(untitled) (Journal of Mathematical Study, 2015)
In (T. Ma and S. Wang. Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34(2): 335-366, 2014; arXiv:1206.5078v2), a new set of gravitational field equations are derived based only on 1) the Einstein principle of general relativity, and 2) the principle of interaction dynamics, due to the the presence of dark energy and dark matter. With the field equations, we show that gravity can display both attractive and repulsive behavior, and the dark matter and dark energy are just a property of gravity caused by the nonlinear interactions of the gravitational potential g_μv and its dual field. The main objectives of this paper are two-fold. The first is to study the PID-induced cosmological model, and to show explicitly, as addressed in (T. Ma and S. Wang, Astrophysical dynamics and cosmology, Journal of Mathematical Study, 47(4): 305-378, 2014), that 1) dark matter is due to the curvature of space, and 2) dark energy corresponds to the negative pressure generated by the dual gravitational potential in the field equations, and maintains the stability of geometry and large scale structure of the Universe. Second, for the gravitational field outside of a ball of centrally symmetric matter field, there exist precisely two physical parameters dictating the two-dimensional stable manifold of asymptotically flat space-time geometry, such that, as the distance to the center of the ball of the matter field increases, gravity behaves as Newtonian gravity, then additional attraction due to the curvature of space (dark matter effect), and repulsive (dark energy effect). This also clearly demonstrates that both dark matter and dark energy are just a property of gravity.