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February  2014, 34(2): 335-366. doi: 10.3934/dcds.2014.34.335

Gravitational Field Equations and Theory of Dark Matter and Dark Energy

Tian Ma 1, and  Shouhong Wang 2,

1. 

Department of Mathematics, Sichuan University, Chengdu
2. 

Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  January 2013 Revised  May 2013 Published  August 2013

The main objective of this article is to derive new gravitational field equations and to establish a unified theory for dark energy and dark matter. The gravitational field equations with a scalar potential $\varphi$ function are derived using the Einstein-Hilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{\mu\nu}$, the scalar potential $\varphi$ and their interactions, unified by the new field equations. From quantum field theoretic point of view, the vector field $\Phi_\mu=D_\mu \varphi$, the gradient of the scalar function $\varphi$, is a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle. Both particles can be considered as gravitational force carriers, and as they are massless, the induced forces are long-range forces. The (nonlinear) interaction between these bosonic particle fields leads to a unified theory for dark energy and dark matter. Also, associated with the scalar potential $\varphi$ is the scalar potential energy density $\frac{c^4}{8\pi G} \Phi=\frac{c^4}{8\pi G} g^{\mu\nu}D_\mu D_\nu \varphi$, which represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M \Phi dM=0$. The sum of this potential energy density $\frac{c^4}{8\pi G} \Phi$ and the coupling energy between the energy-momentum tensor $T_{\mu\nu}$ and the scalar potential field $\varphi$ gives rise to a unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8\pi G}{c^4} T + \Phi$. Furthermore, the proposed field equations resolve a few difficulties encountered by the classical Einstein field equations.
Keywords: gravitational eld equations, spin-1 massless particle, spin-2 massless graviton, gravitational force formula., scalar potential energy, dark matter, Dark energy.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Tian Ma, Shouhong Wang. Gravitational Field Equations and Theory of Dark Matter and Dark Energy. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 335-366. doi: 10.3934/dcds.2014.34.335
References:
[1]

H. A. Atwater, "Introduction to General Relativity," International Series of Monographs in Natural Philosophy, Vol. 63, Pergamon Press, Oxford-New York-Toronto, Ont., 1974.

[2]

G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints, Physics Reports, 405 (2005), 279-390. doi: 10.1016/j.physrep.2004.08.031.

[3]

C. H. Brans and R. H. Dicke, Mach's principle and a relativistic theory of gravitation, Physical Review (2), 124 (1961), 925-935. doi: 10.1103/PhysRev.124.925.

[4]

H. A. Buchdahl, Non-linear Lagrangians and cosmological theory, Monthly Notices of the Royal Astronomical Society, 150 (1970), 1-8.

[5]

R. Caldwell, R. Dave and P. J. Steinhardt, Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett., 80 (1998), 1582-1585. doi: 10.1103/PhysRevLett.80.1582.

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R. Caldwell and E. V. Linder, The limits of quintessence, Phys. Rev. Lett., 95 (2005), 141301. doi: 10.1103/PhysRevLett.95.141301.

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S. Capozziello and M. De Laurentis, Extended theories of gravity, Phys. Rept., 509 (2011), 167-320. doi: 10.1016/j.physrep.2011.09.003.

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B. Chow, P. Lu and L. Ni, "Hamilton's Ricci Flow," Graduate Studies in Mathematics, Vol. 77, American Mathematical Society, Providence, RI, 2006.

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T. Damour and G. Esposito-Farse, Tensor-multi-scalar theories of gravitation, Class. Quantum Grav., 9 (1992), 2093-2176. doi: 10.1088/0264-9381/9/9/015.

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M. L. Kutner, "Astronomy: A Physical Perspective," Second edition, Cambridge University Press, 2003. doi: 10.1017/CBO9780511802195.

[14]

L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics, Vol. 2. The Classical Theory of Fields," Fourth edition, Pergamon Press, Oxford-New York-Toronto, Ont., 1975.

[15]

T. Ma, "Manifold Topology," (in Chinese) Science Press, Beijing, 2010.

[16]

_______, "Theory and Methods of Partial Differential Equations," (in Chinese) Science Press, Beijing, 2011.

[17]

T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, Vol. 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812701152.

[18]

_______, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics," Mathematical Surveys and Monographs, Vol. 119, American Mathematical Society, Providence, RI, 2005.

[19]

_______, "Phase Transition Dynamics," Springer-Verlag, October, 2013.

[20]

_______, Unified field equations coupling four forces and principle of interaction dynamics, arXiv:1210.0448, 2012.

[21]

_______, Unified field theory and principle of representation invariance, arXiv:1212.4893, 2012; part (the earliest version) of this preprint is to appear in Applied Mathematics and Optimization.

[22]

R. Massey, J. Rhodes, R. Ellis, N. Scoville, A. Leauthaud, et al., Dark matter maps reveal cosmic scaffolding, Nature, 445 (2007), 286-290.

[23]

P. Peebles and B. Ratra, The cosmological constant and dark energy, Rev. Mod. Phys., 75 (2003), 559-606. doi: 10.1103/RevModPhys.75.559.

[24]

S. Perlmutter, et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae, Astrophys. J., 517 (1999), 565-586.

[25]

Nikodem J. Popławski, Cosmology with torsion: An alternative to cosmic inflation, Phys. Lett. B, 694 (2010), 181-185. doi: 10.1016/j.physletb.2010.09.056.

[26]

B. Ratra and P. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D, 37 (1988), 3406-3427. doi: 10.1103/PhysRevD.37.3406.

[27]

A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J., 116 (1998), 1009-1038. doi: 10.1086/300499.

[28]

V. Rubin, W. K. Ford, Jr., Rotation of the Andromeda nebula from a spectroscopic survey of emission regions, Astrophysical Journal, 159 (1970), 379-404. doi: 10.1086/150317.

[29]

C. Wetterich, Cosmology and the fate of dilatation symmetry, Nucl. Phys. B, 302 (1988), 668-696. doi: 10.1016/0550-3213(88)90193-9.

[30]

C. M. Will, "Theory and Experiment in Gravitational Physics," Second edition, Cambridge University Press, Cambridge-New York, 1993.

[31]

I. Zlatev, L.-M. Wang and P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant, Phys. Rev. Lett., 82 (1999), 896-899. doi: 10.1103/PhysRevLett.82.896.

[32]

F. Zwicky, On the masses of nebulae and of clusters of nebulae, Astrophysical Journal, 86 (1937), 217-246.

show all references

References:
[1]

H. A. Atwater, "Introduction to General Relativity," International Series of Monographs in Natural Philosophy, Vol. 63, Pergamon Press, Oxford-New York-Toronto, Ont., 1974.

[2]

G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints, Physics Reports, 405 (2005), 279-390. doi: 10.1016/j.physrep.2004.08.031.

[3]

C. H. Brans and R. H. Dicke, Mach's principle and a relativistic theory of gravitation, Physical Review (2), 124 (1961), 925-935. doi: 10.1103/PhysRev.124.925.

[4]

H. A. Buchdahl, Non-linear Lagrangians and cosmological theory, Monthly Notices of the Royal Astronomical Society, 150 (1970), 1-8.

[5]

R. Caldwell, R. Dave and P. J. Steinhardt, Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett., 80 (1998), 1582-1585. doi: 10.1103/PhysRevLett.80.1582.

[6]

R. Caldwell and E. V. Linder, The limits of quintessence, Phys. Rev. Lett., 95 (2005), 141301. doi: 10.1103/PhysRevLett.95.141301.

[7]

S. Capozziello and M. De Laurentis, Extended theories of gravity, Phys. Rept., 509 (2011), 167-320. doi: 10.1016/j.physrep.2011.09.003.

[8]

Élie Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, C. R. Acad. Sci. (Paris), 174 (1922), 593-595.

[9]

B. Chow, P. Lu and L. Ni, "Hamilton's Ricci Flow," Graduate Studies in Mathematics, Vol. 77, American Mathematical Society, Providence, RI, 2006.

[10]

D. Clowe, et al., A direct empirical proof of the existence of dark matter, Astrophys. J., 648 (2006), L109-L113. doi: 10.1086/508162.

[11]

T. Damour and G. Esposito-Farse, Tensor-multi-scalar theories of gravitation, Class. Quantum Grav., 9 (1992), 2093-2176. doi: 10.1088/0264-9381/9/9/015.

[12]

Joshua A. Frieman, Michael S. Turner and Dragan Huterer, Dark energy and the accelerating universe, Annu. Rev. Astro. Astrophys., 46 (2008), 385-432.

[13]

M. L. Kutner, "Astronomy: A Physical Perspective," Second edition, Cambridge University Press, 2003. doi: 10.1017/CBO9780511802195.

[14]

L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics, Vol. 2. The Classical Theory of Fields," Fourth edition, Pergamon Press, Oxford-New York-Toronto, Ont., 1975.

[15]

T. Ma, "Manifold Topology," (in Chinese) Science Press, Beijing, 2010.

[16]

_______, "Theory and Methods of Partial Differential Equations," (in Chinese) Science Press, Beijing, 2011.

[17]

T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, Vol. 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812701152.

[18]

_______, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics," Mathematical Surveys and Monographs, Vol. 119, American Mathematical Society, Providence, RI, 2005.

[19]

_______, "Phase Transition Dynamics," Springer-Verlag, October, 2013.

[20]

_______, Unified field equations coupling four forces and principle of interaction dynamics, arXiv:1210.0448, 2012.

[21]

_______, Unified field theory and principle of representation invariance, arXiv:1212.4893, 2012; part (the earliest version) of this preprint is to appear in Applied Mathematics and Optimization.

[22]

R. Massey, J. Rhodes, R. Ellis, N. Scoville, A. Leauthaud, et al., Dark matter maps reveal cosmic scaffolding, Nature, 445 (2007), 286-290.

[23]

P. Peebles and B. Ratra, The cosmological constant and dark energy, Rev. Mod. Phys., 75 (2003), 559-606. doi: 10.1103/RevModPhys.75.559.

[24]

S. Perlmutter, et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae, Astrophys. J., 517 (1999), 565-586.

[25]

Nikodem J. Popławski, Cosmology with torsion: An alternative to cosmic inflation, Phys. Lett. B, 694 (2010), 181-185. doi: 10.1016/j.physletb.2010.09.056.

[26]

B. Ratra and P. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D, 37 (1988), 3406-3427. doi: 10.1103/PhysRevD.37.3406.

[27]

A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J., 116 (1998), 1009-1038. doi: 10.1086/300499.

[28]

V. Rubin, W. K. Ford, Jr., Rotation of the Andromeda nebula from a spectroscopic survey of emission regions, Astrophysical Journal, 159 (1970), 379-404. doi: 10.1086/150317.

[29]

C. Wetterich, Cosmology and the fate of dilatation symmetry, Nucl. Phys. B, 302 (1988), 668-696. doi: 10.1016/0550-3213(88)90193-9.

[30]

C. M. Will, "Theory and Experiment in Gravitational Physics," Second edition, Cambridge University Press, Cambridge-New York, 1993.

[31]

I. Zlatev, L.-M. Wang and P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant, Phys. Rev. Lett., 82 (1999), 896-899. doi: 10.1103/PhysRevLett.82.896.

[32]

F. Zwicky, On the masses of nebulae and of clusters of nebulae, Astrophysical Journal, 86 (1937), 217-246.

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