Abstract
The main objective of this article is to derive a new set of gravitational
field equations and to establish a new unified theory for dark energy and dark
matter. The new gravitational field equations with scalar potential $\varphi$
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
$g_ij$, the scalar potential $\varphi$ and their interactions, unified by the
new gravitational field equations. Associated with the scalar potential
$\varphi$ is the scalar potential energy density $c^48G
\Phi=c^48G g^ijD_iD_j \varphi$, 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: $ınt_M \Phi dM=0$. The sum of this new potential
energy density $c^48G \Phi$ and the coupling energy between the
energy-momentum tensor $T_ij$ and the scalar potential field $\varphi$ gives
rise to a new 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
$R=8Gc^4 T + \Phi$. Furthermore, the new field equations resolve a
few difficulties encountered by the classical Einstein field equations.
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