# American Institute of Mathematical Sciences

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

## Gravitational Field Equations and Theory of Dark Matter and Dark Energy

 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.
Citation: Tian Ma, Shouhong Wang. Gravitational Field Equations and Theory of Dark Matter and Dark Energy. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 335-366. doi: 10.3934/dcds.2014.34.335
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show all references

##### References:
 [1] H. A. Atwater, "Introduction to General Relativity,", International Series of Monographs in Natural Philosophy, (1974). Google Scholar [2] G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints,, Physics Reports, 405 (2005), 279. doi: 10.1016/j.physrep.2004.08.031. Google Scholar [3] C. H. Brans and R. H. Dicke, Mach's principle and a relativistic theory of gravitation,, Physical Review (2), 124 (1961), 925. doi: 10.1103/PhysRev.124.925. Google Scholar [4] H. A. Buchdahl, Non-linear Lagrangians and cosmological theory,, Monthly Notices of the Royal Astronomical Society, 150 (1970), 1. Google Scholar [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. doi: 10.1103/PhysRevLett.80.1582. Google Scholar [6] R. Caldwell and E. V. Linder, The limits of quintessence,, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.141301. Google Scholar [7] S. Capozziello and M. De Laurentis, Extended theories of gravity,, Phys. Rept., 509 (2011), 167. doi: 10.1016/j.physrep.2011.09.003. Google Scholar [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. Google Scholar [9] B. Chow, P. Lu and L. Ni, "Hamilton's Ricci Flow,", Graduate Studies in Mathematics, (2006). Google Scholar [10] D. Clowe, et al., A direct empirical proof of the existence of dark matter,, Astrophys. J., 648 (2006). doi: 10.1086/508162. Google Scholar [11] T. Damour and G. Esposito-Farse, Tensor-multi-scalar theories of gravitation,, Class. Quantum Grav., 9 (1992), 2093. doi: 10.1088/0264-9381/9/9/015. Google Scholar [12] Joshua A. Frieman, Michael S. Turner and Dragan Huterer, Dark energy and the accelerating universe,, Annu. Rev. Astro. Astrophys., 46 (2008), 385. Google Scholar [13] M. L. Kutner, "Astronomy: A Physical Perspective,", Second edition, (2003). doi: 10.1017/CBO9780511802195. Google Scholar [14] L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics, Vol. 2. The Classical Theory of Fields,", Fourth edition, (1975). Google Scholar [15] T. Ma, "Manifold Topology,", (in Chinese) Science Press, (2010). Google Scholar [16] _______, "Theory and Methods of Partial Differential Equations,", (in Chinese) Science Press, (2011). Google Scholar [17] T. Ma and S. Wang, "Bifurcation Theory and Applications,", World Scientific Series on Nonlinear Science, (2005). doi: 10.1142/9789812701152. Google Scholar [18] _______, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics,", Mathematical Surveys and Monographs, (2005). Google Scholar [19] _______, "Phase Transition Dynamics,", Springer-Verlag, (2013). Google Scholar [20] _______, Unified field equations coupling four forces and principle of interaction dynamics,, , (2012). Google Scholar [21] _______, Unified field theory and principle of representation invariance,, , (2012). Google Scholar [22] R. Massey, J. Rhodes, R. Ellis, N. Scoville, A. Leauthaud, et al., Dark matter maps reveal cosmic scaffolding,, Nature, 445 (2007), 286. Google Scholar [23] P. Peebles and B. Ratra, The cosmological constant and dark energy,, Rev. Mod. Phys., 75 (2003), 559. doi: 10.1103/RevModPhys.75.559. Google Scholar [24] S. Perlmutter, et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae,, Astrophys. J., 517 (1999), 565. Google Scholar [25] Nikodem J. Popławski, Cosmology with torsion: An alternative to cosmic inflation,, Phys. Lett. B, 694 (2010), 181. doi: 10.1016/j.physletb.2010.09.056. Google Scholar [26] B. Ratra and P. Peebles, Cosmological consequences of a rolling homogeneous scalar field,, Phys. Rev. D, 37 (1988), 3406. doi: 10.1103/PhysRevD.37.3406. Google Scholar [27] A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant,, Astron. J., 116 (1998), 1009. doi: 10.1086/300499. Google Scholar [28] V. Rubin, W. K. Ford, Jr., Rotation of the Andromeda nebula from a spectroscopic survey of emission regions,, Astrophysical Journal, 159 (1970), 379. doi: 10.1086/150317. Google Scholar [29] C. Wetterich, Cosmology and the fate of dilatation symmetry,, Nucl. Phys. B, 302 (1988), 668. doi: 10.1016/0550-3213(88)90193-9. Google Scholar [30] C. M. Will, "Theory and Experiment in Gravitational Physics,", Second edition, (1993). Google Scholar [31] I. Zlatev, L.-M. Wang and P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant,, Phys. Rev. Lett., 82 (1999), 896. doi: 10.1103/PhysRevLett.82.896. Google Scholar [32] F. Zwicky, On the masses of nebulae and of clusters of nebulae,, Astrophysical Journal, 86 (1937), 217. Google Scholar
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