Unified Theory of Dark Energy and Dark Matter
Tian Ma,1 Shouhong Wang 2∗
1
2
∗
Department of Mathematics, Sichuan University, Chengdu, P. R. China
Department of Mathematics, Indiana University, Bloomington, IN 47405
To whom correspondence should be addressed; E-mail: showang@indiana.edu.
June 24, 2012
New field equations of gravitations with a scalar potential ϕ are discovered,
and are uniquely derived from the Einstein-Hilbert functional using the principle of Lagrangian dynamics subject to the divergence-free constraint of the
Riemannian metric of space-time. The scalar potential ϕ induces a scalar potential energy density Φ, unifying dark energy and dark matter. In fact, Φ
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 Φ
represents the dark matter, which produces attraction, and the positive part
represents the dark energy, which drives the acceleration of expanding galaxR
ies. In addition, Φ is conserved with mean zero: M ΦdM = 0.
This report is based on a new discovery on the theory of dark energy and dark matter, and on
field equations, developed by the authors [1].
1. Field equations with scalar potential. Motivated by the great mystery of the dark matter
and dark energy, a careful fundamental level of examination of the Einstein equations enable us
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to establish a new field equations of gravitation with scalar potential ϕ:
1
8πG
Rij − gij R = − 4 Tij − Di Dj ϕ,
2
c
(1)
where R is the scalar curvature, Rij is the Ricci curvature tensor, G is the gravitation constant,
c is the speed of light, Tij is the energy momentum tensor of matter, and gij is the Riemannian
metric of space time.
2. New scalar potential energy.
From the above new field equations, the conservations of
matter, energy and momentum are given by
div (Di Dj ϕ +
8πG
Tij ) = 0,
c4
(2)
and the momentum density T = g ij Tij and the scalar potential energy density Φ = g ij Di Dj ϕ
satisfy
8πG
R = 4 T + Φ,
c
Z
√
Φ −gdx = 0.
(3)
(4)
M
The scalar potential energy density Φ has a number of important physical properties:
(1) This scalar potential energy density Φ represents a new type of energy caused by the nonuniform distribution of matter in the universe. This scalar potential energy varies as the
galaxies move and matter of the universe redistributes. Like gravity, it affects every part
of the universe as a field.
(2) This scalar potential energy density Φ consists of both positive and negative energies. The
negative part of this quantity Φ represents the dark matter, which produces attraction, and
the positive part represents the dark energy, which drives the acceleration of expanding
galaxies.
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(3) The conservation law (4) amounts to saying that the the total scalar potential energy is
conserved.
(4) The universe with uniform distributed matter leads to identically zero scalar potential
energy density, and is unstable. It is this instability that leads to the existence of the dark
matter and dark energy, and consequently the high non-homogeneity of the universe.
(5) The curvature R is always balanced by this scalar potential energy Φ in the entire space
by (3), and the entire space is no longer flat. Namely the entire space is also curved
and is filled with dark energy and dark matter. In particular, the discontinuities of R induced by the discontinuities of the momentum energy density T , dictated by the Einstein
field equations, are no longer present thanks to the balance of the scalar potential energy
density Φ.
(6) This scalar potential energy density should be viewed as the main cause for the nonhomogeneous distribution of the matter/galaxies in the universe, as the dark matter (negative scalar potential energy) attracts and dark energy (positive scalar potential energy)
repels different galaxies.
3. Interaction force formula. Consider a central matter field with the total mass M and radius
r0 and spherical symmetry. With the new field equations, the force excerted on an object with
mass m is given by
1
1
F = mM G − 2 −
r
δ
δ dϕ Rr
2+
+
,
r dr
δ
R=Φ
for r > r0 .
(5)
where δ = 2GM/c2 , R is the scalar curvature, and ϕ is the scalar potential. The first term is the
classical Newton gravitation, the second term is the coupling interaction between matter and the
scalar potential ϕ, and the third term is the interaction generated by the scalar potential energy
density Φ (R = Φ for r > r0 ), as indicated in (3) . In this formula, the negative and positive
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values of each term represent respectively the attracting and repelling forces. It is then clear
that the combined effect of the second and third terms in the above formula represent the force
generated by dark matter, dark energy and their interaction with normal matter.
Also, importantly, this formula is a direct representation of the Einstein’s equivalence principle. Namely, the curvature of space-time induces interaction forces between matter.
In addition, one can derive a more detailed version of the above formula:
Z
1
Rr 1
δ
δ
2
2
−2
F = mM G − 2 + 2 +
εr +
2+
r
r Rdr ,
+
r
r
δ
δ
r
(6)
where ε > 0. The conservation law (4) of Φ suggests that R behaviors as r−2 for r sufficiently
large. Consequently the second term in the right hand side of (6) must dominate and be positive,
indicating the existence of dark energy.
To illustrate the main ideas, the above formula can be further simplified to derive the following approximate formula for r0 < r < r1 ≈ 1021 − 1022 km:
1
k0
+ k1 r ,
F = mM G − 2 −
r
r
(7)
k0 = 4 × 10−18 km−1 ,
(8)
k1 = 10−57 km−3 .
Again, in (7), the first term represents the Newton gravitation, the attracting second term stands
for dark matter and the repelling third term is the dark energy.
4. Derivation of the field equations.
It is clear that any modification of the Einstein field
equations should obey three basic principles: the principle of equivalence, the principle of
general relativity, and the principle of Lagrangian dynamics. The first two principles tell us
that the spatial and temporal world is a 4-dimentional Riemannian manifold (M, gij ), where
the metric {gij } represents gravitational potential, and the third principle determines that the
Riemannian metric {gij } is an extremum point of the Lagrangian action. There is no doubt that
the most natural Lagrangian action is the Einstein-Hilbert functional.
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The key point for our study is a well-known fact that the Riemannian metric gij is divergencefree. This suggests two important postulates for deriving a new set of gravitational field equations:
• The momentum tensor of matter need not to be divergence-free due to the presence of
dark energy and dark matter; and
• The field equations obey the Euler-Lagrange equation of the Einstein-Hilbert functional
under the natural divergence-free constraint:
F (gij + λXij ) − F (gij )]
= (δF (gij ), X) = 0
λ→0
λ
lim
∀X = {Xij } with div X = 0. (9)
Here δF (gij ) is the Euler-Lagrange variation of the Einstein-Hilbert functional. As the variational elements X are divergence-free, (9) do not imply δF (gij ) = 0, which is the classical
Einstein equations. In fact, (9) amounts to saying that δF (gij ) is orthogonal to all divergencefree tensor fields X, and consequently it is balanced by scalar potential term Di Dj ϕ for a scalar
potential function ϕ. The existence and uniqueness of ϕ are based on the orthogonal decomposition of tensor fields into gradient and divergence parts proved in [1]. Namely, the new field
equations (1) are then uniquely determined by the principle of Lagrangian dynamics applied to
the Einstein-Hilbert functional with divergence-free constraint.
5. Summary. In a nutshell, our theory indicates that dark energy and dark matter are respectively the positive and negative parts of the scalar potential energy caused by the non-uniform
distribution of the universe, which fills every part of the universe. The total scalar potential
energy density is conserved with mean zero. In other works, the unified energy for dark energy
and dark matter is conserved. The new field equations are the unique set of equations derivable
from the Einstein-Hilbert functional following the principle of Lagrangian dynamics, and can
be used to resolve a number of difficulties encountered by the classical Einstein equations.
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References and Notes
1. Tian Ma and Shouhong Wang, Gravitational Field Equations and Theory of Dark Matter
and Dark Energy, arXiv:1206.5078v1 [gr-qc], 2012.
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