7.6 Theory of Dark Matter and Dark energy
In summary, we have derived new law of gravity based solely onfirst principles, the PE, the
PGR, and the constraint variational principle (1.2.6):
Law of gravity (Ma and Wang,2014e)1) (Einstein’s PE). The space-time is a 4D Riemannian manifold{M,gμ ν},
with the metric{gμ ν}being the gravitational potential;
2) The Einstein PGR dictates the Einstein-Hilbert action (1.2.2);
3) The gravitational field equations (1.2.7) and (1.2.8) are derived using PID,
and determine gravitational potential{gμ ν}and its dual vector fieldΦμ=
∇μφ;
4) Gravity can display both attractive and repulsive effect, caused by the dual-
ity between the attracting gravitational field{gμ ν}and the repulsive dual
vector field{Φμ}, together with their nonlinear interactions governed by
the field equations (1.2.7) and (1.2.8).We remark that it is the duality and both attractive and repulsive behavior of gravity that
maintain the stability of the large scale structure of our Universe.
Modified Newtonian formula from first principles (Ma and Wang,2014e)
Consider a central matter field with total massMand with spherical symmetry. We derive
an approximate gravitational force formula:
(1.2.9) F=mMG
[
−
1
r^2−
k 0
r+k 1 r]
, k 0 = 4 × 10 −^18 km−^1 , k 1 = 10 −^57 km−^3.Here the first term represents the Newton gravitation, the attracting second term stands for
dark matter and the repelling third term is the dark energy. We note that the dark matter
property of the gravity and the approximate gravitational interaction formula are consistent
with the MOND theory proposed by (Milgrom, 1983 ); see also (Milgrom, 2014 ) and the
references therein. In particular, our modified new formulais derived from first principles.
Dark matter and dark energy: a property of gravity (Ma and Wang,2014e)
We have shown that it is the duality between theattractinggravitational field{gμ ν}and
therepulsivedual field{Φμ=∇μφ}in (1.2.7), and their nonlinear interaction that gives rise
to gravity, and in particular the gravitational effect of dark energy and dark matter.
Also, we show in (Hernandez, Ma and Wang, 2015 ) that consider the gravitational field
outside of a ball of centrally symmetric matter field. There exist precisely two physical
parameters dictating the two-dimensional stable manifoldof asymptotically flat space-time
geometry, such that, as the distance to the center of the ballof 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.
Of course, one can consider dark matter and dark energy as theenergy carried by the
gravitons and the dual gravitons, to addressed in the unifiedfield theory later in this book.