Dark Matter: The Problem of Motion

Abstract

Equations of non-geodesic and non-geodesic deviations for different particles are obtained, using a specific type of classes of the Bazanski Lagrangian. Such type of paths has been found to describe the problem of variable mass in the presence of Riemannian geometry. This may give rise to detect the effect of dark matter which reveals the mystery of motion of celestial objects that are not responding neither to Newtonian nor Einsteinian gravity. An important link between non-geodesic equations and the dipolar particle or fluids has been introduced to apply the concept of geometization of physics. This concept has been already extended to represent the hydrodynamic equations in a geometric way. Such an approach, demands to seek for an appropriate theory of gravity able to describe different regions, eligible for detecting dark matter. Using different versions of bi-metric theory of gravity, to examine their associate non-geodesic paths. Due to implementing the geometrization concept, the stability problem of non-geodesic equations are essential to be studied for detecting the behavior of those objects in the presence of dark matter.

Full text

arXiv:1801.08396v1 [gr-qc] 25 Jan 2018
Dark Matter: The Problem of Motion
January 26, 2018
Magd E. Kahil 1 2 3
Abstract
Equations of non-geodesic and non-geodesic deviations for different particles are
obtained, using a specific type of classes of the Bazanski Lagrangian. Such type of
paths has been found to describe the problem of variable mass in the presence of
Riemannian geometry. This may give rise to detect the effect of dark matter which
reveals the mystery of motion of celestial objects that are not responding neither to
Newtonian nor Einsteinian gravity. An important link between non-geodesic equa-
tions and the dipolar particle or fluids has been introduced to apply the concept of
geometization of physics. This concept has been already extended to represent the
hydrodynamic equations in a geometric way. Such an approach, demands to seek
for an appropriate theory of gravity able to describe different regions, eligible for
detecting dark matter. Using different versions of bi-metric theory of gravity, to ex-
amine their associate non-geodesic paths. Due to implementing the geometrization
concept, the stability problem of non-geodesic equations are essential to be studied
for detecting the behavior of those objects in the presence of dark matter .
1 Introduction
The quest of flat rotational curves for spiral galaxies cannot be explained neither classical
nor general relativistic gravity, such a violation can be regarded to the existence of dark
matter. In our Galaxy, several meticulous observations have confirmed that, the rotational
velocities are ranging between 200 ∼300km/s, based on considering it as a function of the
distance rfrom the SgrA*. Taking into account that those clouds are moving in circular
orbits with velocity vg(r), leading to the relationship between the mass profile M(r) and
rotational velocities in the following way [1],
M(r) = rv2
g(r)/G, (1)
1Modern Sciences and Arts University , Giza , Egypt
2Nile University,Giza, Egypt
e.mail: mkahil@msa.eun.eg
magdelias.kahil@gmail.com
3Egyptian Relativity Group. Cairo, Egypt
1

Consequently, the problem now is clarified on studying this stringent motion and
revealing its violation in different characteristics which gives some elusiveness to obtain a
unique explanation for this type of discrepancy between theory and observation.
Accordingly, one may find out that dark matter (DM) can be detected as a mass excess
quantity, expressed in non-geodesic equation due to the projection of the fifth component
of the geodesic of non-compact higher dimensional theory of gravity on its 4-dimensions
[1], or considering it as a result of the motion of dipolar particles rather than test particles
on these clouds. In more detail, dark matter can be presented, by a set of equations of
linear momentum and internal momentum :the evolution equation. In this approach, we
suggest a corresponding Lagrangian, similar to the spinning Lagrangian 4, having the
flavor of the Bazanski-like Lagrangian [3],[4]. Following this trend, we can utilize this
Lagrangian for examining the stability of a system affected by dark matter, by solving
the deviation equations of the path and evolution respectively.
Nevertheless, the notation of DM has not fixed only in the region of halos, but there are
some traces that may confirm its existence at different regions in the universe or inside the
core of the Galaxy. It is attributed to determine the precise value of the cosmic microwave
spectrum, providing a reliable scenario of the abundances of elementary particles of large
scale structures from Big Bang nucleosynthesis [5]. From this perspective, it is really
essential to find a consistent theory of gravity able to reveal these variant regions, having
some features of strong field theory of gravitation.
Thus, we focus on different versions of bi-metric theories of gravity. These theories
have the ability to express dark matter as a twin matter appeared in the Halo , by means
of a ghost free bi-gravity theory or a mass-excess term appeared in a fluid circumventing
the core of the galaxy, as in case of SgrA*, using a specific bi-metric theory having geodesic
mapping to be acting as gauge transformation [6].
Thus, it is well known that DM is a matter of elusiveness. Some authors consider it as
weak interacting massive particles, others may view it not only a stream of perfect fluids,
but also dipolar fluids [7] to be matched with descriptions of MOND - weak gravitational
fields and BIMOND- strong gravitational fields; which can be detected on the halos and
nearby the core of the galaxy within the accretion disk [8].
Yet, there is a rival approach, should be considered , rejecting the existence of dark
matter and dark energy [DE] revealed as expressed in terms of MOND [9]and BIMOND
paradigms [10]. This type of illustration led Blanchet et al to modify the appearance of
dark matter in their model to behave from a particle content into a fluid-like behavior.[2]
From this perspective, it is vital in our study to derive the candidate equations of mo-
tion showing the mass excess term is due to the existence of dark matter. Nevertheless,
a vital question should be addressed: What is dark matter?
In our present work, we present three rival explanations for its contents:
Firstly: The projection of a higher dimension spatial dimension on the 4-dim manifold?
[1]
Secondly: Motion of dipolar particles /fluids as claimed in spiral galaxies? [2]
Thirdly: The existence of a scalar field associated with the Galaxy’s gravitational field?
[10]
4See Apendix A
2

Moreover, there are several remedies to modify Newtonian and/or Einsteinan Gravity
by changing their gravitational potentials to become like φ=−GM[1+αexp (−r/r0)]/(1+
α) , with α=−0.9 and r0≈30kpc , ini order to explain the behavior of the flat rotational
curves for spiral galaxies.[1]
These varieties of explanations may let us argue about its dominance, Recently,
some authors have indicated that DM occupies 26% of invisible matter of the Universe,
while 73% of Universe’s composition is made of DE due to expansion with acceleration
[11][Iroiro]. Yet, the link between DM and DE is becoming one of vital problems, it has
been found that MOND paradigm has a link between DM and DE which in terms of
ΛCDM . Such an argument may lead to a crucial speculation that the existence of DM is
not localized in the halos of the galaxy, but it could be visualized differently in accretion
disk or through very minute discrepancies of perihelion motion for some planets [12].
In our work we are not localizing dark matter to be detected only on the halos of
spiral galaxies , but also in the core of the galaxy. Accordingly, the need to search for
an eligible theory of gravity able to describe all these ones. One of the candidates is
examining the motion of bi-gravity type theories. These theories have adopted two times
of matter: ordinary matter and twin matter. The interaction between twin matter and
ordinary matter may give rise to the significance of the existence of dark matter.
Consequently, the problem of realizing the notation of dark matter remains unsolved
unless, one presents its associate gravitation field theory. In our work, we are going to
apply different versions bi-metric theories [13] as candidates to examine its existence, the
first is the Hassan-Rosen [14] version for galactic region; while, the second, a special case of
conformal type, is presented by Verzub’s description of bi-metric [6], which is successfully
related to strong gravitational field as in the core of the galaxy is based on the strength
of the gravitational field, nearby the core can be expressed as mass excess appeared in
hydrostatic equation of the flow of fluid on the accretion [15].
Accordingly, in the present work , we are going to deal with expressing, the behavior
of dark matter in terms of non-geodesic equations, these equations are derived using the
Lagrangian formalism using the Bazanski-like Lagrangian. This type of equation may
give rise to geometrize all trajectories associated with the appearance of dark matter. In
other words, the appropriate path equations as described in the Riemannian geometry to
represent dipolar particles or fluids of the halos; and the corresponding path equations
that represents the hydrostatic stream of fluids in of accretion disk.due to solving the
non-geodesic deviation equation, we can give rise to examine stability conditions, which
means an indication of remaining DM effect on each observed regions. Thus, we are able to
examine the equivalence of non-geodesic trajectories with each of the following equations,
dipolar moment and dipolar fluids and hydrostatic stream of motion as described in
General Relativity is explained in section 2 . We extend the previous equations to be
expressed in different versions of bi-metric theories of gravity as shown in sec 3. Finally,
it turns out that, the problem of detecting the existence of DM is expressed in terms
studying the behavior of the stream of fluids in different gravitational fields.
This may raise the necessity to examine the stability of these systems for being affected
by dark matter. This can be seen, by solving the different corresponding deviation equa-
tion for examining the stability condition, using an independent method of coordinate
3

transformation [16] which is be described in sec 4.
2 Dark Matter: Equations of Motion in GR
2.1 Dark Matter: Non-Geodesic Equations
Paths that follow non-geodesic trajectories, and their corresponding deviation equations
are obtained using the Euler-Lagrange equation of the following Lagrangian:
Ldef.
=m(s)gµν UµDΨν
Ds +m(s),ρΨρ(2)
such that: d∂L
ds∂ ˙
Ψα−∂L
∂Ψα= 0 (3)
One obtains,
dUα
ds + Γα
βδ UβUδ=m(s),β
m(s)(gαβ −UαUβ) (4)
and using the commutation relation (A.4) and the condition (A.5) we obtain its corre-
sponding deviation equation;
D2Ψµ
Ds2=Rµ
νρσ UνUρΨσ+ ( m(s),β
m(s)(gαβ −UαUβ));ρΨρ(5)
Assuming that the effective mass m(s)∼exp(g(ψ)ψ) , ,which may contribute to describe
the behavior of the parallel force , f|| =∇[g(ψ)ψ] as shown on the right hand side of
equation.
It is well known that this force is responsible for mass variation of paths in the presence
of the Riemaniann geometry. Yet, it has been found [17] by taking σas another parameter
, such that s∼σto obtain the following relation
1
m
dm
dσ ≡qΛ/2
6(6)
in which to be expressed as, 1
m
dm
dσ ≈2a0/c (7)
where a0is a constant of acceleration, a0∼2×10−10m/sec2, as known of the MOND
and c is the speed of light. Thus, we can find that the non-geodesic equation can be
related to MOND [9] in the following way:
dˆ
Uα
dσ + Γα
βδ ˆ
Uβˆ
Uδ= 2a0
cˆ
Uβ(gαβ −ˆ
Uαˆ
Uβ) (8)
where, ˆ
Uα=dxα
dσ and its corresponding deviation equation becomes
D2ˆ
Ψµ
Dσ2=Rµ
νρσ ˆ
Uνˆ
Uρˆ
Ψσ+ 2a0
c(ˆ
Uβ(gαβ −ˆ
Uαˆ
Uβ));ρˆ
Ψρ(9)
such that ˆ
Ψαis its associated deviation vector.
4

2.2 Dark Matter: An Extra-dimensional Effect
The non-geodesic equations are expressed as , the four components of a geodesic equations
for a test particle [1] in a non-compact space-time gAB,56= 0 following Wesson’s approach
of space-time-matter[17]. Thus, the characteristics of dark matter can be appeared within
solving the geodesic equation in 5-dim., provided that
dS
ds =q(1 + ǫˆ
Φ2(U5)2)
such that ˆ
Φ is a scalar function, and ǫ=±1.
Thus, it can possible to suggest the following Lagrangian:
L=gABUADΨA
DS (10)
By taking the variation with respect to ΨCand UCrespectively, one can find
(i) Equation of Geodesic:
DU C
DS = 0,(11)
(2) Equation of Geodesic Deviation:
D2ΨC
DS2=RC
BDE UBUDΨE(12)
With taking into account that the force appeared on its right hand side is expressed within
the component of the fifth dimension of a 5-dim manifold. Accordingly, equation (4) may
be expressed as
d2xµ
dS + Γµ
AB
xA
dS
xB
dS = 0
while,the fifth component of (11) plays as a vital to affect the behavior of dark matter
particles as present in the other components .
d2x5
dS + Γ5
AB
xA
dS
xB
dS = 0
2.3 Dark Matter: Equations of Motion Dipolar Moment Par-
ticles in The Halo
The rotational curves of the galaxy can be expressed by the presence of dipolar dark
Matter particles [18]. From this perspective, it has been found that these particles are
described of two equations, one may describe the (passive) linear momentum and the
other describes the microscopic (active) momentum, in terms of the evolution equation.
The Lagrangian formalism for motion of the dipole moment in a gravitational field is
analogous to the its counterpart the motion of spinning with precession [19]5.
5see Appendix A
5

Thus, we suggest the following Lagrangian:
Ldef.
=gαβPαDΨβ
(1)
Ds + Ωα
DΨβ
(2)
Ds +fαΨα
(1) +ˆ
fαΨα
(2) (13)
such that Ψµ
(1) is the non-geodesic deviation from the world line and Ψµ
(2) is the evolution
deviation due to dipole moment, with taking into account that the projector tensor is
responsible for raising and lowering indices for the Dipole moment vector i.e.
Ωµ=hµν Ων
By taking the variation with respect to Φµ
1and Φµ
2separately we obtain the following
set of equation of motion and evolution respectively:
DP µ
Ds =fµ(14)
and DΩµ
Ds =ˆ
fµ(15)
such that
fµ= 2mF µ
and ˆ
fµ=Rµ
νρσ ΠσUρUν
in which
Πµ=hµν πν
Using (A.4) and (A.5) as in the previous section, we obtain their corresponding
geodesic deviation equations
D2Ψµ
(1)
DS2=Rµ
νρσ PνUρΨσ
(1) +fµ
;ρΨρ
(1),(16)
And,
D2Ψµ
(2)
DS2=Rµ
νρσ ΠνUρΨσ
(2) +ˆ
fµ
;ρΨρ
(2) (17)
Equations (16), (17) are essentially vital to solve the problem of stability for different
celestial objects in various gravitational fields.
2.4 Equations of Motion of Dipolar Fluid in The Halos
Recently, it has been the involvement of cosmological constant is vital to identify the
mystery of dark matter. This led Blanchet et al to revisit the description of of dipolar
dark matter from particle contents into fluid-like description [2] , this can be found by
imposing the effect of polarization vector as a candidate to examine the interaction of
dark energy on the system .
6

From this perspective, Blanchet and Le Tiec [8] have postulated that the dynamics
of the dipolar fluid in a prescribed gravitational field gµν is derived from an action of the
type found
S=Zd4x√−gL[Jµ, ξµ˙
ξ, gµν ] (18)
Provided that the density current Jµand the polarization vector Πµare new quantities
added in dipolar fluids: such that: Jµ=ρUµ, and Πµ=ρξµApplying the least action
principle on (18) to obtain their corresponding set of path equations
DJµ
Ds = 0 (19)
and DΩ
Ds =1
σ∇µ(W−ˆ
Πˆ
W)−Rµ
ρνλ uρξνKλ
where, The above set of equations can be obtained using its associated Bazanski-Like
Lagrangian, if we take the variation with respect to Ψµ
(1) and Ψµ
(2) simultaneously
L=gµν JµDΨν
(1)
Ds + Ωµ
DΨν
(2)
Ds +¯
fµΨµ
2.(20)
Also, using the commutation rule (A.4) and the condition (A.5) we obtain their cor-
responding path deviation and evolution deviation equations respectively
D2Ψµ
(1)
Ds2=Rµ
νρσ JνUρΨσ
1,(21)
and D2Ψµ
(2)
Ds2=Rµ
νρσ ΩνUρΨσ
2+¯
fµ
;ρΨρ
2.(22)
2.5 Equations of Motion of Fluids in The Accretion Disk
The role of non-geodesic equations are acting to describe geometrically the behavior of
dark matter particles in the accretion disk, as existed as a collision-less fluid. In this
section, we are going to focus on its contribution to mass of the accretion desk and
consequently, the accretion process is less efficient than that expected from dissipative fluid
; dark matter gives a significant contribution to the mass of the accretion desk producing
an important inflow as in our Galaxy, e.g. a mass growth scaling as Mbh =const.t9/16 [8].
Thus, we can find out that the equivalence between non-geodesic motions and hydro-
dynamics flows appears in following two sets of equations
dUα
ds + Γα
βδ UβUδ=fα(23)
wherefαis described as non-gravitational force, in which its vanishing turns the equation
into a geodesic, which becomes
dUα
ds + Γα
βδ UβUδ=1
E+Phαβ P,β (24)
7

where hαβ is the projection tensor defined as:
hµν =gµν −UµUν,(25)
If equation (23) satisfies the first law of thermodynamics
P,β =ρc2(E+P
ρc2),β (26)
then it becomes,
dUα
ds + Γα
βδ UβUδ=(E+P
ρc2),β
(E+P
ρc2)hαβ (27)
Thus, we find out that from ...
1
E+P/ρ
dE +P/ρ
dσ ≈2a0/c (28)
Such a result is inevitable to ensure that the stream of hydrodymamics equations may
be expressed with respect to the MOND constant, for arbitrary parameters defining the
motion.
Meanwhile, in case of isobaric pressure, the equation of stream becomes conditionally
equivalent to geodesic. Thus, the appearance of the extra term on the right hand side of
equation (...) inspire many authors to interrelate it with the problem of dark matter as
an excess of mass due to the Lagrangian suggested by Kahil and Harko (2009) [1]:
From the above equations, we can find that the excess of mass for a test particle is
equivalent to the hydrodynamic equation of motion for a perfect fluid satisfying the first
law of thermodynamics. Such an analogy is required to describe the behavior of cluster
of fluid circumventing the AGN it has detected that annihilation of dark matter particles
in terms of increase γray density in the accretion disk)[21]
Accordingly, we can obtain the hydrodynamic flow of accretion desk by applying the
Euler-Lagrange equation on (1) with taking into account that
m(s)def.
=(P+E)
ρc2(29)
3 Dark Matter : Equations of Motion in Bimetric
Theories
In this section, owing to implementing the concept of geometrization of physics, it is es-
sential to express the motion of non-geodeisc equations and their corresponding deviation
equation to regulate the behavior of as expressed in particle content or fluid-like in the
presence of different bi-metric gravitational fields, able to explain its presence at different
scales.
8

3.1 Non-Geodesic Trajectories for Bi-gravity
Hossenfelder [22] has introduced an alternative version of bi-metric theory, having two
different metrics gand hof Lorentzian signature on a manifold Mdefining the tangential
space TM and co-tangential space T*M respectively. These can be obtained in terms of
two types of matter and twin matter; existing individually , each of them has its own
field equations as defined within Riemannian geometry. It is well known that implement-
ing bi-gravity theory, without cosmological constants, will be vital to describe motion of
dipolar objects in the halos [23]; while the conformal type may be able to describe dark
matter as mass excess quantities found in as in accretion disk circumventing the center of
the Galaxy, as described by strong gravitational fields.
Meanwhile, theories of bi-metric theories, have one metric combining the two metrics,
with cosmological constant, describing variable speed of light to replace the effect dark
energy in big bang scenario [24].
From the previous versions of bi-metric theories [25], we are going to present a generalized
form which can be present different types of path and path deviation which can be ex-
plained for any bi-metric theory which has two different metrics and curvatures as defined
by Riemannian geometry [26] . Their Corresponding Lagrangian can be expressed in the
following way [27]
Ldef.
=mggµν Ψ;νUµUν+mffµν Φ|νVµVν+(mg(s),β
mg(s)(gαβ−UαUβ));ρΨρ+(mf(τ),β
mf(τ)(gαβ−VαVβ));ρΨρ,
(30)
Thus, considering
(1) dτ
ds = 0 .
This will give to two separate sets of path equations owing to each parameter by applying
the following Bazanski-like Lagrangian:
DU α
DS =m(g)(s),β
m(g)(s)(gαβ −UαUβ),(31)
and DV α
Dτ =m(f)(τ),β
m(f)(τ)
(fαβ −VαVβ) (32)
and their corresponding path deviation equations:
D2Ψα
DS2=Rα
βγδ UγUβΨδ+ (m(g)(s),β
m(g)
(gαβ −UαUβ))ρΨρ,(33)
And,
D2Φα
Dτ 2=Sα
βγδVγVβΦδ+ (m(f)(τ),β
m(f)
(fαβ −VαVβ));ρΦρ,(34)
(2) dτ
dS 6= 0 [26]
9

The two metrics can be related to each other by means of a quasi-metric one [28].
˜gµν =gµν −fµν +αg(gµν −UµUν) + αf(fµν −VµVν).(35)
Such an assumption may give rise to define its related Lagrangian of Bazanski’s flavor
to describe the geodesic and geodesic deviation equation due to this version of bi-gravity
theory.
Ldef.
= ˜gαβ Uα˜
DΨβ
˜
DS ,(36)
˜
Γα
βσ =1
2˜gαδ(˜gσδ,β + ˜gδβ,σ −˜gβ σ,δ)
and its corresponding Lagrangian:
L=˜
m(s)˜gµν ˜
Uµ(d˜
Ψν
d˜
S+˜
Γν
ρδ ˜
Ψρ˜
Uδ) + ˜
fµΨµ(37)
Thus, equation of its path equation can be obtained by taking the variation respect to
Ψµto obtain:
d˜
Uα
d˜
S+˜
Γα
βδ ˜
Uβ˜
Uδ=m˜
(S),β
m˜
(S)(˜gαβ −˜
Uα˜
Uβ) (38)
and using the commutation relation (A.4) and the condition (A.5), we obtain its corre-
sponding deviation equation;
D2Ψµ
˜
DS2=˜
Rµ
νρσ ˜
Uν˜
Uρ˜
Ψσ+ (
˜
m(˜
S),β
m(˜
S)(˜gαβ −˜
Uα˜
Uβ));ρ˜
Ψρ(39)
where ˜
Rα
.µνρ =˜
Γα
µρ,ν −˜
Γα
µν,ρ +˜
Γσ
µρ ˜
Γα
σρ −˜
Γσ
µρ ˜
Γα
σρ
3.2 Equations of Dipolar Moment in Bi-gravity Theory
Equation of motion of diplar moment in the presence of bi-metric theory as a candidate to
represent DM as interaction between ordinary and twin matter as described by bi-gravity
ghost-free theory.
Accordingly, we suggest the following Lagrangian;
Ldef.
=def.
=gαβ PαDΨβ
(1)
Ds +Ωα
DΨβ
(2)
Ds +fαΨα
(1)+ˆ
fαΨα
(2)+fαβ QαDΦβ
(1)
Dτ +∆α
DΦβ
(2)
Dτ +kαΦα
(1)+ˆ
kαΨα
(2)
(40)
where, Qtwin matter momentum vector ∆ twin matter dipole moment vector ,jTwin
non-gravitational force to momentum ,Twin non-gravitational force of dipole moment
Thus, taking the variation with respect to Ψ1, Ψ2, Φ1and Φ2we obtain: the dipolar
momentum of ordinary matter, the evolution equation of ordinary matter, the equation
of twin dipolar momentum and the equation of twin evolution dipolar moment
DP µ
Ds =fµ(41)
10

Evolution equation of dipolar moment
DΩµ
Ds =ˆ
fµ(42)
While, for twin matter Equation of dipolar twin matter
DQµ
Dτ =kµ(43)
and Evolution equation of twin dipolar moment
D∆µ
Dτ =ˆ
kµ(44)
Also, in order to obtain their corresponding deviation equations following the same pro-
cedures in for both metrics gand findependently, we get after some manipulations the
following set of deviation equations for ordinary matter and twin matter as follows; for
the ordinary matter.
D2Ψµ
(1)
DS2=Rµ
νρσ PνUρΨσ
(1) +fµ
;ρΨρ
(1),(45)
and D2Ψµ
(2)
DS2=Rµ
νρσ ΠνUρΨσ
(2) +ˆ
fµ
;ρΨρ
(2) (46)
and for the twin matter
D2Φµ
(1)
Dτ 2=Sµ
νρσ QνVρΦσ
(1) +kµ
;ρΦρ
(1),(47)
and D2Φµ
(2)
Dτ 2=Sµ
νρσ ˆ
ΠνVρΦσ
(2) +ˆ
kµ
;ρΦρ
(2) (48)
where, Sα
βγδ ,Vα,ˆ
Παtheir associated curvature, for vector velocity, Polarization vector
for particles defined as twin matter respectively.
3.3 Dipolar Fluid in Bi-gravity Theory
Extending the previous section to be expressed in bi-gravity ghost free theory to de-
scribe both ordinary fluid and twin fluid as described in bi-gravity theory, we suggest the
following Lagrangian;
Ldef.
=gµν JµDΨν
(1)
Ds + Ωµ
DΨν
(2)
Ds +fµν ˆ
JµDΨν
(1)
Dτ + ∆µ
DΨν
(2)
Dτ (49)
taking the variation with respect to Ψ1, Ψ2, Φ1and Φ2we obtain
for Ordinary fluid DJµ
Ds = 0
11

and DΩ
Ds =1
σ∇µ(W−ˆ
Πˆ
W)−Rµ
ρνλ uρξνKλ
and for twin fluid
Dˆ
Jµ
Dτ = 0
and D∆
Dτ =1
σ∇µ(˜
W−˜
Π˜
W)−Sµ
ρνλ Vρ˜
ξν˜
Kλ
3.4 Non-Geodesic Euations in AGN: Bimetric theory
The bi-metric version of equation (4) can be obtained by obtaining the Euler-lagrange
equation on the following Lagrangian
˜
L= ˜gαβ ˜
UαD˜
Ψβ
D˜s(50)
To obtain the corresponding path equation
d˜
Uα
d˜s+˜
Γα
βδ ˜
Uβ˜
Uδ=˜m(s),β
˜m(˜s)(˜gαβ −˜
Uα˜
Uβ) (51)
and using the commutation relation (A.4) and the condition (A.5) we obtain its corre-
sponding deviation equation;
D2˜
Ψµ
D˜s2=˜
Rµ
νρσ ˜
Uν˜
Uρ˜
Ψσ+ ( ˜m(s),β
˜
m˜
(s)
(˜gαβ −˜
Uα˜
Uβ));ρ˜
Ψρ(52)
4 Dark Matter: Problem of Stability
4.1 The Relationship between Stability and Geodesic Deviation
The importance of solving deviation equations that are obtained with its path equation
for an object whether is counted to be a test particle or not is inevitably used for testing
the stability of the system by a perturbation to the geodesic equations [29]is dependent
of the style of coordinate system .
This approach has been applied previously in examining the stability of some cosmo-
logical models using two geometric structures [30]. Recently, Then above approach has
been modified by introducing an approach introduced by [18]] , based on stability condi-
tion as a result of by obtaining the scalar value of the deviation vector which gives rise
to become independent of any coordinate system which works for examining the stability
problem for any planetary system, being a covariant coordinate independent which can
be explained in the following way. More over this condition has been implemented to test
the stability of stellar systems orbiting strong gravitational fields [32]
12

This approach is shown in the following way:
Let Ψα(S) is obtained from the solutions of the deviation equation in a given interval [a,b]
in which Ψα(S) behave monotonically. These quantities can become sensors for measuring
the stability of the system are
qdef.
= lim
s→bqΨαΨα.(53)
If
q→ ∞
then the system is unstable, otherwise it is always stable.
where Cαare constants and f(S) is a function known from the metric. If f(S)→ ∞ ,
the system becomes unstable otherwise it is stable. For geodesic or non-geodesic deviation
equations
qdef.
= lim
s→bqΨαΨα.(54)
If q→ ∞ then the system is unstable, otherwise it is always stable.
Now, in case of dipolar moment equation there will be another condition for the
evolution equation, we suggest the above condition be extended to include deviation
tensor Φµν
(2) as
q(2)
def.
= lim
s→bqΨαΨ(2)α.(55)
Thus, for such a member in stellar/planetary system is stable, if and only if the magnitude
of the scalar value of both spin deviation vectors Φ2αand evolution deviation tensors Φαβ
2
to be real numbers respectively. i.e. either q1→ ∞ or q1→ ∞ the assigned member is
unstable. Accordingly, a strong stability condition must be admitted if both q(1) and q(2)
are satisfying the following conditions :
lim
s(1)→∞(ΨαΨα) = 0,(56)
And,
lim
τ→∞(ΦαΦα) = 0.(57)
Special Cases:
[1]Non-Geodesic Stability conditions: Higher Dimensions
In an approach to examine stability problems as a direct application of [18] .
Let us obtain the solution of the equation ΨA(S) as exerted from the deviation equation
in a given interval [A,B] , such that ΨA(S) behave monotonically. Thus, such quantities
describe the deviation vector may be sensors for measuring the stability in the following
way
ˆqdef.
= lim
S→BqΨAΨA.(58)
If ˆq→ ∞ then the system is unstable, otherwise it is always stable.
where CAare constants and f(S) is a function known from the metric. If f(S)→ ∞
, the system becomes unstable otherwise it is stable.
13

[2] Non-geodesic Stability conditions in Bi-metric Theories : On the Halos
For case dτ
ds 6= 0
The solution of the set deviation equations (...) and (...) are
Ψaα=Cα
1f(S) (59)
And,
Φaα=Cα
2f(S) (60)
We must obtain two stability conditions in the following way:
qdef.
= lim
s→bqΨαΨα.(61)
and
ˆqdef.
= lim
τ→bqΦαΦα.(62)
If q→ ∞ or q→ ∞ then the system is unstable, otherwise it is always stable.
[3] Non-geodesic Stability conditions in Bi-metric Theories : At the Accretion Disk
For case dτ
ds = 0
we have only one stability condition; condition as there exists one family of deviation
equations ˜
Ψα
˜
Ψα˜
Cαf(˜
S)
in which
˜qdef.
= lim
˜
S→bq˜
Ψα˜
Ψα.
thus, the stability condition becomes ˜q→ ∞ then the system is unstable, otherwise it is
always stable. Also, in a similar way , the case of diople moment trajectoiies as express
in GR, we may have two simultaneous conditions one for linear passive momentum, as
similar as for spinning particle with precession [32], and the other is for evolution deviation
equation. Such a previous condition in case of its analogue in Bimetric theory such that
ds
dτ = 0 we obtain four conditions due to existence of twin matter equations. But it reduces
to two conditions if we apply ds
dτ 6== 0.
5 Discussion and Conclusion
Dark matter may be regarded as a fluid with different characteristics behavior based on
its position from the source of the gravitational field.
In view of the above mentioned speculation, we may give rise to emphasize previous
ideas, such as of activated as dipolar fluid in the halo. The notation of a solely dipolar
activity is analogous is inspired from the spinning motion of particles. Accordingly, in
this study, we have developed the associated Lagrangian that presented simultaneously
the linear equation and precession equation using a modified Bazanski Lagrangian. This
14

can be employed in our case, by suggesting the motion dipolar moment be represented in
linear momentum and evolution equations from one single Lagrangian [...].
Yet, such a tendency to express to include DM in the halo as dipolar particles is
constraining this effect al galactic level, which is a violation to apparant observation that
DM is also existed in the Universe as well as being felt to to associated incidences of
excess of γ-ray radiation as an indicator of self annihilation DM-particles in the AGNs, as
well as nearby neutron stars, binary pulsars. This can be revealed by applying Blanchet’s
approach of replacing the dipolar particles by dipolar fluids. In favor of this idea, the
dipolar fluid may feel the interaction with dark energy and that itself quite reasonable
due to the dominance of dark energy 74% while dark matter occupies 23% due to the well
known observations, while baryonic matter is about 4%. Recently, due to ESA’s Planck
mission DM has been found to be 26%, which causes the other readings to be revised[11];
such an involvement may be consistent with an MOND-paradigm.
Now, the arising question is related to type of geometry and its associated gravitational
field theory which is expressing this situation. Thus, it has been found that bi-metric
theory of gravity can be regarded as a good representative to express this situation.
However, the representation of bi-metric theory can be expressed differently from one
region into another.
Accordingly it has led us to describe bi-metric theory behaving as bi-gravity ghost
free massive theory at the galactic level to explain the behavior of DM as twin matter
in a fluid, and near by the center of strong gravitational fields the conformal version is
presented to be matched with Verozub’s version of bi-metric theory of gravity. Owing to
the equation of motion, it is vital to examine the stability of these regions, by solving the
geodesic deviation equations, due to inter-relation between geodesic deviation equation
and stability conditions.
In the present work, we extend Wanas-Bakry method [10] to include strong gravita-
tional systems rather than its application on planetary systems. The advantage of this
method is its invariance of coordinate systems.
Equations of relativistic hydrodynamics are expressed as non-geodesic equations, with
taking into account that the mass excess term due to the existence of dark matter. Such a
speculation, is an extension to similar vision by Kahil and Harko (2009)[1], who presented
its existence in terms of a Lagrangian having a scalar defining variable mass.
The problem of motion as described in the Riemanian geometry will be extended
to be explained by different geometries, admitting non vanishing curvature and torsion
simultaneously. This work will be expressed in our future work to emphasize the concept
of geometrization of physics i.e. the significance of the existence of DM and DE may be
expressed within some quantities related to the richness of these geometries. This is a
step towards revealing the cloud of mystery which is always associated with the notations
of DM and DE.
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16

Appendix (A)
The Papapertrou Equation in General Relativity: Lagrangian
Formalism
It is well known that equation of spinning objects in the presence of gravitational field have
been studied extensively []. This led us to suggest its corresponding Lagrangian formalism
, using a modified Bazanski Lagrangian [20], for a spinning and precessing object and their
corresponding deviation equation in Riemanian geometry in the following way
L=gαβ PαDΨβ
Ds +Sαβ
DΨαβ
Ds +FαΨα+Mαβ Ψαβ (A.1)
where
Pα=mUα+Uβ
DSαβ
DS .
Taking the variation with respect to Ψµand Ψµν simultaneously we obtain
DP µ
DS =Fµ,(A.2)
DSµν
DS =Mµν (A.3),
where Pµis the momentum vector, Fµ=1
2Rµ
νρδ Sρδ Uν,and Rα
βρσ is the Riemann curvature,
D
Ds is the covariant derivative with respect to a parameter S,Sαβ is the spin tensor, Mµν =
PµUν−PνUµ, and Uα=dxα
ds is the unit tangent vector to the geodesic.
Using the following identity on both equations (1) and (2)
Aµ
;νρ −Aµ
;ρν =Rµ
βνρAβ,(A.4)
where Aµis an arbitrary vector. Multiplying both sides with arbitrary vectors, UρΨνas
well as using the following condition [19]
Uα
;ρΨρ= Ψα
;ρUρ,(A.5)
and Ψαis its deviation vector associated to the unit vector tangent Uα. Also in a similar
way:
Sαβ
;ρΨρ= Φαβ
;ρUρ,(A.6)
one obtains the corresponding deviation equations [34]
D2Ψµ
DS2=Rµ
νρσ PνUρΨσ+Fµ
;ρΨρ,(A.7)
and DΨµν
DS =Sρ[µRν]
ρσǫUσΨǫ+Mµν
;ρΨρ.(A.8)
17