Gravitational Field Equations and Theory of Dark Matter and Dark Energy
Tian Ma  Department of Mathematics, Sichuan University, Chengdu, China (email) Abstract: 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 EinsteinHilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergencefree 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 spin1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin2 massless bosonic particle) and this spin1 massless bosonic particle. Both particles can be considered as gravitational force carriers, and as they are massless, the induced forces are longrange 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 nonuniform 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 energymomentum 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 spacetime 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: Dark energy, dark matter, gravitational eld equations, spin1 massless
particle, spin2 massless graviton, scalar potential energy, gravitational force formula.
Received: January 2013; Revised: May 2013; Available Online: August 2013. 
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