Gravitational Field Equations and Theory of Dark Matter and
Dark Energy
Tian Ma & Shouhong Wang (Supported by NSF, ONR and Chinese NSF)
http://www.indiana.edu/˜fluid
arXiv: 1206.5078, 1210.0448, & 1212.4893
I. Unified Theory of Dark Energy and Dark Matter
II. Principle of Representation Invariance
III. Unified Field Theory of Four Interactions
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I. Unified Theory of Dark Energy and Dark Matter
Principle of Interaction Dynamics (PID) (Ma-Wang, 2012): Least action with
energy-momentum conservation constraints.
With PID, we derive the following gravitational field equations:
Z √
8πG
1
8πG
−gdx
δLEH = Rij − gij R + 4 Tij
LEH =
R+ 4 S
c
2
c
M
(δLEH (gij ), X) = 0 ∀ DiXij = 0 =⇒
1
8πG
Rij − gij R = − 4 Tij − DiDj ϕ
2
c
8πG
Di
Tij + DiDj ϕ = 0
c4
Note: The new term DiDj ϕ can not be derived from any existing f (R) theories
and from any scalar field theories.
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New gravitational vector bosonic field:
• The new vector particle field Φµ = Dµϕ is massless with spin s = 1.
• This particle field corresponds to the scalar potential field ϕ in (??), caused by
the non-uniform distribution of matter in the universe.
• The nonlinear interaction between this particle field Φµ and the graviton leads
to a unified theory of dark matter and dark energy and explains the acceleration
of expanding universe.
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Consider a spherically symmetric central matter field with mass M and radius r0.
With the new field equations, we derive that the force exerted on an object with
mass m is given by
2
2
c
2GM 1 dϕ
c
1
(1) F = mM G − 2 −
2+ 2
+
Φr ,
for r > r0.
r
2GM
c r dr 2GM
The first term is the classical Newton gravitation, the 2nd and 3rd terms are the
coupling interaction between matter and the scalar potential ϕ.
This formula can be further simplified to derive the following approximate formula
for r0 < r < r1 ≈ 1021 − 1022km:
1
k0
(2)
F = mM G − 2 −
+ k1 r ,
r
r
where k0 = 4 × 10−18km−1 and k1 = 10−57km−3, which are estimated using
rotation curves of galactic motion.
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II. Principle of Representation Invariance (PRI)
PRI (Ma-Wang 2012): Physical laws for an SU (N ) gauge theory should be
covariant under different representations of SU (N ). Namely, for
(3)
Ψ̃(x) = U (x)Ψ(x),
U (x) = e
iθ a (x)τa
∈ SU (N )
∀x ∈ M,
the Lagrangian and gauge equations must be representation {τa} invariant.
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1 c
s
d
,
G
=
λ
Unified field model based on PID & PRI: Gw
λ
ab =
ab
8 ad cb
(4)
(5)
(6)
(7)
(8)
1 c d
12 fad fcb
1
eαE
8πG
gw αaw a gsαks k
Rµν − gµν R + 4 Tµν = ∇µ −
Aµ −
Wµ −
S Φν ,
2
c
~c
~c
~c µ
w
s
E
gw αa a gsαk k E
eα
Aµ −
Wµ −
Sµ φ ,
∂ ν Fνµ − eψ̄γµψ = ∇µ −
~c
~c
~c
h
i
g
w
ν
b
c
Gw
λbcdg αβ Wαµ
Wβd − gw L̄γµσaL
ab ∂ Wνµ −
~c
eαE
gw αbw b gsαks k w
1 mH c 2
= ∇µ +
xµ −
Aµ −
Wµ −
Sµ φa ,
4
~
~c
~c
~c
h
gs j αβ c d i
s
ν j
Gkj ∂ Sνµ − Λcdg SαµSβ − gsq̄γµτk q
~c
s
E
w
2
1 mπ c
eα
gw αa a gsαj j s
= ∇µ +
xµ −
Aµ −
Wµ −
S φ ,
4
~
~c
~c
~c µ k
(iγ µD̃µ − m̃)Ψ = 0.
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Conclusions and Predictions of the Unified Field Model
1. Duality: The unified field model induces a natural duality:
(9)
{gµν } (massless graviton)
←→ Φµ,
Aµ
(photon)
←→ φE ,
Wµa
(massive bosons W ± & Z) ←→ φw
a
Sµk
(massless gluons)
←→ φsk
for a = 1, 2, 3,
for k = 1, · · · , 8.
2. Decoupling and Unification:
• An important characteristics is that the unified model can be easily decoupled.
• Both PID and PRI can be applied directly to individual interactions. For gravity
alone, we have derived modified Einstein equations, leading to a unified theory
for dark matter and dark energy.
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3. The two SU (2) and SU (3) constant vectors {αaw } and {αks }, containing 11
parameters, represent the portions distributed to the gauge potentials by the weak
and strong charges.
4. Origin of mass: We obtained a much simpler mechanism for mass generation
and energy creation, completely different from the classical Higgs mechanism. This
new mechanism offers new insights on the origin of mass.
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5. Strong and weak interaction potentials: For the first time, we derive three
levels of strong interaction potentials: the quark potential Sq , the nucleon potential
Sn and the atom/molecule potential Sa:
1 Bk02 −k0r
Sq = gs
−
e
ϕ(r) ,
r
ρ0
3 2
ρ0
1 Bnk1 −k1r
Sn = 3
gs
e
ϕ(r) ,
−
ρ1
r
ρ1
3 3 2
ρ0
ρ1
1 Bnk1 −k1r
gs
e
ϕ(r) ,
Sa = 3N
−
ρ1
ρ2
r
ρ2
(10)
(11)
(12)
where ϕ(r) ∼ r/2, B, Bn are constants, k0 = mc/~, k1 = mπ c/~, m is mass of
the strong interaction Higgs particle, mπ is the mass of the Yukawa meson, ρ0
is the effective quark radius, ρ1 is the radius of a nucleon, ρ2 is the radius of an
atom/molecule, and N is the number of nucleons in an atom/molecule.
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Φ
r̄
r
10
These potentials match very well with experimental data, and offer an explanation
of e.g. quark confinement, asymptotic freedom, and the short-range nature of
strong and weak interactions.
Summary: Theoretical and experimental studies on models based only on
first principles, such as PID and PRI, are crucial for particle physics in the
next decade, and for understanding the deepest secrets of Nature.
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