, E. I. Lashin, Energy-momentum complex in Møller’s tetrad theory of gravitation, Int. J. Theor. Phys., 32 (1993), 1627.
  • [18] T. Vargas, The energy of the universe in teleparallel gravity Gen. Rel. Grav., 36 (2004), 1255.
  • [19] M. Salti, A. Havare, Energy-Momentum in viscous Kasner-type universe in Bergmann-Thomson formulations Int. J. Mod. Phys., A20 (2005), 2169.
  • [20] M. Salti, Different Approaches for Moller’s Energy in the Kasner-type Space-time, Mod. Phys. Lett., A20 (2005), 2175.
  • [21] M. Salti, Energy?Momentum In The Viscous Kasner-Type Universe In Teleparallel Gravity, Astrophy. Space Sci., 229 (2005), 159.
  • [22] O. Aydogdu, M. Salti, Energy of the universe in Bianchi-Type I models in Møller’s tetrad theory of gravity, Astrophys. Space Sci., 229 (2005), 227.
  • [23] O. Aydogdu, M. Salti, M. Korunur, Energy in Reboucas-Tiomno-Korotkii-Obukhov and G¨odel-type Space-times in Bergmann-Thomson’s Formulations, Acta Phys. Slov., 55 (2005), 537.
  • [24] M. Sharif, M. J. Amir, Teleparallel energy?momentum distribution of lewis-papapetrou spacetimes Mod. Phys. Lett., A22 (2007), 425 .
  • [25] M. J. Amir, S. Ali, Energy-Momentum Distribution of Non-Static Plane Symmetric Spacetimes in General Relativity and Teleparallel Theory, Chinese Joun. of Phys., 50 (2012), 14 .
  • [26] M. Sharif, M. J. Amir, Teleparallel versions of Friedmann and Lewis-Papapetrou spacetimes, Gen. Relat. Gravit., 38(2006), 1735.
  • [27] M. Sharif, M. J. Amir, Teleparallel killing vectors of the Einstein universe, Mod. Phys. Lett., A22 (2007), 425.
  • [28] M. Sharif, M. J. Amir, Teleparallel Version of the Stationary Axisymmetric Solutions and their Energy Contents, Gen. Relat. Gravit., 39 (2007), 989.
  • [29] M. Sharif, M. J. Amir, Teleparallel Version of the Levi-Civita Vacuum Solutions and their Energy Contents, Canadian J. Phys., 86 (2008), 1091 .
  • [30] M. Sharif, M. J. Amir, Teleparallel Energy-Momentum Distribution of Static Axially Symmetric Spacetimes, Mod. Phys. Lett., A23 (2008), 3167 .
  • [31] M. Sharif, M. J. Amir, Energy-Momentum of the Friedmann Models in General Relativity and Teleparallel Theory of Gravity, Canadian J. Phys., 86 (2008), 1297.
  • [32] M. Sharif, M. J. Amir, Teleparallel Energy-Momentum Distribution of Spatially Homogeneous Rotating Spacetimes, Int. J. Theor. Phys., 47 (2008), 1742.
  • [33] T. Multam¨aki, A. Putaja1, L. Vilja1 , E.C. Vagenas, . Energy-momentum complexes in f(R) theories of gravity, Class. Quantum Gravity, 25 (2008), 075017
  • [34] M. Sharif, M. F. Shamir, Energy distribution in f ( R) gravity, Gen. Relativ. Gravit., 42 (2010) 1557.
  • [35] V. Faraoni, S. Nadeau, The Stability of modified gravity models, Phys. Rev., D72 (2005), 124005 .
  • [36] M. J. Amir, S. Naheed, Spatially Homogeneous Rotating Solution in f(R) Gravity and Its Energy Contents, Int. J. Theor. Phys., 52 (2013), 1688.
  • [37] M. J. Amir, S. Sattar, Locally Rotationally Symmetric Vacuum Solutions in f(R) Gravity, Int. J. Theor. Phys. 53 (2013), 773.
  • [38] J. E. G. Silva, V. Santos, C. A. S. Almeida, Gravity localization in a string-cigar braneworld, Class. Quantum Grav., 30 (2013), 025005 .
  • [39] W. D. Linch, G. Tartaglino-Mazzucchelli, Six-dimensional supergravity and projective superfields, JHEP, 08 (2012), 075.
  • [40] N. Popov, Geometric Model of the Gravitational Field, Gravitation and Cosmology, 4 (1998), 151.
  • [41] C. W. Misner, K. S. Thorne , J. A. Wheeler, Gravitation, W. H. Freeman and Co., NY 1973.
  • [42] P. Freud, The energy-momentum problem and the theory of gravitation, Ann. of Math., 40 (1938), 417 .

Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field

Year 2019, , 141 - 147, 30.09.2019
https://doi.org/10.32323/ujma.553017

Abstract

Much work has been done in exploring the energy-momentum distribution of different four-dimensional spacetimes using different prescriptions. In this paper, we intend to explore the energy and momentum density of six-dimensional geometric model of the gravitational field. The model was constructed by postulating a six-dimensional spacetime manifold with a structure of spacetime of absolute parallelism. For this purpose, we consider the metric representing the geometric model and use five prescriptions, namely, Einstein, Landau-Lifshitz, Bergmann-Thomson, Papapetrou, and Möller in the framework of General Relativity. The energy and momentum turn out to be well defined and finite. The comparison of the results shows that Einstein and Bergmann-Thomson prescriptions yield same energy-momentum densities but different from the other three prescriptions. It is mentioning here that the energy vanishes in the case of Möller's prescription and the momentum densities become zero in all the cases.

References

  • [1] A. Einstein, Sitzungsber, On the general theory of relativity, Preus. Akad. Wiss. Berlin (Math. Phys.) 778 (1915).
  • [2] L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, (Addison-Wesley Press, New York), (1962).
  • [3] C. Møller, On the localization of the energy of a physical system in the general theory of relativity, Ann. Phys., 4 (1958), 347.
  • [4] P. G. Bergmann, R. Thomson, Conservation laws in general relativity as the generators of coordinate transformations, Phys. Rev., 89 (1958), 400.
  • [5] S. Weinberg, Anthropic bound on the cosmological constant, Phys. Rev. Lett., 59 (1987), 2607.
  • [6] A. Papapetrou, Einstein’s theory of gravitation and flat space, Proc. R. Irish Acad., A52 (1948), 11.
  • [7] R. C. Tolman, Relativity, Thermodynamics and Cosmology (Oxford University Press, Oxford, 1934).
  • [8] C. W. Misner, D. H. Sharp, Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse, Phys. Rev. B, 136 (1964), 571.
  • [9] F. I. Cooperstock, R. S. Sarracino, The localisation of energy in general relativity, J. Phys. A: Math. Gen., 11 (1978), 877.
  • [10] K. S. Virbhadra, J. C. Parikh, Gravitational energy of a stringy charged black hole, Phys. Lett., B317 (1993), 312.
  • [11] K. S. Virbhadra, Naked singularities and Seifert?s conjecture, Phys. Rev., D60(1999), 104041
  • [12] K. P. Tod, Some examples of Penrose?s quasi-local mass construction, Proc. Roy. Soc., London A388(1983),1795.
  • [13] R. Penrose, Penrose?s quasi-local mass, Proc. Roy. Soc. London, A381 (1982), 53.
  • [14] J. W. Maluf, Hamiltonian formulation of the teleparallel description of general relativity, J. Math. Phys., 35 (1994), 335.
  • [15] M. Blagojevic, I. A. Nikolic, Hamiltonian structure of the teleparallel formulation of general relativity, Phys. Rev., D62 (2000), 024021.
  • [16] J. Schwinger, Quantized gravitational field, Phys. Rev., 130 (1963), 1253.
  • [17] F. I. Mikhail, M. I. Wanas, A. Hindawi, E. I. Lashin, Energy-momentum complex in Møller’s tetrad theory of gravitation, Int. J. Theor. Phys., 32 (1993), 1627.
  • [18] T. Vargas, The energy of the universe in teleparallel gravity Gen. Rel. Grav., 36 (2004), 1255.
  • [19] M. Salti, A. Havare, Energy-Momentum in viscous Kasner-type universe in Bergmann-Thomson formulations Int. J. Mod. Phys., A20 (2005), 2169.
  • [20] M. Salti, Different Approaches for Moller’s Energy in the Kasner-type Space-time, Mod. Phys. Lett., A20 (2005), 2175.
  • [21] M. Salti, Energy?Momentum In The Viscous Kasner-Type Universe In Teleparallel Gravity, Astrophy. Space Sci., 229 (2005), 159.
  • [22] O. Aydogdu, M. Salti, Energy of the universe in Bianchi-Type I models in Møller’s tetrad theory of gravity, Astrophys. Space Sci., 229 (2005), 227.
  • [23] O. Aydogdu, M. Salti, M. Korunur, Energy in Reboucas-Tiomno-Korotkii-Obukhov and G¨odel-type Space-times in Bergmann-Thomson’s Formulations, Acta Phys. Slov., 55 (2005), 537.
  • [24] M. Sharif, M. J. Amir, Teleparallel energy?momentum distribution of lewis-papapetrou spacetimes Mod. Phys. Lett., A22 (2007), 425 .
  • [25] M. J. Amir, S. Ali, Energy-Momentum Distribution of Non-Static Plane Symmetric Spacetimes in General Relativity and Teleparallel Theory, Chinese Joun. of Phys., 50 (2012), 14 .
  • [26] M. Sharif, M. J. Amir, Teleparallel versions of Friedmann and Lewis-Papapetrou spacetimes, Gen. Relat. Gravit., 38(2006), 1735.
  • [27] M. Sharif, M. J. Amir, Teleparallel killing vectors of the Einstein universe, Mod. Phys. Lett., A22 (2007), 425.
  • [28] M. Sharif, M. J. Amir, Teleparallel Version of the Stationary Axisymmetric Solutions and their Energy Contents, Gen. Relat. Gravit., 39 (2007), 989.
  • [29] M. Sharif, M. J. Amir, Teleparallel Version of the Levi-Civita Vacuum Solutions and their Energy Contents, Canadian J. Phys., 86 (2008), 1091 .
  • [30] M. Sharif, M. J. Amir, Teleparallel Energy-Momentum Distribution of Static Axially Symmetric Spacetimes, Mod. Phys. Lett., A23 (2008), 3167 .
  • [31] M. Sharif, M. J. Amir, Energy-Momentum of the Friedmann Models in General Relativity and Teleparallel Theory of Gravity, Canadian J. Phys., 86 (2008), 1297.
  • [32] M. Sharif, M. J. Amir, Teleparallel Energy-Momentum Distribution of Spatially Homogeneous Rotating Spacetimes, Int. J. Theor. Phys., 47 (2008), 1742.
  • [33] T. Multam¨aki, A. Putaja1, L. Vilja1 , E.C. Vagenas, . Energy-momentum complexes in f(R) theories of gravity, Class. Quantum Gravity, 25 (2008), 075017
  • [34] M. Sharif, M. F. Shamir, Energy distribution in f ( R) gravity, Gen. Relativ. Gravit., 42 (2010) 1557.
  • [35] V. Faraoni, S. Nadeau, The Stability of modified gravity models, Phys. Rev., D72 (2005), 124005 .
  • [36] M. J. Amir, S. Naheed, Spatially Homogeneous Rotating Solution in f(R) Gravity and Its Energy Contents, Int. J. Theor. Phys., 52 (2013), 1688.
  • [37] M. J. Amir, S. Sattar, Locally Rotationally Symmetric Vacuum Solutions in f(R) Gravity, Int. J. Theor. Phys. 53 (2013), 773.
  • [38] J. E. G. Silva, V. Santos, C. A. S. Almeida, Gravity localization in a string-cigar braneworld, Class. Quantum Grav., 30 (2013), 025005 .
  • [39] W. D. Linch, G. Tartaglino-Mazzucchelli, Six-dimensional supergravity and projective superfields, JHEP, 08 (2012), 075.
  • [40] N. Popov, Geometric Model of the Gravitational Field, Gravitation and Cosmology, 4 (1998), 151.
  • [41] C. W. Misner, K. S. Thorne , J. A. Wheeler, Gravitation, W. H. Freeman and Co., NY 1973.
  • [42] P. Freud, The energy-momentum problem and the theory of gravitation, Ann. of Math., 40 (1938), 417 .
There are 42 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sarfraz Ali 0000-0002-9299-2611

M. Jamil Amir This is me 0000-0002-9299-2611

Publication Date September 30, 2019
Submission Date April 12, 2019
Acceptance Date August 8, 2019
Published in Issue Year 2019

Cite

APA Ali, S., & Amir, M. J. (2019). Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field. Universal Journal of Mathematics and Applications, 2(3), 141-147. https://doi.org/10.32323/ujma.553017
AMA Ali S, Amir MJ. Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field. Univ. J. Math. Appl. September 2019;2(3):141-147. doi:10.32323/ujma.553017
Chicago Ali, Sarfraz, and M. Jamil Amir. “Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field”. Universal Journal of Mathematics and Applications 2, no. 3 (September 2019): 141-47. https://doi.org/10.32323/ujma.553017.
EndNote Ali S, Amir MJ (September 1, 2019) Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field. Universal Journal of Mathematics and Applications 2 3 141–147.
IEEE S. Ali and M. J. Amir, “Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field”, Univ. J. Math. Appl., vol. 2, no. 3, pp. 141–147, 2019, doi: 10.32323/ujma.553017.
ISNAD Ali, Sarfraz - Amir, M. Jamil. “Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field”. Universal Journal of Mathematics and Applications 2/3 (September 2019), 141-147. https://doi.org/10.32323/ujma.553017.
JAMA Ali S, Amir MJ. Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field. Univ. J. Math. Appl. 2019;2:141–147.
MLA Ali, Sarfraz and M. Jamil Amir. “Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field”. Universal Journal of Mathematics and Applications, vol. 2, no. 3, 2019, pp. 141-7, doi:10.32323/ujma.553017.
Vancouver Ali S, Amir MJ. Energy-Momentum Distribution of Six-Dimensional Geometric Model of Gravitational Field. Univ. J. Math. Appl. 2019;2(3):141-7.

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