Modified general relativity

  • Gary NashEmail author
Research Article


A modified Einstein equation of general relativity is obtained by using the principle of least action, a decomposition of symmetric tensors on a time oriented Lorentzian manifold, and a fundamental postulate of general relativity. The decomposition introduces a new symmetric tensor \( \varPhi _{\alpha \beta } \) which describes the energy-momentum of the gravitational field itself. It completes Einstein’s equation and addresses the energy localization problem. The positive part of \( \Phi \), the trace of the new tensor with respect to the metric, describes dark energy. The cosmological constant must vanish and is dynamically replaced by \( \Phi \). A cyclic universe which developed after the Big Bang is described. The dark energy density provides a natural explanation of why the vacuum energy density is so small, and why it dominates the present epoch of the universe. The negative part of \( \Phi \) describes the attractive self-gravitating energy of the gravitational field. \(\varPhi _{\alpha \beta } \) introduces two additional terms into the Newtonian radial force equation: the force due to dark energy and the \(\frac{1}{r}\) “dark matter” force. When the dark energy force balances the Newtonian force, the flat rotation curves and the baryonic Tully–Fisher relation are obtained. The Newtonian rotation curves for galaxies with no flat orbital curves and those with rising rotation curves for large radii are described as examples of the flexibility of the orbital rotation curve equation.


General relativity Dark energy Dark matter 



I would like to thank the anonymous referee for his/her constructive comments.


  1. 1.
    Einstein, A.: Die Feldgleichungen der Gravitation. Königlich Preußische Akademie der Wissenschaften (Berlin), Sitzungsberichte, pp. 844–847 (1915)Google Scholar
  2. 2.
    Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 354(7), 769–822 (1916)ADSCrossRefGoogle Scholar
  3. 3.
    Einstein, A.: Note on E. Schrödinger’s Paper: the energy components of the gravitational field. Phys. Z 19, 115–116 (1918)Google Scholar
  4. 4.
    Xulu, S.S.: The Energy-Momentum Problem in General Relativity. arXiv:hep-th/0308070v1 (2003)
  5. 5.
    Baryshev, Y.: Foundation of Relativistic Astrophysics: Curvature of Riemannian Space versus Relativistic Quantum Field in Minkowski Space. arXiv:1702.02020v1 [physics.gen-ph] (2017)
  6. 6.
    Dupré, M.J.: The Fully Covariant Energy Momentum Stress Tensor for Gravity and the Einstein Equation in General Relativity. arXiv:0903.5225 [gr-qc] (2009)
  7. 7.
    Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differ. Geom. 3, 379–392 (1969)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lovelock, D.: The Einstein tensor and its generalizations. J. Math. Phys. 12, 498–501 (1971)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Misner, C., Thorne, C.K., Wheeler, J.: Gravitation, p. 487. Freeman, San Francisco (1973)Google Scholar
  10. 10.
    Ashtekar, A.: Loop Quantum Cosmology: An Overview. arXiv:0812.0177v1 [gr-qc] (2008)
  11. 11.
    Steinhardt, P.J., Turok, N.: Cosmic evolution in a cyclic universe. Phys. Rev. D 65, 126003 (2002)ADSCrossRefGoogle Scholar
  12. 12.
    Steinhardt, P.J., Turok, N.: The Cyclic Model Simplified, arXiv:astro-ph/0404480v1 (2004)
  13. 13.
    Baum, L., Frampton, P.: Turnaround in Cyclic Cosmology. arXiv:hep-th/0610213v2 (2006)
  14. 14.
    Penrose, R.: Before the Big Bang: an outrageous new perspective and its implications for particle physics. In: Proceedings of the EPAC 2006, Edinburgh, Scotland (2006)Google Scholar
  15. 15.
    Caldwell, R.R., Dave, R., Steinhardt, P.J.: Cosmological Imprint of an Energy Component with General Equation-of-State. arXiv:astro-ph/9708069 (1998)
  16. 16.
    Steinhardt, P.J.: A quintessential introduction to dark energy. Philos. Trans. Math. Phys. Eng. Sci. 361(1812), 2497–2513 (2003)ADSCrossRefGoogle Scholar
  17. 17.
    Cai, Y.-F., Saridakis, E.N., Setare, M.R., Xia, J.-Q.: Quintom Cosmology: theoretical implications and observations. arXiv:0909.2776v2 [hep-th] (2010)
  18. 18.
    Markus, L.: Line element fields and Lorentz structures on differentiable manifolds. Ann. Math. 62(3), 411–417 (1955)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time, vol. 39. Cambridge University Press, Cambridge (1973)CrossRefGoogle Scholar
  20. 20.
    Deser, S.: Covariant decomposition of symmetric tensors and the gravitational Cauchy problem. Ann. Inst. H. Poincaré 7, 149 (1967)zbMATHGoogle Scholar
  21. 21.
    York Jr., J.W.: Covariant decomposition of symmetric tensors in the theory of gravitation. Ann. Inst. H. Poincaré 20, 319 (1974)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ma, T., Wang, S.: Gravitational field equations and theory of dark matter and dark energy. Discrete Contin. Dyn. Syst. 34(2), 354 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Choquet-Bruhat, Y.: General Relativity and Einstein’s Equations, vol. 373. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  24. 24.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, Vol. 2, p. 250. Publish or Perish Inc., Houston (1999)Google Scholar
  25. 25.
    Weinberg, S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, pp. 390–395. Wiley, New York (1972). 409–415Google Scholar
  26. 26.
    Hubble, E.P.: A relation between distance and radial Velocity among extra-galactic nebulae. Proc. Nat. Acad. Sci. 15, 168–173 (1929)ADSCrossRefGoogle Scholar
  27. 27.
    Weinberg, S.: The cosmological constant problem. Rev. Mod. Phys. (1989).
  28. 28.
    Perlmutter, S., et al.: (The Supernova Cosmology Project), Measurements of Omega and Lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565–586 (1999)ADSCrossRefGoogle Scholar
  29. 29.
    Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astrophys. J. 116, 1009–1038 (1998)Google Scholar
  30. 30.
    Riess, A.G., et al.: Type Ia Supernova Discoveries at \(z>1\) from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution. arXiv:astro-ph/0402512v2 (2004)
  31. 31.
    Anderson, L. et al.: The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations in the Data Release 10 and 11 Galaxy Samples. arXiv:1312.4877v2 [astro-ph.CO] (2014)
  32. 32.
    Milgrom, M.: A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J. I 270, 365–370 (1983)ADSCrossRefGoogle Scholar
  33. 33.
    McGaugh, S.: A Tale of Two Paradigms: the Mutual Incommensurability of \(\Lambda CDM\) and MOND. arXiv:1404.7525v2 [astro-ph.CO] (2014)
  34. 34.
    Mannheim, P.D.: Alternatives to Dark Matter and Dark Energy. arXiv:astro-ph/0505266v2 (2005)
  35. 35.
    Moffat, J.W.: Modified Gravitational Theory as an Alternative to Dark Energy and Dark Matter. arXiv:astro-ph/0403266v5 (2004)
  36. 36.
    Milgrom, M.: Bimetric MOND gravity. Phys. Rev. D 80, 123536 (2009)ADSCrossRefGoogle Scholar
  37. 37.
    Bernard, L., Blanchet, L., Heisenberg, L.: Bimetric Gravity and Dark Matter. arXiv:abs/1507.02802v1 [gr-qc] (2015)
  38. 38.
    Verlinde, E.: Emergent Gravity and the Dark Universe. arXiv:1611.02269v2 [hep-th] (2016)
  39. 39.
    Campigotto, M.C., Diaferio, A., Fatibenec, L.: Conformal gravity: light deflection revisited and the galactic rotation curve failure. arXiv:1712.03969v1 [astro-ph.CO] (2017)
  40. 40.
    Kroupa, P.: The Dark Matter Crisis: Falsification of the Current Standard Model of Cosmology. arXiv:1204.2546v2 [astro-ph.CO] (2016)
  41. 41.
    Lelli, F., McGaugh, S.S., Schombert, J.M.: Testing Verlinde’s Emergent Gravity with the Radial Acceleration Relation. arXiv:1702.04355v1 [astro-ph.GA] (2017)
  42. 42.
    Lelli, F., McGaugh, S.S., Schombert, J.M., Pawlowski, M.S.: One Law to Rule Them All: The Radial Acceleration Relation of Galaxies. arXiv:1610.08981v2 [astro-ph.GA] (2017)
  43. 43.
    van Dokkum, P., et al.: A galaxy lacking dark matter. Nature 555, 629–632 (2018)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of AlbertaEdmontonCanada
  2. 2.EdmontonCanada

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