Motion in Bimetric Type Theories of Gravity

Abstract

The problem of motion of different test particles, charged and spinning objects with a constant spin tensor in different versions of the bimetric theory of gravity is considered by deriving their corresponding path and path deviation equations using a modified Bazanski Lagrangian. Such a Lagrangian, as in the framework of Riemannian geometry, has a capability to obtain path and path deviations of any object simultaneously. This method enables us to derive the path and path deviation equations of different objects orbiting in very strong gravitational fields.

Full text

arXiv:1502.04415v1 [gr-qc] 16 Feb 2015
Motion in Bimetric Type Theories of Gravity
February 17, 2015
M.E.Kahil1 2
Abstract
The problem of motion for different test particles , charged and spinning objects
of constant spinning tensor in different versions of bimetric theory of gravity is
obtained by deriving their corresponding path and path deviation equations, using
a modified Bazanski in presence of Riemannian geometry. This method enables us
to find path and path deviation equations of different objects orbiting very strong
gravitational fields.
1 Bimetric Theories: A brief introduction
General relativity is considered a land mark in history of science of being, during the last
century, a pioneer non-linear theory of gravity[1]. Yet, some problems have been remained
unsolved due to applying Riemannian geometry in its explanation. One of these difficul-
ties is related to the law of conservation of energy and momentum [2], which is dealt by
considering the metrical tensor as flat one at great distances from the gravitational source.
Rosen [3,4]introduced a remedy to this problem by proposing two different metrics gµν
representing the gravitational source and giving a curved space and γµν describing a phys-
ical one expressing an inertial frame and becoming a flat space. Using this assumption,
it can be found that the field equations of Einstein imply to a theory of gravitation in
flat space . A problem arises because the pseudo-tensor quantities in Orthodox General
Relativity then turn out to be tensorial ones [2]. This led Yalmoz [5] to examine a new
class of solutions for the field equations of Rosen’s theory of gravitation to solve the dif-
ficulty of dealing with the flat metric. But, a slight problem has emerged in that, the
speed of light is no longer constant as confirmed experimentally in the realm of Special
Relativity. Such a problem is counted to be a virtue by expressing Moffat’s approach of
a bi-metric theory of gravity for a variable speed of light, as it helps to reveal the puzzle
of dark energy problem. This approach may be expressed by means of two metrics are
likened to each other in terms of gradients of scalar /biscalar fields to explain the rapid
expansion of galaxies is due to change of speed of light from one epoch to another that
1Scientific Thinking Division, The American University in Cairo, New Cairo, Egypt
e.mail: kahil@aucegypt.edu
2Egyptian Relativity Group. Cairo, Egypt
1

i.e. dark energy can be interpreted This type of theories are expressing how inflation
senecio of the universe is due to Bi-metric version of variable speed of light [7] . Some
applications using the Moffat formalism of bi-metric theory of gravity are explaining the
causal description of quantum entanglement [8] and testing the propagation of neutrinos
using OPERA experiment [9].
Another problem in GR is inability to explain the rotation curves of spiral galaxies.
Milgram [10] proposed a specific treatment by performing a modified Newtonian Dynam-
ics paradigm (MOND)that was able to explain its causality, apart from appealing to dark
matter problem. A new step has been taken successively to extend MOND to be expressed
in terms of bi-metric theory to be produce BIMOND [11] by having two field equations
describing matter and twin-matter[12] may be used for examining the existence of gravita-
tional waves and explaining two interacting 4D membranes[13]. Recently, Hassan-Rosen
[14] developed an extension of the present bi-metric theory using the concept of bi-gravity,
which used two metrics describing their gravitational fields. In doing so, they discard the
earlier bimetric theories metrics in which one metric describes gravity while the other is
physical. Accordingly, a new massive gravity theory for spin-2 , free from ghosts, has been
obtained [15]. Several applications of these types of theories are viable in dealing with
obtaining field equations for very strong gravitational fields of neutron stars [16], which
suggests the possibility to examine black holes and super-massive black holes e.g Sgr A*
by studying the stability of objects orbiting in these fields. This hypothesis is essentially
to be examined.
2 Motion in Bimetric theories
Theories of gravity may help to explain the functioning of different objects. For example,
Rosen[3] obtained the equation of motion in bimetric theory of gravitation for a test
particle. These equations were solved by Isrealit[17] in order to examine their behavior
in the case of small velocities and weak fields using past Newtonian approximation .
Also, Falik and Opher [18] applied the bimetric theory of gravity to find the field
equations associated with spinning neutron stars as an example of strong gravitational
field . This achievement opens the way to examine motion of different charged objects
and spinning ones in presence of a strong gravitational field as defined by Bi-metric type
theories. This led us to obtain the corresponding path and path deviation of different
objects , such as test particles, charged particle, spinning objects. These results are
determined by introducing a Lagrangian with a specific feature for obtaining the path
and path deviations. Therefore, the key role to this approach is based to obtain path and
path deviation equations for each of these objects using a specified Lagrangian for each
case. The aim of our study is obtaining equations of motion for objects using bimetric
theory originated from two metrics in one stands for gravitational field and the other
defines physical matter or both represent gravity.
2

2.1 Path and Path Deviation Equations: The Bazanski Ap-
proach
Geodesic and geodesic deviation equations can be obtained from the following Bazanski
Lagrangian [19]:
L=gαβUαDΨβ
Ds ,(1)
where Uαis a four vector velocity, Ψβits deviation vector, D
DS is a covariant derivative
with respect to gµν . If one takes the variation with respect to the deviation vector Ψρto
obtain geodesic equations:
dUα
dS + Γα
µν UµUν= 0,(2)
where Γα
µν is the Levi-Civita affine connection. Also, the same technique can be applied to
obtain the variation with respect to the tangent vector Uρto obtain the geodesic deviation
equations: DΨα
DS2=Rα
.βγδΨγUβUδ(3)
Where Rα
.βγδ the curvature of space time defined by the affine connection Γα
µν . The
above method has been applied in different geometries than the Riemannian one e.g.
non-Riemannian geometries admitting non-vanishing curvature and torsion tensors si-
multaneously [20-22]. This approach helps to implement the concept of geometrization
to include not only physics but also biological epidemic curves [23] as well as economic
complex systems in terms of information geometry [24]. Also, this Lagrangian has been
modified to describe the path equation of charged object to take the following form [25];
L=gαβUαDΨβ
Ds +e
mFαβ UαΨβ(4)
to give
dUα
d¯
S+ Γα
µν UµUν=e
mFµ
.ν Uν(5)
where Fµν is an electromagnetic tensor, e
mthe ratio between charge to mass of an object.
and its corresponding deviation equation becomes:
D2Ψα
DS2=Rα
.µνρ UµUνΨρ+e
m(Fα.ν DΨν
Ds +Fα
.ν;ρUνΨρ).(6)
In the mean time the corresponding Papapetrou Equation for spinning objects with con-
stant spinning tensor [26] is obtained from the following Lagrangian :
L=gαβUαDΨβ
DS +1
2mRαβγσUαΨβSγσ (7)
where Sµν is a spin tensor of a spinning object. By taking variation with respect to the
Ψαto obtain dUα
dS + Γα
µν UµUν=1
2Rα
.µνρ Sρν Uµ(8)
3

and taking the variation with respect to Uαto obtain its deviation equation:
D2Ψα
DS2=Rα
.µνρ UµUνΨρ+1
2m(Rα
.µνρ Sνρ DΨν
Ds +Rα
µνλ Sµλ
.;ρUνΨρ+Rα
µνλ;ρSν λUµΨρ),(9)
as well as the Dixon equation for spinning charged objects is expressed as [27]
dUα
dS + Γα
µν UµUν=e
mFµ
.ν Uν+1
2mRα
.µνρ Sρν Uµ,(10)
while, its corresponding deviation equation becomes
D2Ψα
Ds2=Rα
.µνρ UµUνΨρ+e
m(Fα.ν DΨν
Ds +Fα
.ν;ρUνΨρ) + 1
2mRα
.µνρ UµUνΨρ
+1
2m(Rα
.µνρ Sνρ DΨν
Ds +Rα
µνλ Sµλ
.;ρUνΨρ+Rα
µνλ;ρSν λUµΨρ) (11)
The Papapetrou equation of a spinning object with precession [28]is obtained by a
modified Bazanski Lagrangian [29] :
L=gαβ(mUα+Uβ
DSαβ
DS )DΨβ
Ds +1
2RαβγδSγ δ UβΨα(12)
to obtain equation of a spinning object by taking the variation with respect to the devi-
ation vector Ψα
D
DS (mU α+Uβ
DSαβ
DS ) = 1
2Rα
.µνρ Sρν Uµ(13)
and its deviation equation can be obtained by taking the variation with respect to Uαto
become:
D2Ψα
Ds2=Rα
.µνρ Uµ(mUν+Uβ
DSνβ
Ds )Ψρ+gασ gνλ (mUλ+Uβ
DSλβ
Ds );σ
DΨν
Ds
+1
2(Rα
.µνρ Sνρ DΨµ
Ds +Rα
µνλ Sνλ
.;ρUµΨρ+Rα
µνλ;ρSν λUµΨρ).(14)
2.2 Path and Path Deviation Equations in Weyl geometry
It is well known that in Weyl geometry the gravitational potential tensor is associated
with such a scalar field. From this perspective one can define a combined gravitation
potential tensor in the following manner [30]:
¯gµν =eφgµν ,(15)
Where ¯gµν is the Weyl gravitational potential, and φa scalar field, which may give raise to
introduce disformal transformation of any gravitational theory having two metrics defined
in the following way [31]
¯gµν = [Agµν +¯
Aφ,µφ,ν ] (16)
where Aand ¯
Aare arbitrary constants .
4

Thus, in this type of geometry it can be defined its corresponding affine connection to
become :
¯
Γα
βσ = Γα
βσ +1
2gαδ(gσδφ,β +gδβ φ,σ −gβσ φ,δ ) (17)
In order to obtain the geodesic equation in the following which can be obtained by applying
the action principle on the following Lagrangian:
L= ¯gµν UµUν(18)
to give
dUα
dS +¯
Γα
βσ UβUσ= 0,(19)
which is obtained by taking the variation with respect to Ψµon its a developed Lagrangian
mentioned in [32] :
L= ¯gµν Uµ(dΨν
dS +¯
Γν
ρδΨρUδ).(20)
And its corresponding deviation equation is obtained by taking the variation with respect
to Uµto become: ¯
D2Ψα
¯
DS2=¯
Rα
.µνρ UµUνΨρ(21)
where ¯
Rα
.µνρ =¯
Γα
µρ,ν −¯
Γα
µν,ρ +¯
Γσ
µρ ¯
Γα
σρ −¯
Γσ
µρ ¯
Γα
σρ
(ii) Dixon-like Equation for spinning charged objects of Weyl geometry: Similarely,
we can obtain the Dixon -like path equation
dUα
dS +¯
Γα
µν UµUν=e
mFµ
.ν Uν+1
2m¯
Rα
.µνρ Sρν Uµ(22)
and its corresponding deviation equation becomes:
¯
D2Ψα
¯
DS2=¯
Rα
.µνρ UµUνΨρ+e
m(Fα.ν
¯
DΨν
¯
DS +Fα
.ν;ρUνΨρ) + 1
2m¯
Rα
.µνρ UµUνΨρ
+1
2m(¯
Rα
.µνρ Sνρ DΨν
Ds +¯
Rα
µνλ Sµλ
.;ρUνΨρ+¯
Rα
µνλ;ρSν λUµΨρ) (23)
2.3 Path and Path Deviation Equations of MOND
In this part, it is worth mentioning the path and path deviation of MOND paradigm
due to its vital role in explaining the vague regions due to dark matter problem that
are unknown by Newtonian/Einsteinain formulations i.e. revealing the nature of rotation
curves of spiral galaxies [33] Accordingly, some authors have studied motion of a test in
MOND [34]. This has led us to apply the Bazanski method in order to obtain the path
and path deviation equations for any test particle related to this paradigm.
L=gµν UαDΨβ
Ds +1
2mφ,µΨµ(24)
5

to give its path equation
dUα
dS + Γα
µν UµUν=1
2mgαµφ,µ (25)
and its deviation equation in the following way:
D2Ψα
DS2=Rα
βγδUβUγΨδ+gαρφρ;σΨσ+gαρφρUν
DΨν
DS (26)
3 Path and Path deviation of Bimetric Theories
3.1 Path Equation and Path Deviation of Rosen’s Approach
Equations of motion of test particles subject to bimetric theory of gravity were obtained
by Rosen [4] in the following way:
dUµ
dS + ∆µ
νσ UνUσ= 0,(27)
where
∆µ
νσ = [Γµ
νσ −γµ
νσ ],
and γµ
νσ is an affine connection defined by γµν .
Following the Rosen approach, Isrealit (1976) solved this type of motion using PPN ap-
proximation to be compared with the previous findings in GR. Lately, Foukzon et al [35]
studied the bimetric theory of gravitational inertial field in Riemannain and its relation-
ship with Finsler-Lagrange geometry and obtained the same path equation, which can be
obtained from the following Lagrangian:
L= (gµν −γµν )UµUν.(28)
In an alternative way, we suggest its corresponding lagrangian which follows the Bazanki
Lagrangian to obtain the sets of equations of geodesic and geodesic deviation by taking
the ovation with respect to Ψ to obtain
L= (gµν −γµν )UµDΨν
DS (29)
While, taking the variation with respect Uαwe obtain
∇2Ψν
∇S2= (Rα
βγσ −Pα
βγσ)UβUγΨσ
where ∇Ψν
∇S=dΨµ
dS + ∆µ
νσ ΨνUσ
and Pα
βγσ is the curvature tensor obtained by the affine connection γα
βδ [3].
Due to Rosen’s approach the curvature tensor Pα
βγσ = 0 which let equation (..) reduces
to ∇2Ψν
∇S2=Rα
βγσUβUγΨσ.
6

Also, for charged objects in bimetric theory of gravity, Falik and Rosen [36] obtained
their corresponding field equations , which led us to introduces the following Lagrangian
to obtain their corresponding path and path deviation equations:
L= (gµν −γµν )Uµ∇Ψν
∇S+e
mFµν UµΨν(30)
to give ∇Uα
∇S=e
mFα
.ν Uν(31)
and its corresponding deviation equation becomes:
∇2Ψα
∇S2=Rα
.µνρ UµUνΨρ+e
mFα
.ν
∇Ψν
∇S+e
m(Fα
.ν;ρ−Fα
.ν|ρ)UνΨρ(32)
Moreover, Avakian et al.[16] studied the field equations of a spinning bodies in the
presence of Bimetric theory. Accordingly its the corresponding spinning equation can be
obtained from the following Lagrangian:
L= (gαβ −γαβ)Uα∇Ψβ
∇S+1
2m(Rαβγσ −Pαβγσ )UαΨβSγ σ (33)
we can apply its Bazanski approach to obtain its path equation:
dUα
dS + ∆α
µν UµUν=1
2m(Rα
.µνρ −Pα
.µνρ )Sρν UµUν(34)
and its corresponding deviation equation:
∇2Ψα
∇S2= (Rα
.µνρ −Pα
.µνρ )UµUνΨρ+1
2m(Rα
.µνρ −Pα
.µνρ )Sνρ DΨν
Ds
+ (Rα
µνλ Sµλ
.;ρ−Pα
µνλ Sµλ
.|ρ)UνΨρ+ (Rα
µνλ;ρ−Pα
µνλ|ρ)Sν λUµΨρ(35)
Thus, if we take into consideration that Pα
βγδ = 0, path equation becomes:
dUα
dS + ∆α
µν UµUν=1
2mRα
.µνρ Sρν UµUν(36)
and its corresponding deviation equation becomes:
∇2Ψα
∇S2=Rα
.µνρ UµUνΨρ+1
2mRα
.µνρ Sνρ DΨν
Ds +1
2mRα
µνλ (Sµλ
.;ρ−Sµλ
.|ρ)UνΨρ+1
2mRα
µνλ;ρSν λUµΨρ
(37)
7

3.2 Path and Path Deviation Equations of Moffat’s Approach
Moffat [6] presented the framework of VSL satisfying bimetric theory and its causality
to reveal the problem of dark energy due to VSL by introducing such a metric in the
following way.
ˆgµν =gµν +B∂µφ∂νφ(38)
where ˆgµν defines a specific matter metric tensor of a given matter field, Bis an arbitrary
constant has a dimension of [length]2and chosen to be positive and φis a biscalar field .
The inverse metrics gµν ˆgµν satisfy
gµνgµρ =δν
ρ(39)
ˆgµν =gµν +B∂µφ∂νφ(40)
ˆgµν ˆgµρ =δν
ρ(41)
Yet, the modification processes to control the casual propagation of the biscalar field led
to redefine (40) to become:
ˆgµν =gµν +B
K∇µφ∇νφ+KBqTµν ,(42)
where Kis an arbitrary constant and Tµν is a given energy-momentum tensor to control
the causal propagation of the biscalar field [9].
Consequently, the equation of geodesic can be obtained from taking the action on the
following Lagrangian
L= ˆgµν UµUν(43)
to become dUν
dS +ˆ
Γν
µρUρUµ= 0,(44)
where
ˆ
Γν
µρ =1
2ˆgσν (ˆgρσ,µ + ˆgµσ,ρ −ˆgρµ,σ)
Also, we suggest its corresponding lagrangian which follows the Bazanki lagrangian to
obtain the sets of equations of geodesic and geodesic deviation by taking the ovation with
respect to Ψαto obtain (44)
L= ˆgµν Uµˆ
DΨν
ˆ
DS (45)
While, taking the variation with respect to Uαto obtain:
ˆ
D2Ψν
ˆ
DS2=ˆ
Rα
βγδUγUβΨδ
where ˆ
Rα
βγδ =ˆ
Γα
βδ,γ −ˆ
Γα
βγ,δ +ˆ
Γν
βδ ˆ
Γα
νγ −ˆ
Γν
βγ ˆ
Γα
νδ
8

3.3 Path equations and Path deviation of BIMOND Type The-
ories
In this section, we present the corresponding path and path deviation equation for test
particles or spinning objects in the presence of BIMOND theories. Accordingly , it is worth
mentioning at the beginning the above corresponding paths and there deviation equation
in MOND paradigm to be extended in case of its BIMOND version its corresponding path
equation becomes:
dUα
dS + (Γα
βγ −¯
Γα
βγ )UβUγ= 0
which can be formed from the following lagrangian by taking the variation with respect
to Ψα:
L= ˆgµν
ˆ
DΨν
Ds
where: DΨα
DS =dΨα
ds + (Γα
βγ −¯
Γα
βγ )ΨβUγ
and its corresponding deviation equation becomes;
D2Ψα
DS2= (Rα
βγδ −¯
Rα
βγ δ )ΨγUδUβ
In case of BIMOND , Milgram [14] introduced the relationship between the two affine
connections as defined by gµν and γµν to become:
Cα
βρ = Γαβρ −¯
Γαβρ ,
such that [,,, ]:
, gµν;ρ=gδν Cδ
µρ +gδµCδ
νρ
and
γµν|ρ=−γδν Cδ
µρ −γδµCδ
νρ .
Thus, we suggest the following Lagrangian path and path deviation equations becomes
L=gµν Uµ¯
D2Ψν
DS2+γµν UµD2Φν
Dτ 2+1
2m(Rα.µνρ −¯
Rα.µνρ )Sνρ UµΨα
In case of spinning object
DU α
DS =1
2m(Rα
.µνρ −¯
Rα
.µνρ )Sρν Uµ(46)
and its deviation equation can be obtained by taking the variation with respect to Uαto
become: D2Ψα
DS2=Rα
.µνρ UµUνΨρ+1
2m(Rα
.µνρ −¯
Rα
.µνρ )
+1
2mRα
.µνρ Sνρ DΨν
Ds +(Rα
.µνρ Sµλ
.;ρUνΨρ−¯
Rα
.µνρ Sµλ
.|ρUνΨρ)+(Rα
µνλ;ρ−¯
Rα
µνλ|ρ)Sν λUµΨρ
(47)
9

3.4 Generalized Path equations and Path Deviation of Bimetric
Theories
Hossenfelder [35] introduced an alternative version of bi-metric theory, having two different
metrics gand hof Lorentzian signature on a manifold Mone is defined in tangential space
TM and the other is in its co-tangential space T*M respectively. These can be regarded as
two sorts of matter and twin matter, existing individually , each of them has its own field
equations as defined within Riemannian geometry. In this part we are gong to present a
generalized form which can be present different types of path and path deviation which
can be explained for any bimetric theory which has two different metrics and curvatures
as defined by Riemannian geometry . Their Corresponding Lagrangian can be expressed
in the following way:
L=gµν Ψ;νUν+γµν Φ|νVµ,(48)
By considering dτ
ds = 0 it will lead to two separate sets of path equations owing to each
parameter by applying the following Bazanski-like lagrangian:
L=gµν Ψ;νUν−γµν Φ|νVµVν
DU α
DS = 0,(49)
and DV α
Dτ = 0 (50)
and their corresponding path deviation equations:
D2Ψα
DS2=Rα
βγ δ UγUβΨδ,(51)
and D2Φα
Dτ 2=Sα
βγ δ VγVβΦδ,(52)
Thus we suggest, the corresponding lagrangian to describe two independent sets of a
generalized path and path deviation equations:
L=gµν Ψ;νUν−γµν Φ|νVµVν+fµΨµ+ˆ
fµΦµ(53)
where,
fµ=1
m(eFµν +1
2Rµνρσ Sρσ )Uν
and
¯
fµ=1
m(eFµν +1
2Sµνρσ Sρσ )Vν
. By taking the variation of Ψαand Φαwe obtain the generalized set of path deviation
DU α
DS =fα,(54)
10

and DV α
Dτ =¯
fα(55)
and taking the variation with respect to Uαand Vαto obtain the set of their corresponding
path deviation equations:
D2Ψα
DS2=Rα
βγ δ UγUβΨδ+fα
;ρΨρ+gαρfρUν
DΨν
DS (56)
and D2Φα
Dτ 2=Sα
βγδVγVβΦδ+¯
fα
|ρΦρ+γαρ ¯
fρVν
DΨν
Dτ (57)
3.5 Path Equations and Path Deviation of Bi-gravity Type The-
ories
Recently, Arkami et al[30] have suggested two independent metrics to explain bi-gravity
phenomena,
ds2=gµν dxµdxν
and
dτ2=hµν dxµdxν
.
Thus, the variational method to obtain geodesic-like equations of bigravity theory is
expressed in the following way [36] :
(d
dS
∂L
∂˙
Φα−∂L
∂Φα) + ( dτ
dS )2(d
dτ
∂L
∂¯
Φα−∂L
∂Φα) = 0
to give the same results as mentioned by Arkani et al (2014)
gµν
DU µ
DS +hµν (dτ
dS )DU µ
Dτ = 0 (58)
Applying the same technique of the Bazanski approach, we obtain its deviation equa-
tions to obtain:
gµα[D2Ψα
DS2+Rα
βδγ UγUβΨδ] + ( dτ
dS )
2
γµα[D2Φα
Dτ 2+Rα
βδγ VγVβΦδ],= 0 (59)
If one considers dτ
ds 6= 0, the two metrics can be related to each other by means of a
quasimetric one [31].
˜gµν =gµν −hµν +αg(gµν −UµUν) + αh(hµν −VµVν),(60)
such that
L= ˜gαβUα˜
DΨβ
˜
DS ,(61)
˜
Γα
βσ =1
2˜gαδ(˜gσδ,β + ˜gδβ ,σ −˜gβσ,δ)
11

and its corresponding Lagrangian:
L= ˜gµν Uµ(dΨν
dS +˜
Γν
ρδΨρUδ) (62)
Thus, equation of its path equation can be obtained by taking the variation respect to ψµ
to obtain: ˜
DU α
˜
DS2= 0 (63)
while taking the variation with respect to Uµto obtain its corresponding path deviation
equation: ˜
D2Ψα
˜
DS2=˜
Rα
.µνρ UµUνΨρ(64)
where ˜
Rα
.µνρ =˜
Γα
µρ,ν −˜
Γα
µν,ρ +˜
Γσ
µρ ˜
Γα
σρ −˜
Γσ
µρ ˜
Γα
σρ
Consequently, the following path and path deviation of charged and spinning objects of
constant spinning tensor are explained as follows
L= ˜gαβ Uα˜
DΨβ
˜
DS +e
mFαβ UαΨβ(65)
to give ˜
DU α
˜
DS =e
mFµ
.ν Uν(66)
and its corresponding deviation equation becomes:
˜
D2Ψα
˜
DS2=˜
Rα
.µνρ UµUνΨρ+e
m(Fα.ν
˜
DΨν
˜
Ds +Fα
.ν||ρUνΨρ).(67)
where || represents the covariant derivative with respect to affine connection Γα
βσ . Also,
the generalized path and path deviation equations for spinning objects are obtained from
the following Lagrangian:
L= ˜gµν
˜
DΨα
˜
DS +1
2m˜
Rαµνρ Sνρ UµΨα(68)
By taking the variation with respect Ψαthe to obtain its corresponding path equation:
˜
DU α
˜
DS =1
2m˜
Rα
βµν Sµν Uβ,(69)
and taking the variation with respect Uαto obtain its path deviation equation:
˜
D2Ψα
˜
DS2=˜
Rα
βγ δ UγUβΨδ+1
2m(˜
Rα
βµν Sµν Uβ)||ρΨρ+1
2m˜
Rα
βµν Sµν UβUρ
˜
DΨρ
˜
DS (70)
12

4 Discussion and Concluding Remarks
In this study, we have obtained the corresponding equations of path and path deviation
equations for test particles, charged and spinning objects -constant spinning tensor- in
different versions of Bimetric theories of gravity using a modified Bazanski Lagrangian.
This type of study has imposed us to determine prior to its procedure some relevant path
and path deviation for different objcts equations in Weyl geometry and MOND paradigm
to be counted as an introductory step to visualize the different stages of path and path
deviation equations that must be included before dealing with different bimetric theories
of gravity. The study may give rise to search of a possible geometry able to express
bimetric theory of gravity. It can be sought that Finslerian geometry is a good candidate
to express bi-metric theory of gravity as an extension of Riemannian geometry [33]. In the
mean time, path and path deviation equations using the Bazanski Lagrangian in Finsler
geometry is in preparation [37]. Also, the above treatment of utilizing symmetric affine
connection, can be extended into another version of bimetric theory of gravity following
Einstein-Cartan geometry,as an extended approach of Drummond [38] which gives rise to
different types of torsion and how does it propagate with respect to metric propagation
due bimetric formalism. Finally, this work will enable us to examine, the stability of
objects orbiting very strong gravitational field by solving the spin and spin deviation
equations.
Acknowledgement
The author would like to thank Professors T.Harko , G. De Young, M.I.Wanas , M. Abdel
Megied and his colleague Dr. E. Hassan for their remarks and comments.
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