Remarks on gravitational interaction in Kaluza-Klein models

Abstract

In these remarks, we clarify the problematic aspects of gravitational
interaction in a weak-field limit of Kaluza-Klein models. We explain why some
models meet the classical gravitational tests, while others do not. We show
that variation of the total volume of the internal spaces generates the fifth
force. This is the main reason of the problem. It happens for all considered
models (linear with respect to the scalar curvature and nonlinear $f(R)$, with
toroidal and spherical compactifications). We explicitly single out the
contribution of the fifth force to nonrelativistic gravitational potentials. In
the case of models with toroidal compactification, we demonstrate how tension
(with and without effects of nonlinearity) of the gravitating source can fix
the total volume of the internal space, resulting in the vanishing fifth force
and consequently in agreement with the observations. It takes place for latent
solitons, black strings and black branes. We also demonstrate a particular
example where non-vanishing variations of the internal space volume do not
contradict the gravitational experiments. In the case of spherical
compactification, the fifth force is replaced by the Yukawa interaction for
models with the stabilized internal space. For large Yukawa masses, the effect
of this interaction is negligibly small, and considered models satisfy the
gravitational tests at the same level of accuracy as general relativity.

Full text

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arXiv:1201.1756v1 [gr-qc] 9 Jan 2012
Comments on gravitational interaction in Kaluza-Klein models
Maxim Eingorn∗and Alexander Zhuk†
Astronomical Observatory and Department of Theoretical Physics,
Odessa National University, Street Dvoryanskaya 2, Odessa 65082, Ukraine
In these comments, we clarify the problematic aspects of gravitational interaction in a weak-field
limit of Kaluza-Klein models. We explain why some models meet the classical gravitational tests,
while others do not. We show that variation of the total volume of the internal spaces generates the
fifth force. This is the main reason of the problem. It happens for all considered models (linear with
respect to the scalar curvature and non-linear f(R), with toroidal and spherical compactifications).
In the case of models with toroidal compactification, we demonstrate how tension (with and without
effects of non-linearity) of the gravitating source can eliminate the fifth force, resulting in agreement
with the observations. It takes place for latent solitons, black strings and black branes. In the case
of spherical compactification, the fifth force is replaced by the Yukawa interaction for models with
the stabilized internal space. For large Yukawa masses, the effect of this interaction is negligibly
small, and considered models satisfy the gravitational tests at the same level of accuracy as General
Relativity.
PACS numbers: 04.50.Kd, 04.50.Cd, 04.25.Nx, 04.80.Cc
I. INTRODUCTION
In our recent papers [1]-[7], we investigated classical
gravitational tests (the perihelion shift, the deflection of
light and the time delay of radar echoes) in Kaluza-Klein
(KK) models. We paid attention mainly to models with
toroidal compactification of extra dimensions, i.e. with
compact and flat internal spaces. On the one hand, these
theories are very popular in the literature devoted to KK
models. On the other hand, it is well known that the
gravitational tests in General Relativity are calculated
on the background of a flat metrics. This background
metrics is perturbed by a point-like mass. In a weak-
field limit, such matter source has a dust-like equation of
state. In General Relativity, this formulation of the prob-
lem leads to formulas for gravitational tests, which are in
good agreement with the experimental data [8]. There-
fore, to generalize this approach, we also supposed that
the background metrics (when the matter source is ab-
sent) is flat for our external four-dimensional space-time
and internal spaces, and a point-like matter source has a
dust-like equation of state in all spatial dimensions. To
our surprise, this approach does not work in KK models.
Obtained formulas strongly contradict the observations
[1]. It turned out that to satisfy the experimental data,
the matter source should have negative equations of state
(tension) in the internal spaces [2, 3]. For example, la-
tent solitons satisfy the gravitational tests at the same
level of accuracy as General Relativity [3]. The similar
situation takes place for non-linear (with respect to the
scalar curvature) KK theories with toroidal compactifica-
tion [4, 5]. Here, a point-like mass contradicts the obser-
vations [4], but there are two classes of asymptotic latent
∗Electronic address: maxim.eingorn@gmail.com
†Electronic address: ai˙zhuk2@rambler.ru
solitons, which are in agreement with the observations at
the same level of accuracy as General Relativity [5]. The
main drawback of latent solitons consists in tension in
the internal spaces. It is very difficult to provide a rea-
sonable physical explanation for such equation of state
for ordinary astrophysical objects, e.g., for our Sun.
Then, we considered KK models with spherical com-
pactification of the internal space [6, 7]. Here, the back-
ground metrics is not flat because the internal space (e.g.,
the two-sphere) is curved. To create such background,
we need to introduce a background matter source, e.g.,
in the form of a perfect fluid. A point-like mass disturbs
this background.We demonstrate that there are two
different models with and without a bare cosmological
constant. In the former case, a point-like mass contra-
dicts the observations [6]. However, in the latter case,
the perturbed metric coefficients have the Yukawa type
corrections with respect to the usual Newtonian gravi-
tational potential [7]. These corrections are negligible
in the Solar system. Therefore, the considered model
satisfies the gravitational tests. In the present paper, we
provide an explanation of why some models with toroidal
and spherical compactifications failed with the observa-
tions, but others are in good agreement with the experi-
mental data.
The paper is structured as follows. In Sec. II, we con-
sider linear models with toroidal compactification of the
internal spaces and different types of gravitating sources.
These models are generalized to non-linear f(R) ones in
Sec. III. In Sec. IV, we investigate models with spherical
compactification of the internal space. The main results
are summarized in the concluding Sec. V.

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II. LINEAR MODELS WITH TOROIDAL
COMPACTIFICATION
In this section we analyze linear with respect to the
scalar curvature KK models with toroidal compactifica-
tion of the internal spaces.
A. Point-like mass
First, we investigate a model with a point-like massive
source. We consider a weak-field limit. It means that the
gravitational field is weak, i.e. the metrics is only slightly
perturbed from its flat space-time value:
gik≈ ηik+ hik. (1)
Here, the metric perturbations hik∼ O(1/c2), where c is
the speed of light, i,k = 0,1,...,D, and D is a total num-
ber of the spatial dimensions. In the weak-field limit, the
only non-zero component of the energy-momentum ten-
sor for a point-like mass at rest is T00≈ ρDc2∼ O(c2).
ρDis a D-dimensional rest mass density, and for a point-
like mass m we have ρD = mδ(rD). Usually, we deal
with the case of matter sources, which are uniformly
smeared over the extra dimensions [9].
the metric coefficients may depend only on coordinates
of the external space1. For the smeared extra dimen-
sions, the non-relativistic three-dimensional mass density
ρ3 is connected with the D-dimensional one as follows:
ρD= ρ3/VD′ = mδ(r3)/VD′, where D′= D−3 is a total
number of the extra dimensions and VD′ is a total volume
of the unperturbed internal spaces. For example, if aiare
periods of tori, then VD′ =?D′
In this case,
i=1ai. For such setup, the
Einstein equation
Rik=2SD˜GD
c4
?
Tik−
1
D − 1gikT
?
,(2)
where SD = 2πD/2/Γ(D/2) is a total solid angle (a
surface area of the (D − 1)-dimensional sphere of the
unit radius) and˜GDis the gravitational constant in the
(D = D+1)-dimensional space-time, is reduced to a sys-
tem of linearized equations with the following non-zero
1For the smeared extra dimensions, KK modes are absent. Our
following analysis can be easily generalized to the case of non-
smeared extra dimensions. In this case, KK modes are present in
considered models. However, in the Solar system, they are neg-
ligible compared with the zero mode [1], and such generalization
does not change the main conclusions of our paper.
solutions [1]:
h00 = −2(D − 2)
D − 1
2
D − 1
2
D − 1
2GNm
c2r3
2GNm
c2r3
2GNm
c2r3
, (3)
hαα = −
,α = 1,2,3, (4)
hµµ = −,µ = 4,5,...,D, (5)
where we introduce the Newton’s gravitational constant
GN=SD
4π
˜GD
VD′. (6)
Hereafter, the Latin indices i,k = 0,...,D, the Greek
indices α,β = 1,2,3 and the Greek indices µ,ν =
4,5,...,D.
It is well known that to satisfy the gravitational ex-
periments (the deflection of light, the time delay of radar
echoes) at the same level of accuracy as General Relativ-
ity, the metric coefficients h00 and hαα should coincide
with each other. However, Eqs. (3) and (4) show that
for the considered model h00/hαα= D − 2 and this ra-
tio does not depend on the size of the internal space.
So, we can not make it equal to unity. On the other
hand, h00 defines the non-relativistic gravitational po-
tential: h00= 2ϕ/c2. For example, in General Relativity
˜h00 = 2ϕN/c2= −2GNm/?c2r3
a prefactor 2(D − 2)/(D − 1). What is the reason for
this prefactor and for inequality between h00 and hαα?
It can be easily seen that we can rewrite h00and hααin
the following form:
?
and˜h00 =˜hαα. In
our case, the Newtonian gravitational potential acquires
h00 =˜h00+1
2δVD′ ,(7)
hαα =˜hαα−1
2δVD′ ,(8)
where δVD′ is a relative value of the internal space volume
variation:
δVD′ =
D
?
µ=4
hµµ= −2(D − 3)
D − 1
2GNm
c2r3
.(9)
Eqs. (7) and (8) demonstrate explicitly that the prefactor
2(D−2)/(D−1) in h00as well as inequality between h00
and hααoriginate from the admixture of the variation of
the internal space volume. Comparing δVD′ with 2ϕN/c2
shows that it coincides with the Newtonian expression
up to the prefactor 2(D − 3)/(D − 1) and satisfies the
equation similar to the Poisson one. Therefore, in this
model, the variation of the internal space volume results
in the fifth force. This is the reason of the contradictions
with the experimental data.
B.Latent solitons, black strings and black branes
Above, we consider the case of a point-like mass with
the dust-like equations of state in all spatial dimensions.

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However, there is a class of exact soliton solutions (see,
e.g., [2, 3]) with non-zero equations of state in the extra
dimensions:
1
VD′ρ3(r3)c2≈ ωµT00,
Tµµ≈ ωµ
µ = 4,5,...,D. (10)
These solutions are defined by the parameters γµ, which
are connected with the equation of state parameters ωµ
[3]:
ωµ=γµ− 1 + τ
2 − τ
, (11)
where τ =?D
these solutions read [3]:
µ=4γµ.
In the weak-field limit, the metric perturbations for
h00 = −2(D − 2)
D − 1
2
D − 1
2
D − 1
2GNm
c2r3
2GNm
c2r3
2GNm
c2r3
−
2Ω
D − 1
2Ω
D − 1
− 2
2GNm
c2r3
2GNm
c2r3
Ω
D − 1
, (12)
hαα = −
+,(13)
hµµ = −
?
ωµ−
?2GNm
c2r3
,
(14)
where Ω =
Eqs. (12)-(14) are due to non-zero equations of state in
the extra dimensions. From (14), for the variation of the
internal space volume we have
?D
µ=4ωµ. Obviously, the second terms in
δVD′ =
D
?
µ=4
hµµ= −2
?D − 3
D − 1+
2Ω
D − 1
?2GNm
c2r3
, (15)
which shows that the variation arises due to two effects.
The first term is due to the extra dimensions (D > 3),
and the second term originates from non-zero equations
of state in the extra dimensions (Ω ?= 0). There is a very
interesting special class of solutions, where both of these
effects cancel each other:
Ω = −D − 3
We call these solutions latent solitons. In this case, the
internal space volume is constant:
2
. (16)
δVD′ = 0, (17)
and h00 and hαα exactly coincide with the Newtonian
expressions and with each other: h00 =˜h00 = hαα =
˜hαα. Black stings (D = 4) and black branes (D > 4)
are particular cases of the latent solitons with the same
equations of state ωµ = −1/2 in all extra dimensions.
For these particular cases, each hµµ= 0, i.e. each scale
factor of the internal spaces is constant. However, in the
general case of the latent solitons, to be in agreement
with the observations it is sufficient to hold constant the
total volume of the internal spaces. In this case the fifth
force is absent. Since Ω < 0, all or some of ωµ should
be negative, i.e. such equations of state correspond to
tension.
III. NON-LINEAR MODELS WITH TOROIDAL
COMPACTIFICATION
In this section we analyze non-linear f(R) KK models
with toroidal compactification of the internal spaces. In
the case of non-zero equations of state (10) in the extra
dimensions, solutions (up to O(1/c2)) read [5]
h00 = −2(D − 2)
D − 1
4a
D − 1R,
2
D − 1
4a
D − 1R,
2
D − 1
4a
D − 1R,
2GNm
c2r3
−
2Ω
D − 1
2GNm
c2r3
−
(18)
hαα = −
2GNm
c2r3
+
2Ω
D − 1
2GNm
c2r3
+
(19)
hµµ = −
2GNm
c2r3
− 2
?
ωµ−
Ω
D − 1
?2GNm
c2r3
+
(20)
where a ≡ (1/2)f′′(0) and the scalar curvature is
R =1 − Ω
2aD
2GNm
c2r3
exp
?
−
?D − 1
4|a|D
?1/2
r3
?
. (21)
It is clear that the second terms in (18)-(20) take place
due to the non-zero equations of state in the extra dimen-
sions (ωµ,Ω ?= 0) and the third terms originate from the
non-linearity of the model (a ?= 0). The Eq. (21) shows
that non-linearity generates the Yukawa interaction with
the mass [(D − 1)/(4|a|D)]1/2[4].
A.Point-like mass
Let us first consider the case of a point-like matter
source at rest, i.e. with the dust-like equation of state
in all spatial dimensions: ωµ = 0, µ = 4,5,...,D, ⇒
Ω = 0. In this case, the second terms in Eqs. (18)-(20)
disappear and we arrive at equations of the subsection
IIA with the admixture of the Yukawa terms. It can be
easily seen that it is impossible to make constant the total
volume of the internal spaces due to the Yukawa term in
the Eq. (20). In other words, the effect of non-linearity
can not reduce δVD′ to zero. Similar to the linear case,
the perturbations of the internal spaces (the first term in
the Eq. (20)) result in the fifth force, which leads to the
contradiction to the observational data [4].
B.Asymptotic latent solitons, black strings and
black branes
Now, we want to consider solutions, which are in agree-
ment with the gravitational tests (the deflection of light
and the time delay of radar echoes). In the case of the

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linear models, it takes place for the latent solitons. That
is we should take into account tension in the internal
spaces: Ω ?= 0. Unfortunately, it is impossible to get
exact soliton solutions in the case of an arbitrary func-
tion f(R). Therefore, in the paper [5], it was proposed
two types of asymptotic solutions, where h00=˜h00and
hαα =˜hαα
⇒
solitons exist in the regions r3≫
Let us consider these two regions separately.
h00 = hαα. These asymptotic latent
?|a| and r3≪
?|a|.
1.
r3 ≫?|a|
In this asymptotic region, the exponent in (21) is neg-
ligible and we can drop the third terms in Eqs. (18)-(20).
That is the effect of non-linearity is negligibly small and
we arrive back at the case of the subsection IIB. Here,
Ω = −(D − 3)/2 and δVD′ = 0 in the Eq. (15), because
the effects of multidimensionality (the fifth force effect)
and tension in the extra dimensions cancel each other.
2.
r3 ≪
?|a|
In this case, we can replace the exponent in (21) by
unity. Here, the effect of non-linearity is not negligible.
After substitution (21) into (18) and (19) we get
h00 = −2GNm
c2r3
?
1 +D − 3
D − 1+
2Ω
D − 1+2(1 − Ω)
D(D − 1)
?
,
(22)
?
(23)
hαα = −2GNm
c2r3
?
1 −D − 3
D − 1−
2Ω
D − 1−2(1 − Ω)
D(D − 1)
,
where we have split the first terms in (18) and (19) into
the Newtonian expression (the first terms in (22) and
(23)) and the fifth force contribution (the second terms).
The third and fourth terms appear due to the combined
effect of tension in the extra dimensions and non-linearity
of the model.
It can be easily seen that for
Ω = −D − 2
2
(24)
the fifth force contribution is eliminated by the combined
effect of tension and non-linearity. For this value of Ω, we
have h00=˜h00and hαα=˜hαα ⇒ h00= hαα. An inter-
esting feature of this case is that the value of the internal
spaces total volume is not constant and its variation up
to a sign coincides with the Newtonian expression:
δVD′ =
D
?
µ=4
hµµ=2GNm
c2r3
. (25)
To understand the reason why we have agreement with
the experiments for non-zero δVD′, we consider the sum
(25) in more detail:
1
2
D
?
6Ω(D − 1) + (D − 2)2+ D + 2
2D(D − 1)
where the first term arises due to the fifth force contri-
bution (see (9)) and the second term is the combined
effect of tension and non-linearity, from which we have
singled out 1/2. It can be easily verified that exactly
for Ω = −(D − 2)/2, the combined effect of tension and
non-linearity eliminates the fifth force contribution.
µ=4
hµµ=2GNm
c2r3
?
−D − 3
D − 1
−
+1
2
?
, (26)
IV. SPHERICAL COMPACTIFICATION OF
THE INTERNAL SPACE
In this section we consider a model with spherical com-
pactification of the internal space, where the background
metrics is defined on a product manifold M4×M2. Here,
M4 describes the external four-dimensional flat space-
time and M2 corresponds to a two-dimensional sphere
with the radius (the internal space scale factor) a. To
create such space-time with the curved internal space,
we should introduce a background matter. As we have
shown in our papers [6, 7], this matter simulates a perfect
fluid with the vacuum-like equation of state in the exter-
nal/our space. In the internal space (the two-sphere) the
parameter of equation of state is
ω1=
Λ6
1/[(2S5˜G6/c4)a2] − Λ6
, (27)
where Λ6 is a bare multidimensional cosmological con-
stant. Different forms of matter can simulate such per-
fect fluid. For example, ω1= 1 and ω1= 2 correspond
to the monopole form-fields (the Freund-Rubin scheme
of compactification) and the Casimir effect, respectively.
In the case Λ6= 0 we get the dust-like equation of state
ω1= 0. It is worth noting that for ω1> 0 the internal
space is stabilized [7]. In this model, the Eq. (6) takes
the form 4πGN = S5˜G6/(4πa2), where we take into ac-
count that the unperturbed volume of the internal space
VD′ ≡ V2= 4πa2. The background metrics and matter
are perturbed by a point-like massive source with dust-
like equations of state in all spatial dimensions. In the
case ω1> 0 solutions (up to O(1/c2)) read [7]
h00 = −2GNm
hαα = −2GNm
c2r3
+1
2δV2,
−1
(28)
c2r3
2δV2,(29)
where the relative value of the conformal variation of the
volume of the two-sphere is
δV2= −2GNm
c2r3
exp
?
−
√ω1
a
r3
?
. (30)

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Therefore, this conformal variation generates the Yukawa
interaction with the mass squared ω1/a2. Obviously, the
admixture of such interaction to h00and hααis negligible
for sufficiently large Yukawa mass. Exactly this situation
takes place for the gravitational tests in the Solar system
[7]. Here, h00 = hαα with very high accuracy, and we
achieve a good agreement with the gravitational tests for
the considered model.
Obviously, models with ω1≤ 0 do not satisfy the ex-
perimental data. For example, if ω1= 0, then Eqs. (28)
and (29) exactly reduce to Eqs. (7) and (8) for D = 5
[6]. Here, the fifth force spoils the picture.
V. CONCLUSION
In this paper we investigated the cause of failure with
the classical gravitational tests for models with toroidal
compactification of the internal spaces. Similar to Gen-
eral Relativity, the matter source is taken in the form of
a point-like mass at rest. In this case T00is the only non-
zero component of the energy-momentum tensor. In the
language of a perfect fluid, it means the dust-like equa-
tion of state in all spatial dimensions. We have shown
that the perturbation of the volume of the internal spaces
generates the fifth force. This is the reason of the prob-
lem. To avoid it, it is necessary to eliminate the admix-
ture of the fifth force to the non-relativistic gravitational
potential. It happens when perturbations of the volume
of the internal spaces is exactly equal to zero: δVD′ = 0.
To achieve it, we should introduce tension in the inter-
nal spaces. For example, in the case of latent solitons
(in particular, black strings and black branes) tension
exactly eliminates the fifth force contribution.
In the case of non-linear f(R) models with toroidal
compactification we again need tension to be in agree-
ment with the observations. Here, we found two differ-
ent types of asymptotic latent solitons.
reproduces the results of the linear models, where ten-
sion cancels the fifth force and the internal space volume
is constant. For the second type of soliton solutions the
combined effect of tension and non-linearity eliminates
the fifth force contribution. However, the variation of
the internal space volume is not vanishing but coincides
up to a sign with the Newtonian gravitational potential.
As we noted in the introduction, it is difficult to give
a reasonable physical explanation for the emergence of
tension. This is the main drawback of the models with
toroidal compactification of the internal spaces.
In the case of spherical compactification of the internal
space the situation is different. Here, instead of tension,
we need a background matter (e.g., monopole form-fields,
the Casimir effect, etc.), which stabilizes the internal
space. Then, the conformal perturbations of the internal
space generate the Yukawa interaction. For large Yukawa
masses, the effect of this interaction is negligibly small,
and the considered model satisfies the gravitational tests
at the same level of accuracy as General Relativity. That
is the case for our model in the Solar system.
One of them
ACKNOWLEDGEMENTS
This work was supported in part by the ”Cosmomicro-
physics” programme of the Physics and Astronomy Di-
vision of the National Academy of Sciences of Ukraine.
A. Zh. acknowledges the hospitality of the Theory Di-
vision of CERN where a part of this work was carried
out.
References
[1] M. Eingorn and A. Zhuk, Class. Quant. Grav. 27, 205014
(2010); (arXiv:gr-qc/1003.5690).
[2] M. Eingorn and A. Zhuk, Phys. Rev. D 83, 044005 (2011);
(arXiv:gr-qc/1010.5740).
[3] M. Eingorn, O. de Medeiros, L. Crispino and A. Zhuk,
Phys. Rev. D 84, 024031 (2011); (arXiv:gr-qc/1101.3910).
[4] M. Eingorn and A. Zhuk, Phys. Rev. D 84, 024023 (2011);
(arXiv:gr-qc/1104.1456).
[5] M. Eingorn and A. Zhuk, Asymptotic latent solitons,
black strings and black branes in f(R)-gravity; (arXiv:gr-
qc/1112.1539).
[6] A. Chopovsky, M. Eingorn and A. Zhuk, Weak-field limit
of Kaluza-Klein models with spherical compactification I:
problematic aspects; (arXiv:gr-qc/1107.3387).
[7] A. Chopovsky, M. Eingorn and A. Zhuk, Weak-field limit
of Kaluza-Klein models with spherical compactification II:
agreement with the observations; (arXiv:gr-qc/1107.3388).
[8] L.D. Landau and E.M. Lifshitz, The Classical Theory of
Fields, Fourth Edition: Volume 2 (Course of Theoretical
Physics Series), (Pergamon Press, Oxford, 2000).
[9] M. Eingorn and A. Zhuk, Class. Quant. Grav. 27, 055002
(2010); (arXiv:gr-qc/0910.3507).