Research Article Kaluza-Klein Multidimensional Models with ...
7
Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 106135, 6 pages http://dx.doi.org/10.1155/2013/106135 Research Article Kaluza-Klein Multidimensional Models with Ricci-Flat Internal Spaces: The Absence of the KK Particles Alexey Chopovsky, 1,2 Maxim Eingorn, 3 and Alexander Zhuk 2 1 Department of eoretical Physics, Odessa National University, Dvoryanskaya Street 2, Odessa 65082, Ukraine 2 Astronomical Observatory, Odessa National University, Dvoryanskaya Street 2, Odessa 65082, Ukraine 3 North Carolina Central University, CREST and NASA Research Centers, Fayetteville Street 1801, Durham, NC 27707, USA Correspondence should be addressed to Maxim Eingorn; [email protected] Received 31 October 2013; Accepted 10 December 2013 Academic Editor: Douglas Singleton Copyright © 2013 Alexey Chopovsky et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a multidimensional Kaluza-Klein (KK) model with a Ricci-flat internal space, for example, a Calabi-Yau manifold. We perturb this background metrics by a system of gravitating masses, for example, astrophysical objects such as our Sun. We suppose that these masses are pressureless in the external space but they have relativistic pressure in the internal space. We show that metric perturbations do not depend on coordinates of the internal space and gravitating masses should be uniformly smeared over the internal space. is means, first, that KK modes corresponding to the metric fluctuations are absent and, second, particles should be only in the ground quantum state with respect to the internal space. In our opinion, these results look very unnatural. According to statistical physics, any nonzero temperature should result in fluctuations, that is, in KK modes. We also get formulae for the metric correction terms which enable us to calculate the gravitational tests: the deflection of light, the time-delay of the radar echoes, and the perihelion advance. 1. Introduction It may turn out that our Universe is multidimensional and has more than three spatial dimensions. is fantastic idea was first proposed in the twenties of the last century in the papers by Kaluza and Klein [1, 2], and since then it has consistently attracted attention of researchers. Obviously, to be compat- ible with our observable four-dimensional spacetime, the extra dimensions should be compact and sufficiently small. According to the recent accelerator experiments, their size should be of the order or less than the Fermi length 10 −17 cm. (In the brane-world models, the extra dimensions can be significantly larger [3, 4]. We do not consider such models in our paper.) If the internal space is very small, the hope to detect them is connected with Kaluza-Klein (KK) particles which correspond to appropriate eigenfunctions of the inter- nal manifold, that is, excitations of fields in the given internal space. Such excitations were investigated in many articles (see, e.g., the classical papers [5–7]). Quite recently, KK particles were considered, for example, in the papers [8, 9]. e internal compact space can have different topology. e Ricci-flat space is of special interest because Calabi- Yau manifolds, which are widely used in the superstring theory [10], belong to this class. A particular case of toroidal compactification was considered in our previous paper [11]. We demonstrate in this paper that metric perturbations depend only on coordinates of the external spacetime; that is, the corresponding KK particles are absent. In our approach, the metric perturbations are caused by a system of gravitating masses, for example, by astrophysical objects. is is the main difference between our investigation and studies in [8, 9] where the metric and form-field perturbations are considered without taking into account the reason of such fluctuations. Our analysis in [11] clearly shows that the inclusion of the matter sources, being responsible for the perturbations, imposes strong restrictions on the model. In the present paper, we investigate this important phenomenon in the case of Ricci-flat internal spaces and arrive again at the same conclusions. at is, KK modes corresponding to the metric fluctuations are absent. We also show that gravitating
Transcript of Research Article Kaluza-Klein Multidimensional Models with ...
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 106135 6 pageshttpdxdoiorg1011552013106135
Research ArticleKaluza-Klein Multidimensional Models with Ricci-FlatInternal Spaces The Absence of the KK Particles
Alexey Chopovsky12 Maxim Eingorn3 and Alexander Zhuk2
1 Department of Theoretical Physics Odessa National University Dvoryanskaya Street 2 Odessa 65082 Ukraine2 Astronomical Observatory Odessa National University Dvoryanskaya Street 2 Odessa 65082 Ukraine3 North Carolina Central University CREST and NASA Research Centers Fayetteville Street 1801 Durham NC 27707 USA
Correspondence should be addressed to Maxim Eingorn maximeingorngmailcom
Received 31 October 2013 Accepted 10 December 2013
Academic Editor Douglas Singleton
Copyright copy 2013 Alexey Chopovsky et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We consider a multidimensional Kaluza-Klein (KK) model with a Ricci-flat internal space for example a Calabi-Yau manifold Weperturb this background metrics by a system of gravitating masses for example astrophysical objects such as our Sun We supposethat these masses are pressureless in the external space but they have relativistic pressure in the internal space We show that metricperturbations do not depend on coordinates of the internal space and gravitating masses should be uniformly smeared over theinternal spaceThismeans first that KKmodes corresponding to themetric fluctuations are absent and second particles should beonly in the ground quantum state with respect to the internal space In our opinion these results look very unnatural According tostatistical physics any nonzero temperature should result in fluctuations that is in KKmodes We also get formulae for the metriccorrection terms which enable us to calculate the gravitational tests the deflection of light the time-delay of the radar echoes andthe perihelion advance
1 Introduction
Itmay turn out that ourUniverse ismultidimensional and hasmore than three spatial dimensions This fantastic idea wasfirst proposed in the twenties of the last century in the papersby Kaluza and Klein [1 2] and since then it has consistentlyattracted attention of researchers Obviously to be compat-ible with our observable four-dimensional spacetime theextra dimensions should be compact and sufficiently smallAccording to the recent accelerator experiments their sizeshould be of the order or less than the Fermi length 10minus17 cm(In the brane-world models the extra dimensions can besignificantly larger [3 4] We do not consider such modelsin our paper) If the internal space is very small the hope todetect them is connected with Kaluza-Klein (KK) particleswhich correspond to appropriate eigenfunctions of the inter-nal manifold that is excitations of fields in the given internalspace Such excitations were investigated in many articles(see eg the classical papers [5ndash7]) Quite recently KKparticles were considered for example in the papers [8 9]
The internal compact space can have different topologyThe Ricci-flat space is of special interest because Calabi-Yau manifolds which are widely used in the superstringtheory [10] belong to this class A particular case of toroidalcompactification was considered in our previous paper [11]We demonstrate in this paper that metric perturbationsdepend only on coordinates of the external spacetime that isthe corresponding KK particles are absent In our approachthemetric perturbations are caused by a system of gravitatingmasses for example by astrophysical objects This is themain difference between our investigation and studies in[8 9] where the metric and form-field perturbations areconsidered without taking into account the reason of suchfluctuations Our analysis in [11] clearly shows that theinclusion of the matter sources being responsible for theperturbations imposes strong restrictions on the model Inthe present paper we investigate this important phenomenonin the case of Ricci-flat internal spaces and arrive again at thesame conclusions That is KK modes corresponding to themetric fluctuations are absent We also show that gravitating
2 Advances in High Energy Physics
masses should be uniformly smeared over the internal spaceFrom the point of quantum mechanics this means thatparticles must be only in the ground state with respect to theinternal space Of course it looks very unnatural Accordingto statistical physics any nonzero temperature should resultin fluctuations that is in KK states
The paper is structured as follows In Section 2 weconsider the multidimensional model with the Ricci-flatinternal space This background spacetime is perturbed bythe system of gravitating masses We demonstrate that themetric perturbations do not depend on the coordinates of theinternal space and gravitating masses are uniformly smearedover the internal space That is KK modes are absent InSection 3 we provide the exact expressions for the metriccorrection terms The main results are briefly summarized inconcluding Section 4
2 Smeared Extra Dimensions
We consider a block-diagonal background metrics
[119892(D)119872119873
(119910)] = [120578(4)120583] ] oplus [119892
(119889)
119898119899
(119910)] (1)
which is determined on (D = 1 + 119863 = 4 + 119889)-dimensionalproduct manifold MD = M
4
times M119889
Here 120578(4)120583] (120583 ] =
0 1 2 3) is the metrics of the four-dimensional Minkowskispacetime and 119892(119889)
119898119899
(119910) (119898 119899 = 4 119863) is the metrics ofsome 119889-dimensional internal (usually compact) Ricci-flatspace with coordinates 119910 equiv 119910119898
(119889)119898119899
[119892(119889)119898119899
] = 0 (2)
In what follows 119883119872 (119872 = 0 1 2 119863) are coordinatesonMD 119909
120583 = 119883120583 (120583 = 0 1 2 3) are coordinates onM4
and119910119898 = 119883119898 (119898 = 4 5 3 + 119889) are coordinates on M
119889
Wewill also use the definitions for the spatial coordinates 119883 119883119872 ( = 1 2 119863) and 119909 119909120583 = 119883120583 (120583 = 1 2 3)
Now we perturb the metrics (1) by a system of119873 discretemassive (with rest masses119898
119901
119901 = 1 119873) bodies We sup-pose that the pressure of these bodies in the external three-dimensional space is much less than their energy densityThis is a natural approximation for ordinary astrophysicalobjects such as our Sun For example in general relativitythis approach works well for calculating the gravitationalexperiments in the solar system [12] In other words thegravitating bodies are pressureless in the externalour spaceOn the other hand we suppose that theymay have pressure inthe extra dimensions Therefore nonzero components of theenergy-momentum tensor of the system can be written in thefollowing form for a perfect fluid [11 12]
119879119872] = 1205881198882 1198891199041198891199090119906119872119906] 119906119872 = 119889119883
119872
119889119904
119879119898119899 = minus119901119898
119892119898119899 + 1205881198882 1198891199041198891199090119906119898119906119899
(3)
with the equation of state (EoS) in the internal space
119901119898
= 1205961205881198882 1198891199041198891199090 119898 = 4 5 119863 (4)
where 120596 is the parameter of EoS (we should note that thepressure (4) is the intrinsic pressure of a gravitating mass inthe extra dimensions and is not connected with its motionin other words 120596 is the parameter of a particle) and the rest-mass density is
120588 equiv119873
sum119901=1
[10038161003816100381610038161003816119892(D)10038161003816100381610038161003816]minus12
119898119901
120575 (X minus X119901
) (5)
where X119901
is a 119863-dimensional ldquoradius vectorrdquo of the 119901thparticle In EoS we include the factor 1198891199041198891199090 just forconvenience We can also eliminate this factor from EoSand include it in the definition of the rest-mass density (5)(see eg [11]) For our purpose it is sufficient to consider asteady-state model (For example we can consider just onegravitating mass and place the origin of a reference frame onit)That is we disregard the spatial velocities of the gravitatingmasses
In the weak-field approximation the perturbed metriccomponents can be written in the following form
119892119872119873
asymp 119892119872119873
+ ℎ119872119873
119892119872119873 asymp 119892119872119873 minus ℎ119872119873 (6)
where ℎ119872119873
sim 119874(11198882) and ℎ119873119872
equiv 119892119873119879ℎ119872119879
Hereafter theldquohatsrdquo denote the unperturbed quantities For example themetric coefficients in (3)ndash(5) are already perturbed quantities
To get the metric correction terms we should solve(in the corresponding order) the multidimensional Einsteinequation
119877119872119873
=2119878119863
119866D1198884
[119879119872119873
minus 1119863 minus 1
119879119892119872119873
] (7)
where 119878119863
= 21205871198632Γ(1198632) is the total solid angle (the surfacearea of the (119863 minus 1)-dimensional sphere of the unit radius)and 119866D is the gravitational constant in the (D = 119863 + 1)-dimensional spacetime For our model and up to the119874(11198882)terms the Einstein equation (7) is reduced to the followingsystem of equations (see [11] for details)
11987700
[119892(D)] asymp minus12nabla
119871
nabla119871ℎ00
=2119878119863
119866D1198882
(119863 minus 2 + 120596119889119863 minus 1
)120588
= 1 2 119863
(8)
119877120583] [119892(D)] asymp minus1
2nabla
119871
nabla119871ℎ120583] =
2119878119863
119866D1198882
(1 minus 120596119889119863 minus 1
)120588120575120583]
120583 ] = 1 2 3(9)
119877119898119899
[119892(D)] asymp minus12nabla
119871
nabla119871ℎ119898119899
= minus2119878119863
119866D1198882
(120596 (119863 minus 1) + 1 minus 120596119889119863 minus 1
)120588119892119898119899
(10)
where the Ricci-tensor components 119877119872119873
[119892(D)] are deter-mined by (A11)
Obviously the solution of this system is determined bythe rest-mass density 120588(119883) Let us suppose for example
Advances in High Energy Physics 3
that the physically reasonable solution ℎ00
(119883) of (8) exists(The exact form of such solution depends on the topologyof the internal space For example in the case of toroidalinternal spaces with the periodic boundary conditions inthe directions of the extra dimensions the correspondingsolutions can be found in [13 14]However aswewill see laterthe exact form of 119863-dimensional solutions is not importantfor our purpose A similar situation took place in our paper[15] where this point was discussed in detail after Equation(256)) Then we can write it in the form
ℎ00
(119883) equiv2120593 (119883)1198882 (11)
It is clear that the function 120593(119883) describes the nonrela-tivistic gravitational potential created by the system ofmasseswith the rest-mass density (5) From (9) and (10) we get
ℎ120583 ] =
1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
120575120583 ]
ℎ119898119899
= minus120596 (119863 minus 1) + 1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
119892119898119899
(12)
It is worth noting that now we can easily determine (viathe gravitational potential 120593(119883)) the functions 120585
1
1205852
and 1205853
introduced in (A10)It is well known (see eg [16]) that to be in agreement
with the gravitational tests (the deflection of light and theShapiro time-delay experiment) at the same level of accuracyas general relativity the metric correction terms shouldsatisfy the equalities
ℎ00
= ℎ11
= ℎ22
= ℎ33
(13)
This results in the black stringbrane condition [17 18]
120596 = minus12 (14)
Thus to be in agreement with observations this parame-ter should not be negligibly small that is 120596 sim 119874(1)
The solutions (11)-(12) should satisfy the gauge condition(A7) Taking into account that ℎ = 119892119872119873ℎ
119872119873
= [2(120596119889 minus1)(119863 minus 2 + 120596119889)]ℎ
00
we can easily get that for 119873 = ] (] =0 1 2 3) this condition is automatically satisfied However inthe case119873 = 119899 (119899 = 4 5 119863) we arrive at the condition
nabla119871
119892119871119879ℎ119879119899
minus 12120597119899
ℎ = minus 120596 (119863 minus 1)119863 minus 2 + 120596119889
120597119899
ℎ00
= 0 (15)
Because 120596 = 0 to satisfy this equation we must demand that
120597119899
ℎ00
= 21198882120597119899
120593 (119883) = 0 (16)
Hence the gravitational potential is the function of thespatial coordinates of the externalour space 120593(119883) = 120593(119909)Obviously it takes place only if the rest-mass density alsodepends only on the spatial coordinates of the external space120588(119883) = 120588(119909) In other words the gravitating masses should
be uniformly smeared over the internal space In short weusually call such internal spaces ldquosmearedrdquo extra dimensions
This conclusion has the following important effect Sup-pose that we have solved for the considered particle amultidi-mensional quantumSchrodinger equation and found itswavefunction Ψ(119883) In general this function depends on allspatial coordinates 119883 = (119909 119910) and we can expand it inappropriate eigenfunctions of the compact internal spacethat is in the Kaluza-Klein (KK) modes The ground statecorresponds to the absence of these particles In this state thewave function may depend only on the coordinates 119909 of theexternal spaceThe classical rest-mass density is proportionalto the probability density |Ψ|2 Therefore the demand thatthe rest-mass density depends only on the coordinates of theexternal space means that the particle can be only in theground quantum state and KK excitations are absent Thislooks very unnatural from the viewpoint of quantum andstatistical physics
3 Metric Components
Now we want to get the expression for the metrics up to 11198882terms Since 1199090 = 119888119905 the component 119892
00
should be calculatedup to 119874(11198884)
11989200
asymp 12057800
+ ℎ00
+ 11989100
(17)
where ℎ00
sim 119874(11198882) and 11989100
sim 119874(11198884) For the rest metriccoefficients we still have (6) 119892
119872
119873
asymp 119892
119872
119873
+ ℎ
119872
119873
ℎ
119872
119873
sim119874(11198882) We recall that we consider the steady-state case1198920
119872
= 0According to the results of the previous section the
gravitating masses are uniformly smeared over the internalspace Therefore the rest-mass density (5) now reads
120588 = 1119881119889
119873
sum119901=1
119898119901
120575 (x minus x119901
)
radicminus119892(4)= 1119881119889
9848583
radicminus119892(4) (18)
where 119881119889
= intM119889119889119889119910radic|119892(119889)| is the volume of the internal
space and
9848583
(119909) equiv119873
sum119901=1
119898119901
120575 (x minus x119901
) (19)
Here x is a three-dimensional radius vector in the externalspace Additionally it is clear that all prefactors 120585
1
1205852
and 1205853
in (A10) are also functions of the external spatial coordinates119909 For convenience we redetermine the function 120585
3
1205853
(119909) equiv120572(119909) Then ℎ
119898119899
(119909 119910) = 120572(119909)119892119898119899
(119910) where 120572(119909) sim 119874(11198882)To get the metric correction terms we should solve the
Einstein equation (7) First we consider the 119874(11198882) termsℎ119872119873
These terms satisfy the system (8)ndash(10) where we mustperform the following replacements 120588 rarr 984858
3
119889
where 119889
=intM119889119889119889119910radic|119892(119889)| is the volume of the unperturbed internal
4 Advances in High Energy Physics
space and minusnabla
119871
nabla119871 rarr Δ = 120575120583]1205972120597119909120583120597119909] Introducing thedefinition ℎ
00
equiv 21205931198882 we can easily get from this system
120593 = minus119866119873
119873
sum119901=1
119898119901
10038161003816100381610038161003816x minus x11990110038161003816100381610038161003816 (20)
where
119866119873
= 119878119863 (119863 minus 2 + 120596119889)2120587119889
(119863 minus 1)119866D (21)
is theNewtonian gravitational constant and120593(119909) is the nonre-lativistic gravitational potential created by 119873 masses At thesame time
ℎ120583 ] =
1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
120575120583 ]
ℎ119898119899
(119909 119910) = 120572 (119909) 119892119898119899
(119910)
120572 = minus120596 (119863 minus 1) + 1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
(22)
Now we intend to determine the 11989100
sim 119874(11198884) correc-tion term Obviously to do it we need to solve only the 00-component of the Einstein equation (7) Keeping in mind theprefactor 11198884 in the right hand side of (7) it is clear thatthe energy-momentum tensor component 119879
00
and the trace119879 must be determined up to 119874(1) Therefore similar to thepaper [11] we get
11987900
asymp9848583
1198882
119889
[1 + 3119863 minus 4 + 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
]
119879 asymp9848583
1198882
119889
[1 + 119863 minus 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
] (1 minus 120596119889)
(23)
To get the Ricci-tensor component 11987700
up to 119874(11198884) wecan use (B2) and Equations (229) (238) and (239) in [15]and arrive at the following formula
11987700
asymp 12Δ(11989100
minus 211988841205932) + 1
1198882Δ120593
+ 119863 minus 1119863 minus 2 + 120596119889
21198884120593Δ120593
(24)
where Δ = 120575120583]1205972120597119909120583120597119909] is the 3-dimensional Laplace oper-ator Therefore the 00-component of the Einstein equationreads
12Δ(11989100
minus 211988841205932) = 1
1198884120593Δ120593
=4120587119866119873
1198884119873
sum119901=1
119898119901
1205931015840 (x119901
) 120575 (x minus x119901
) (25)
where we used the equation Δ120593 = 4120587119866119873
sum119873119901=1
119898119901
120575(x minus x119901
)The function 1205931015840(x
119901
) is the potential of the gravitational field
at a point with the radius vector x119901
produced by all particlesexcept the 119901th
1205931015840 (x119901
) = sum119902 = 119901
120593119902
(x119901
minus x119902
)
Δ120593119901
(x minus x119901
) = 4120587119866119873
119898119901
120575 (x minus x119901
) (26)
Therefore the solution of (25) is
11989100
(x) = 211988841205932 (x) + 2
1198884119873
sum119901=1
120593119901
(x minus x119901
) 1205931015840 (x119901
) (27)
Obviously in the case of the toroidal internal spacethis section reproduces the results of our paper [11] Thefound metric correction terms enable us to calculate thegravitational tests (the deflection of light the time-delay ofthe radar echoes and the perihelion advance) in full analogywith the paper [18]
4 Conclusion
In our paper we have studied the multidimensional KKmodel with the Ricci-flat internal space Such models areof special interest because Calabi-Yau manifolds which arewidely used in the superstring theory [10] belong to this classIn the absence of matter sources the background manifold isa direct product of the four-dimensional Minkowski space-time and the Ricci-flat internal space Then we perturbedthis background metrics by a system of gravitating massesfor example astrophysical objects such as our Sun It is wellknown that pressure inside these objects is much less thanthe energy density Therefore we supposed that gravitatingmasses are pressureless in the external space Howeverthey may have relativistic pressure in the internal spaceMoreover the presence of such pressure is the necessarycondition for agreement with the gravitational tests (thedeflection of light the time-delay of the radar echoes and theperihelion advance) [16ndash18] We clearly demonstrated thatmetric perturbations do not depend on the coordinates of theinternal space and gravitating masses are uniformly smearedover the internal space This means first that KK modescorresponding to the metric fluctuations are absent andsecond the particles corresponding to the gravitating massesshould be only in the ground quantum state with respectto the internal space Of course it looks very unnaturalAccording to statistical physics any nonzero temperatureshould result in fluctuations that is in KK modes On theother hand the conclusion about the absence of KK particlesis consistent with the results of [19 20] where a mechanismfor the symmetries formation of the internal spaces wasproposed According to this mechanism due to the entropydecreasing in the compact subspace its metric undergoesthe process of symmetrization during some time after itsquantum nucleation A high symmetry of internal spacesactually means suppression of the extra degrees of freedom
We also obtained the formulae for the metric correctionterms For the 00-component we got it up to 119874(11198884) Wefound other components up to 119874(11198882) This enables us to
Advances in High Energy Physics 5
calculate the gravitational tests the deflection of light thetime-delay of the radar echoes and the perihelion advanceIt also gives a possibility to construct the Lagrange functionfor the considered many-body system (see eg [11])
Appendices
A Ricci Tensor in the Weak-FieldApproximation
In our paper we use the signature of the metrics (+ minus minusminus ) and determine the Ricci-tensor components as follows
119877119872119873
equiv 119877119878119872119878119873
= 120597119871
Γ119871119872119873
minus 120597119873
Γ119871119872119871
+ Γ119871119872119873
Γ119878119871119878
minus Γ119878119872119871
Γ119871119873119878
(A1)
where the Christoffel symbols are
Γ119871119872119873
= 12119892119871119879 (minus120597
119879
119892119872119873
+ 120597119872
119892119873119879
+ 120597119873
119892119872119879
)
119872119873 119871 = 0 1 2 119863(A2)
The metrics 119892(D)119872119873
equiv 119892119872119873
is determined on (D = 1 + 119863)-dimensional manifoldMD
Now we suppose that the metric coefficients 119892119872119873
cor-respond to the perturbed metrics and in the weak-fieldapproximation they can be decomposed as
119892119872119873
asymp 119892119872119873
+ ℎ119872119873
119892119872119873 asymp 119892119872119873 minus ℎ119872119873 (A3)
where 119892119872119873
is the background unperturbed metrics and themetric perturbations ℎ
119872119873
sim 119874(11198882) (119888 is the speed oflight) As usual ℎ119873
119872
equiv 119892119873119879ℎ119879119872
Here and after the ldquohatsrdquodenote unperturbed quantities Then up to linear terms ℎthe Christoffel symbols read
Γ119871119872119873
asymp Γ119871119872119873
+ 12119892119871119879 [minus120597
119879
ℎ119872119873
+ (120597119872
ℎ119873119879
minus Γ119875119872119873
ℎ119875119879
)
+ (120597119873
ℎ119872119879
minus Γ119875119873119872
ℎ119875119879
)]
= Γ119871119872119873
+ 12119892119871119879 [minusnabla
119879
ℎ119872119873
+ nabla119872
ℎ119873119879
+ nabla119873
ℎ119872119879
]
(A4)
and for the Ricci tensor we get
119877119872119873
asymp 119872119873
+ 12(minusnabla119871
nabla119871ℎ119872119873
+ 119876119872119873
) (A5)
where we introduced a new tensor
119876119872119873
equiv nabla119871
nabla119872
ℎ119871119873
+ nabla119871
nabla119873
ℎ119871119872
minus nabla119873
nabla119872
ℎ119871119871
= [nabla119872
(nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
) + nabla119873
(nabla119871
ℎ119871119872
minus 12120597119872
ℎ119871119871
)]
minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A6)
To get the last expression we used the relation (918) in[12]Thegauge conditions (see eg the paragraph 105 in [12])
nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
= 0 (A7)
simplify considerably this expression
119876119872119873
= minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A8)
Now we suppose that the manifold MD is a productmanifoldMD =M4timesM119889 with a block-diagonal backgroundmetrics of the form (1) However in this appendix we do notassume that the internal manifoldM
119889
is a Ricci-flat one Forthe metrics (1) and an arbitrary internal space M
119889
the onlynonzero components of the Riemann tensor
119871119873119872119875
[119892(D)]and the Ricci tensor
119873119872
[119892(D)] are
119897119899119898119901
[119892(D)] = 119897119899119898119901
[119892(119889)]
119898119899
[119892(D)] = 119898119899
[119892(119889)] (A9)
Typically the matter which perturbs the backgroundmetrics is taken in the form of a perfect fluid with isotropicpressure In the case of the product manifold the pressureis isotropic in each factor manifold (see eg the energy-momentum tensor (3)) Obviously such perturbation doesnot change the topology of the model for example thetopologies of the factor manifolds in the case of the productmanifold Additionally it preserves also the block-diagonalstructure of the metric tensor In the case of a steady-statemodel (our case) the nondiagonal perturbations ℎ
0
119872
( =1 2 3 119863) are also absentTherefore themetric correctionterms are conformal to the background metrics
ℎ00
= 1205851
119892(D)00
= 1205851
12057800
ℎ120583] = 1205852119892
(D)
120583]= minus1205852
120575120583] 120583 ] = 1 2 3
ℎ119898119899
= 1205853
119892(D)119898119899
= 1205853
119892(119889)119898119899
(A10)
where 120585123
are some scalar functions of all spatial coordinates119883 It can be easily verified that under these conditions (it isworth noting that as it follows from (A8)119876
119872119873
is also equalto zero when the following condition holds 119871
119873119875119872
= minus119871119872119875119873
resulting in the Ricci-flat space
119872119873
= 0 (but not vice versa))the tensor119876
119872119873
in (A8) is equal to zero119876119872119873
= 0Thereforeup to 119874(11198882) terms we get
119877119872119873
[119892(D)] asymp 119872119873
[119892(D)] minus 12nabla119871
nabla119871ℎ119872119873
= 119872119873
[119892(D)] minus 12nabla
119871
nabla119871ℎ119872119873
+ 119900 ( 11198882)
(A11)
where = 1 2 119863 and we took into account that1198830 = 119888119905
B Ricci Tensor for a Block-Diagonal Metrics
In this appendix we consider a block-diagonal metrics
[119892(D)119872119873
(119909 119910)] = [119892(1198891)120583] (119909)] oplus [119892
(1198892)
119898119899
(119909 119910)]
= [119892(1198891)120583] (119909)] oplus [119890
120572(119909)119892(1198892)119898119899
(119910)] (B1)
6 Advances in High Energy Physics
Then for the Ricci-tensor components we get the follow-ing expressions
119877120583] [119892119872119873] = 119877120583] [119892
(1198891)
120582120591
] minus 11988922[nabla(1198891)120583
(120597]120572) +12(120597120583
120572) (120597]120572)](B2)
119877119898] [119892119872119873] = 0 (B3)
119877119898119899
[119892119872119873
] = 119877119898119899
[119892(1198892)119896119897
] minus 12119890120572(119909)119892(1198892)
119898119899
119892(1198891)120582120591
times [nabla(1198891)120582
(120597120591
120572) + 11988922(120597120582
120572) (120597120591
120572)] (B4)
Acknowledgments
This work was supported by the ldquoCosmomicrophysics-2rdquoprogramme of the Physics and Astronomy Division of theNational Academy of Sciences of Ukraine The work ofMaxim Eingorn was supported by NSF CREST Award HRD-0833184 and NASA Grant NNX09AV07A
References
[1] T Kaluza ldquoOn the problem of unity in physicsrdquo Sitzungsberichteder Koniglich Preussischen Akademie der Wissenschaften zuBerlin vol 1921 pp 966ndash972 1921
[2] O Klein ldquoQuantum theory and five-dimensional theory ofrelativityrdquoZeitschrift fur Physik vol 37 no 12 pp 895ndash906 1926
[3] NArkani-Hamed SDimopoulos andGDvali ldquoThehierarchyproblem and new dimensions at a millimeterrdquo Physics Letters Bvol 429 no 3-4 pp 263ndash272 1998
[4] I Antoniadis N Arkani-Hamed S Dimopoulos and G DvalildquoNew dimensions at a millimeter to a fermi and superstrings ata TeVrdquo Physics Letters B vol 436 no 3-4 pp 257ndash263 1998
[5] A Salam and J Strathdee ldquoOn Kaluza-Klein theoryrdquo Annals ofPhysics vol 141 no 2 pp 316ndash352 1982
[6] P van Nieuwenhuizen ldquoThe complete mass spectrum of 119889 = 11supergravity compactified on S
4
and a general mass formula forarbitrary cosets M
4
rdquo Classical and Quantum Gravity vol 2 no1 pp 1ndash20 1985
[7] H J Kim L J Romans and P van Nieuwenhuizen ldquoThe massspectrum of chiral 119873 = 2119863 = 10 supergravity on 1198785rdquo PhysicalReview D vol 32 no 2 pp 389ndash399 1985
[8] K Hinterbichler J Levin and C Zukowski ldquoKaluza-Kleintowers on generalmanifoldsrdquo httparxivorgabs13106353
[9] A R Brown and A Dahlen ldquoStability and spectrum of com-pactifications on product manifoldsrdquo httparxivorgabs13106360
[10] M B Green J H Schwarz and E Witten Superstring TheoryLoop Amplitudes Anomalies and Phenomenology vol 2 ofCambridge Monographs on Mathematical Physics CambridgeUniversity Press Cambridge UK 1987
[11] A Chopovsky M Eingorn and A Zhuk ldquoMany-body prob-lem in Kaluza-Klein models with toroidal compactificationrdquoaccepted in European Physical Journal C
[12] L D Landau and E M Lifshitz ldquoThe classical theory of fieldsrdquovol 2 of Course of Theoretical Physics Pergamon Press OxfordUK 4th edition 2000
[13] M Eingorn and A Zhuk ldquoMultidimensional gravity in non-relativistic limitrdquo Physical Review D vol 80 no 12 Article ID124037 5 pages 2009
[14] M Eingorn andA Zhuk ldquoNon-relativistic limit ofmultidimen-sional gravity exact solutions and applicationsrdquo Classical andQuantum Gravity vol 27 no 5 Article ID 055002 p 17 2010
[15] M Eingorn and A Zhuk ldquoClassical tests of multidimensionalgravity negative resultrdquo Classical and Quantum Gravity vol 27no 20 Article ID 205014 p 18 2010
[16] M Eingorn and A Zhuk ldquoRemarks on gravitational interactionin Kaluza-Klein modelsrdquo Physics Letters B vol 713 no 3 pp154ndash159 2012
[17] M Eingorn and A Zhuk ldquoKaluza-Klein models can we con-struct a viable examplerdquo Physical Review D vol 83 no 4 Arti-cle ID 044005 11 pages 2011
[18] M Eingorn O R De Medeiros L C B Crispino and A ZhukldquoLatent solitons black strings black branes and equations ofstate in Kaluza-Klein modelsrdquo Physical Review D vol 84 no 2Article ID 024031 8 pages 2011
[19] A A Kirillov A A Korotkevich and S G Rubin ldquoEmergenceof symmetriesrdquo Physics Letters B vol 718 no 2 pp 237ndash2402012
[20] K A Bronnikov and S G Rubin Black Holes Cosmology andExtra Dimensions World Scientific Singapore 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
2 Advances in High Energy Physics
masses should be uniformly smeared over the internal spaceFrom the point of quantum mechanics this means thatparticles must be only in the ground state with respect to theinternal space Of course it looks very unnatural Accordingto statistical physics any nonzero temperature should resultin fluctuations that is in KK states
The paper is structured as follows In Section 2 weconsider the multidimensional model with the Ricci-flatinternal space This background spacetime is perturbed bythe system of gravitating masses We demonstrate that themetric perturbations do not depend on the coordinates of theinternal space and gravitating masses are uniformly smearedover the internal space That is KK modes are absent InSection 3 we provide the exact expressions for the metriccorrection terms The main results are briefly summarized inconcluding Section 4
2 Smeared Extra Dimensions
We consider a block-diagonal background metrics
[119892(D)119872119873
(119910)] = [120578(4)120583] ] oplus [119892
(119889)
119898119899
(119910)] (1)
which is determined on (D = 1 + 119863 = 4 + 119889)-dimensionalproduct manifold MD = M
4
times M119889
Here 120578(4)120583] (120583 ] =
0 1 2 3) is the metrics of the four-dimensional Minkowskispacetime and 119892(119889)
119898119899
(119910) (119898 119899 = 4 119863) is the metrics ofsome 119889-dimensional internal (usually compact) Ricci-flatspace with coordinates 119910 equiv 119910119898
(119889)119898119899
[119892(119889)119898119899
] = 0 (2)
In what follows 119883119872 (119872 = 0 1 2 119863) are coordinatesonMD 119909
120583 = 119883120583 (120583 = 0 1 2 3) are coordinates onM4
and119910119898 = 119883119898 (119898 = 4 5 3 + 119889) are coordinates on M
119889
Wewill also use the definitions for the spatial coordinates 119883 119883119872 ( = 1 2 119863) and 119909 119909120583 = 119883120583 (120583 = 1 2 3)
Now we perturb the metrics (1) by a system of119873 discretemassive (with rest masses119898
119901
119901 = 1 119873) bodies We sup-pose that the pressure of these bodies in the external three-dimensional space is much less than their energy densityThis is a natural approximation for ordinary astrophysicalobjects such as our Sun For example in general relativitythis approach works well for calculating the gravitationalexperiments in the solar system [12] In other words thegravitating bodies are pressureless in the externalour spaceOn the other hand we suppose that theymay have pressure inthe extra dimensions Therefore nonzero components of theenergy-momentum tensor of the system can be written in thefollowing form for a perfect fluid [11 12]
119879119872] = 1205881198882 1198891199041198891199090119906119872119906] 119906119872 = 119889119883
119872
119889119904
119879119898119899 = minus119901119898
119892119898119899 + 1205881198882 1198891199041198891199090119906119898119906119899
(3)
with the equation of state (EoS) in the internal space
119901119898
= 1205961205881198882 1198891199041198891199090 119898 = 4 5 119863 (4)
where 120596 is the parameter of EoS (we should note that thepressure (4) is the intrinsic pressure of a gravitating mass inthe extra dimensions and is not connected with its motionin other words 120596 is the parameter of a particle) and the rest-mass density is
120588 equiv119873
sum119901=1
[10038161003816100381610038161003816119892(D)10038161003816100381610038161003816]minus12
119898119901
120575 (X minus X119901
) (5)
where X119901
is a 119863-dimensional ldquoradius vectorrdquo of the 119901thparticle In EoS we include the factor 1198891199041198891199090 just forconvenience We can also eliminate this factor from EoSand include it in the definition of the rest-mass density (5)(see eg [11]) For our purpose it is sufficient to consider asteady-state model (For example we can consider just onegravitating mass and place the origin of a reference frame onit)That is we disregard the spatial velocities of the gravitatingmasses
In the weak-field approximation the perturbed metriccomponents can be written in the following form
119892119872119873
asymp 119892119872119873
+ ℎ119872119873
119892119872119873 asymp 119892119872119873 minus ℎ119872119873 (6)
where ℎ119872119873
sim 119874(11198882) and ℎ119873119872
equiv 119892119873119879ℎ119872119879
Hereafter theldquohatsrdquo denote the unperturbed quantities For example themetric coefficients in (3)ndash(5) are already perturbed quantities
To get the metric correction terms we should solve(in the corresponding order) the multidimensional Einsteinequation
119877119872119873
=2119878119863
119866D1198884
[119879119872119873
minus 1119863 minus 1
119879119892119872119873
] (7)
where 119878119863
= 21205871198632Γ(1198632) is the total solid angle (the surfacearea of the (119863 minus 1)-dimensional sphere of the unit radius)and 119866D is the gravitational constant in the (D = 119863 + 1)-dimensional spacetime For our model and up to the119874(11198882)terms the Einstein equation (7) is reduced to the followingsystem of equations (see [11] for details)
11987700
[119892(D)] asymp minus12nabla
119871
nabla119871ℎ00
=2119878119863
119866D1198882
(119863 minus 2 + 120596119889119863 minus 1
)120588
= 1 2 119863
(8)
119877120583] [119892(D)] asymp minus1
2nabla
119871
nabla119871ℎ120583] =
2119878119863
119866D1198882
(1 minus 120596119889119863 minus 1
)120588120575120583]
120583 ] = 1 2 3(9)
119877119898119899
[119892(D)] asymp minus12nabla
119871
nabla119871ℎ119898119899
= minus2119878119863
119866D1198882
(120596 (119863 minus 1) + 1 minus 120596119889119863 minus 1
)120588119892119898119899
(10)
where the Ricci-tensor components 119877119872119873
[119892(D)] are deter-mined by (A11)
Obviously the solution of this system is determined bythe rest-mass density 120588(119883) Let us suppose for example
Advances in High Energy Physics 3
that the physically reasonable solution ℎ00
(119883) of (8) exists(The exact form of such solution depends on the topologyof the internal space For example in the case of toroidalinternal spaces with the periodic boundary conditions inthe directions of the extra dimensions the correspondingsolutions can be found in [13 14]However aswewill see laterthe exact form of 119863-dimensional solutions is not importantfor our purpose A similar situation took place in our paper[15] where this point was discussed in detail after Equation(256)) Then we can write it in the form
ℎ00
(119883) equiv2120593 (119883)1198882 (11)
It is clear that the function 120593(119883) describes the nonrela-tivistic gravitational potential created by the system ofmasseswith the rest-mass density (5) From (9) and (10) we get
ℎ120583 ] =
1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
120575120583 ]
ℎ119898119899
= minus120596 (119863 minus 1) + 1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
119892119898119899
(12)
It is worth noting that now we can easily determine (viathe gravitational potential 120593(119883)) the functions 120585
1
1205852
and 1205853
introduced in (A10)It is well known (see eg [16]) that to be in agreement
with the gravitational tests (the deflection of light and theShapiro time-delay experiment) at the same level of accuracyas general relativity the metric correction terms shouldsatisfy the equalities
ℎ00
= ℎ11
= ℎ22
= ℎ33
(13)
This results in the black stringbrane condition [17 18]
120596 = minus12 (14)
Thus to be in agreement with observations this parame-ter should not be negligibly small that is 120596 sim 119874(1)
The solutions (11)-(12) should satisfy the gauge condition(A7) Taking into account that ℎ = 119892119872119873ℎ
119872119873
= [2(120596119889 minus1)(119863 minus 2 + 120596119889)]ℎ
00
we can easily get that for 119873 = ] (] =0 1 2 3) this condition is automatically satisfied However inthe case119873 = 119899 (119899 = 4 5 119863) we arrive at the condition
nabla119871
119892119871119879ℎ119879119899
minus 12120597119899
ℎ = minus 120596 (119863 minus 1)119863 minus 2 + 120596119889
120597119899
ℎ00
= 0 (15)
Because 120596 = 0 to satisfy this equation we must demand that
120597119899
ℎ00
= 21198882120597119899
120593 (119883) = 0 (16)
Hence the gravitational potential is the function of thespatial coordinates of the externalour space 120593(119883) = 120593(119909)Obviously it takes place only if the rest-mass density alsodepends only on the spatial coordinates of the external space120588(119883) = 120588(119909) In other words the gravitating masses should
be uniformly smeared over the internal space In short weusually call such internal spaces ldquosmearedrdquo extra dimensions
This conclusion has the following important effect Sup-pose that we have solved for the considered particle amultidi-mensional quantumSchrodinger equation and found itswavefunction Ψ(119883) In general this function depends on allspatial coordinates 119883 = (119909 119910) and we can expand it inappropriate eigenfunctions of the compact internal spacethat is in the Kaluza-Klein (KK) modes The ground statecorresponds to the absence of these particles In this state thewave function may depend only on the coordinates 119909 of theexternal spaceThe classical rest-mass density is proportionalto the probability density |Ψ|2 Therefore the demand thatthe rest-mass density depends only on the coordinates of theexternal space means that the particle can be only in theground quantum state and KK excitations are absent Thislooks very unnatural from the viewpoint of quantum andstatistical physics
3 Metric Components
Now we want to get the expression for the metrics up to 11198882terms Since 1199090 = 119888119905 the component 119892
00
should be calculatedup to 119874(11198884)
11989200
asymp 12057800
+ ℎ00
+ 11989100
(17)
where ℎ00
sim 119874(11198882) and 11989100
sim 119874(11198884) For the rest metriccoefficients we still have (6) 119892
119872
119873
asymp 119892
119872
119873
+ ℎ
119872
119873
ℎ
119872
119873
sim119874(11198882) We recall that we consider the steady-state case1198920
119872
= 0According to the results of the previous section the
gravitating masses are uniformly smeared over the internalspace Therefore the rest-mass density (5) now reads
120588 = 1119881119889
119873
sum119901=1
119898119901
120575 (x minus x119901
)
radicminus119892(4)= 1119881119889
9848583
radicminus119892(4) (18)
where 119881119889
= intM119889119889119889119910radic|119892(119889)| is the volume of the internal
space and
9848583
(119909) equiv119873
sum119901=1
119898119901
120575 (x minus x119901
) (19)
Here x is a three-dimensional radius vector in the externalspace Additionally it is clear that all prefactors 120585
1
1205852
and 1205853
in (A10) are also functions of the external spatial coordinates119909 For convenience we redetermine the function 120585
3
1205853
(119909) equiv120572(119909) Then ℎ
119898119899
(119909 119910) = 120572(119909)119892119898119899
(119910) where 120572(119909) sim 119874(11198882)To get the metric correction terms we should solve the
Einstein equation (7) First we consider the 119874(11198882) termsℎ119872119873
These terms satisfy the system (8)ndash(10) where we mustperform the following replacements 120588 rarr 984858
3
119889
where 119889
=intM119889119889119889119910radic|119892(119889)| is the volume of the unperturbed internal
4 Advances in High Energy Physics
space and minusnabla
119871
nabla119871 rarr Δ = 120575120583]1205972120597119909120583120597119909] Introducing thedefinition ℎ
00
equiv 21205931198882 we can easily get from this system
120593 = minus119866119873
119873
sum119901=1
119898119901
10038161003816100381610038161003816x minus x11990110038161003816100381610038161003816 (20)
where
119866119873
= 119878119863 (119863 minus 2 + 120596119889)2120587119889
(119863 minus 1)119866D (21)
is theNewtonian gravitational constant and120593(119909) is the nonre-lativistic gravitational potential created by 119873 masses At thesame time
ℎ120583 ] =
1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
120575120583 ]
ℎ119898119899
(119909 119910) = 120572 (119909) 119892119898119899
(119910)
120572 = minus120596 (119863 minus 1) + 1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
(22)
Now we intend to determine the 11989100
sim 119874(11198884) correc-tion term Obviously to do it we need to solve only the 00-component of the Einstein equation (7) Keeping in mind theprefactor 11198884 in the right hand side of (7) it is clear thatthe energy-momentum tensor component 119879
00
and the trace119879 must be determined up to 119874(1) Therefore similar to thepaper [11] we get
11987900
asymp9848583
1198882
119889
[1 + 3119863 minus 4 + 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
]
119879 asymp9848583
1198882
119889
[1 + 119863 minus 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
] (1 minus 120596119889)
(23)
To get the Ricci-tensor component 11987700
up to 119874(11198884) wecan use (B2) and Equations (229) (238) and (239) in [15]and arrive at the following formula
11987700
asymp 12Δ(11989100
minus 211988841205932) + 1
1198882Δ120593
+ 119863 minus 1119863 minus 2 + 120596119889
21198884120593Δ120593
(24)
where Δ = 120575120583]1205972120597119909120583120597119909] is the 3-dimensional Laplace oper-ator Therefore the 00-component of the Einstein equationreads
12Δ(11989100
minus 211988841205932) = 1
1198884120593Δ120593
=4120587119866119873
1198884119873
sum119901=1
119898119901
1205931015840 (x119901
) 120575 (x minus x119901
) (25)
where we used the equation Δ120593 = 4120587119866119873
sum119873119901=1
119898119901
120575(x minus x119901
)The function 1205931015840(x
119901
) is the potential of the gravitational field
at a point with the radius vector x119901
produced by all particlesexcept the 119901th
1205931015840 (x119901
) = sum119902 = 119901
120593119902
(x119901
minus x119902
)
Δ120593119901
(x minus x119901
) = 4120587119866119873
119898119901
120575 (x minus x119901
) (26)
Therefore the solution of (25) is
11989100
(x) = 211988841205932 (x) + 2
1198884119873
sum119901=1
120593119901
(x minus x119901
) 1205931015840 (x119901
) (27)
Obviously in the case of the toroidal internal spacethis section reproduces the results of our paper [11] Thefound metric correction terms enable us to calculate thegravitational tests (the deflection of light the time-delay ofthe radar echoes and the perihelion advance) in full analogywith the paper [18]
4 Conclusion
In our paper we have studied the multidimensional KKmodel with the Ricci-flat internal space Such models areof special interest because Calabi-Yau manifolds which arewidely used in the superstring theory [10] belong to this classIn the absence of matter sources the background manifold isa direct product of the four-dimensional Minkowski space-time and the Ricci-flat internal space Then we perturbedthis background metrics by a system of gravitating massesfor example astrophysical objects such as our Sun It is wellknown that pressure inside these objects is much less thanthe energy density Therefore we supposed that gravitatingmasses are pressureless in the external space Howeverthey may have relativistic pressure in the internal spaceMoreover the presence of such pressure is the necessarycondition for agreement with the gravitational tests (thedeflection of light the time-delay of the radar echoes and theperihelion advance) [16ndash18] We clearly demonstrated thatmetric perturbations do not depend on the coordinates of theinternal space and gravitating masses are uniformly smearedover the internal space This means first that KK modescorresponding to the metric fluctuations are absent andsecond the particles corresponding to the gravitating massesshould be only in the ground quantum state with respectto the internal space Of course it looks very unnaturalAccording to statistical physics any nonzero temperatureshould result in fluctuations that is in KK modes On theother hand the conclusion about the absence of KK particlesis consistent with the results of [19 20] where a mechanismfor the symmetries formation of the internal spaces wasproposed According to this mechanism due to the entropydecreasing in the compact subspace its metric undergoesthe process of symmetrization during some time after itsquantum nucleation A high symmetry of internal spacesactually means suppression of the extra degrees of freedom
We also obtained the formulae for the metric correctionterms For the 00-component we got it up to 119874(11198884) Wefound other components up to 119874(11198882) This enables us to
Advances in High Energy Physics 5
calculate the gravitational tests the deflection of light thetime-delay of the radar echoes and the perihelion advanceIt also gives a possibility to construct the Lagrange functionfor the considered many-body system (see eg [11])
Appendices
A Ricci Tensor in the Weak-FieldApproximation
In our paper we use the signature of the metrics (+ minus minusminus ) and determine the Ricci-tensor components as follows
119877119872119873
equiv 119877119878119872119878119873
= 120597119871
Γ119871119872119873
minus 120597119873
Γ119871119872119871
+ Γ119871119872119873
Γ119878119871119878
minus Γ119878119872119871
Γ119871119873119878
(A1)
where the Christoffel symbols are
Γ119871119872119873
= 12119892119871119879 (minus120597
119879
119892119872119873
+ 120597119872
119892119873119879
+ 120597119873
119892119872119879
)
119872119873 119871 = 0 1 2 119863(A2)
The metrics 119892(D)119872119873
equiv 119892119872119873
is determined on (D = 1 + 119863)-dimensional manifoldMD
Now we suppose that the metric coefficients 119892119872119873
cor-respond to the perturbed metrics and in the weak-fieldapproximation they can be decomposed as
119892119872119873
asymp 119892119872119873
+ ℎ119872119873
119892119872119873 asymp 119892119872119873 minus ℎ119872119873 (A3)
where 119892119872119873
is the background unperturbed metrics and themetric perturbations ℎ
119872119873
sim 119874(11198882) (119888 is the speed oflight) As usual ℎ119873
119872
equiv 119892119873119879ℎ119879119872
Here and after the ldquohatsrdquodenote unperturbed quantities Then up to linear terms ℎthe Christoffel symbols read
Γ119871119872119873
asymp Γ119871119872119873
+ 12119892119871119879 [minus120597
119879
ℎ119872119873
+ (120597119872
ℎ119873119879
minus Γ119875119872119873
ℎ119875119879
)
+ (120597119873
ℎ119872119879
minus Γ119875119873119872
ℎ119875119879
)]
= Γ119871119872119873
+ 12119892119871119879 [minusnabla
119879
ℎ119872119873
+ nabla119872
ℎ119873119879
+ nabla119873
ℎ119872119879
]
(A4)
and for the Ricci tensor we get
119877119872119873
asymp 119872119873
+ 12(minusnabla119871
nabla119871ℎ119872119873
+ 119876119872119873
) (A5)
where we introduced a new tensor
119876119872119873
equiv nabla119871
nabla119872
ℎ119871119873
+ nabla119871
nabla119873
ℎ119871119872
minus nabla119873
nabla119872
ℎ119871119871
= [nabla119872
(nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
) + nabla119873
(nabla119871
ℎ119871119872
minus 12120597119872
ℎ119871119871
)]
minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A6)
To get the last expression we used the relation (918) in[12]Thegauge conditions (see eg the paragraph 105 in [12])
nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
= 0 (A7)
simplify considerably this expression
119876119872119873
= minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A8)
Now we suppose that the manifold MD is a productmanifoldMD =M4timesM119889 with a block-diagonal backgroundmetrics of the form (1) However in this appendix we do notassume that the internal manifoldM
119889
is a Ricci-flat one Forthe metrics (1) and an arbitrary internal space M
119889
the onlynonzero components of the Riemann tensor
119871119873119872119875
[119892(D)]and the Ricci tensor
119873119872
[119892(D)] are
119897119899119898119901
[119892(D)] = 119897119899119898119901
[119892(119889)]
119898119899
[119892(D)] = 119898119899
[119892(119889)] (A9)
Typically the matter which perturbs the backgroundmetrics is taken in the form of a perfect fluid with isotropicpressure In the case of the product manifold the pressureis isotropic in each factor manifold (see eg the energy-momentum tensor (3)) Obviously such perturbation doesnot change the topology of the model for example thetopologies of the factor manifolds in the case of the productmanifold Additionally it preserves also the block-diagonalstructure of the metric tensor In the case of a steady-statemodel (our case) the nondiagonal perturbations ℎ
0
119872
( =1 2 3 119863) are also absentTherefore themetric correctionterms are conformal to the background metrics
ℎ00
= 1205851
119892(D)00
= 1205851
12057800
ℎ120583] = 1205852119892
(D)
120583]= minus1205852
120575120583] 120583 ] = 1 2 3
ℎ119898119899
= 1205853
119892(D)119898119899
= 1205853
119892(119889)119898119899
(A10)
where 120585123
are some scalar functions of all spatial coordinates119883 It can be easily verified that under these conditions (it isworth noting that as it follows from (A8)119876
119872119873
is also equalto zero when the following condition holds 119871
119873119875119872
= minus119871119872119875119873
resulting in the Ricci-flat space
119872119873
= 0 (but not vice versa))the tensor119876
119872119873
in (A8) is equal to zero119876119872119873
= 0Thereforeup to 119874(11198882) terms we get
119877119872119873
[119892(D)] asymp 119872119873
[119892(D)] minus 12nabla119871
nabla119871ℎ119872119873
= 119872119873
[119892(D)] minus 12nabla
119871
nabla119871ℎ119872119873
+ 119900 ( 11198882)
(A11)
where = 1 2 119863 and we took into account that1198830 = 119888119905
B Ricci Tensor for a Block-Diagonal Metrics
In this appendix we consider a block-diagonal metrics
[119892(D)119872119873
(119909 119910)] = [119892(1198891)120583] (119909)] oplus [119892
(1198892)
119898119899
(119909 119910)]
= [119892(1198891)120583] (119909)] oplus [119890
120572(119909)119892(1198892)119898119899
(119910)] (B1)
6 Advances in High Energy Physics
Then for the Ricci-tensor components we get the follow-ing expressions
119877120583] [119892119872119873] = 119877120583] [119892
(1198891)
120582120591
] minus 11988922[nabla(1198891)120583
(120597]120572) +12(120597120583
120572) (120597]120572)](B2)
119877119898] [119892119872119873] = 0 (B3)
119877119898119899
[119892119872119873
] = 119877119898119899
[119892(1198892)119896119897
] minus 12119890120572(119909)119892(1198892)
119898119899
119892(1198891)120582120591
times [nabla(1198891)120582
(120597120591
120572) + 11988922(120597120582
120572) (120597120591
120572)] (B4)
Acknowledgments
This work was supported by the ldquoCosmomicrophysics-2rdquoprogramme of the Physics and Astronomy Division of theNational Academy of Sciences of Ukraine The work ofMaxim Eingorn was supported by NSF CREST Award HRD-0833184 and NASA Grant NNX09AV07A
References
[1] T Kaluza ldquoOn the problem of unity in physicsrdquo Sitzungsberichteder Koniglich Preussischen Akademie der Wissenschaften zuBerlin vol 1921 pp 966ndash972 1921
[2] O Klein ldquoQuantum theory and five-dimensional theory ofrelativityrdquoZeitschrift fur Physik vol 37 no 12 pp 895ndash906 1926
[3] NArkani-Hamed SDimopoulos andGDvali ldquoThehierarchyproblem and new dimensions at a millimeterrdquo Physics Letters Bvol 429 no 3-4 pp 263ndash272 1998
[4] I Antoniadis N Arkani-Hamed S Dimopoulos and G DvalildquoNew dimensions at a millimeter to a fermi and superstrings ata TeVrdquo Physics Letters B vol 436 no 3-4 pp 257ndash263 1998
[5] A Salam and J Strathdee ldquoOn Kaluza-Klein theoryrdquo Annals ofPhysics vol 141 no 2 pp 316ndash352 1982
[6] P van Nieuwenhuizen ldquoThe complete mass spectrum of 119889 = 11supergravity compactified on S
4
and a general mass formula forarbitrary cosets M
4
rdquo Classical and Quantum Gravity vol 2 no1 pp 1ndash20 1985
[7] H J Kim L J Romans and P van Nieuwenhuizen ldquoThe massspectrum of chiral 119873 = 2119863 = 10 supergravity on 1198785rdquo PhysicalReview D vol 32 no 2 pp 389ndash399 1985
[8] K Hinterbichler J Levin and C Zukowski ldquoKaluza-Kleintowers on generalmanifoldsrdquo httparxivorgabs13106353
[9] A R Brown and A Dahlen ldquoStability and spectrum of com-pactifications on product manifoldsrdquo httparxivorgabs13106360
[10] M B Green J H Schwarz and E Witten Superstring TheoryLoop Amplitudes Anomalies and Phenomenology vol 2 ofCambridge Monographs on Mathematical Physics CambridgeUniversity Press Cambridge UK 1987
[11] A Chopovsky M Eingorn and A Zhuk ldquoMany-body prob-lem in Kaluza-Klein models with toroidal compactificationrdquoaccepted in European Physical Journal C
[12] L D Landau and E M Lifshitz ldquoThe classical theory of fieldsrdquovol 2 of Course of Theoretical Physics Pergamon Press OxfordUK 4th edition 2000
[13] M Eingorn and A Zhuk ldquoMultidimensional gravity in non-relativistic limitrdquo Physical Review D vol 80 no 12 Article ID124037 5 pages 2009
[14] M Eingorn andA Zhuk ldquoNon-relativistic limit ofmultidimen-sional gravity exact solutions and applicationsrdquo Classical andQuantum Gravity vol 27 no 5 Article ID 055002 p 17 2010
[15] M Eingorn and A Zhuk ldquoClassical tests of multidimensionalgravity negative resultrdquo Classical and Quantum Gravity vol 27no 20 Article ID 205014 p 18 2010
[16] M Eingorn and A Zhuk ldquoRemarks on gravitational interactionin Kaluza-Klein modelsrdquo Physics Letters B vol 713 no 3 pp154ndash159 2012
[17] M Eingorn and A Zhuk ldquoKaluza-Klein models can we con-struct a viable examplerdquo Physical Review D vol 83 no 4 Arti-cle ID 044005 11 pages 2011
[18] M Eingorn O R De Medeiros L C B Crispino and A ZhukldquoLatent solitons black strings black branes and equations ofstate in Kaluza-Klein modelsrdquo Physical Review D vol 84 no 2Article ID 024031 8 pages 2011
[19] A A Kirillov A A Korotkevich and S G Rubin ldquoEmergenceof symmetriesrdquo Physics Letters B vol 718 no 2 pp 237ndash2402012
[20] K A Bronnikov and S G Rubin Black Holes Cosmology andExtra Dimensions World Scientific Singapore 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 3
that the physically reasonable solution ℎ00
(119883) of (8) exists(The exact form of such solution depends on the topologyof the internal space For example in the case of toroidalinternal spaces with the periodic boundary conditions inthe directions of the extra dimensions the correspondingsolutions can be found in [13 14]However aswewill see laterthe exact form of 119863-dimensional solutions is not importantfor our purpose A similar situation took place in our paper[15] where this point was discussed in detail after Equation(256)) Then we can write it in the form
ℎ00
(119883) equiv2120593 (119883)1198882 (11)
It is clear that the function 120593(119883) describes the nonrela-tivistic gravitational potential created by the system ofmasseswith the rest-mass density (5) From (9) and (10) we get
ℎ120583 ] =
1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
120575120583 ]
ℎ119898119899
= minus120596 (119863 minus 1) + 1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
119892119898119899
(12)
It is worth noting that now we can easily determine (viathe gravitational potential 120593(119883)) the functions 120585
1
1205852
and 1205853
introduced in (A10)It is well known (see eg [16]) that to be in agreement
with the gravitational tests (the deflection of light and theShapiro time-delay experiment) at the same level of accuracyas general relativity the metric correction terms shouldsatisfy the equalities
ℎ00
= ℎ11
= ℎ22
= ℎ33
(13)
This results in the black stringbrane condition [17 18]
120596 = minus12 (14)
Thus to be in agreement with observations this parame-ter should not be negligibly small that is 120596 sim 119874(1)
The solutions (11)-(12) should satisfy the gauge condition(A7) Taking into account that ℎ = 119892119872119873ℎ
119872119873
= [2(120596119889 minus1)(119863 minus 2 + 120596119889)]ℎ
00
we can easily get that for 119873 = ] (] =0 1 2 3) this condition is automatically satisfied However inthe case119873 = 119899 (119899 = 4 5 119863) we arrive at the condition
nabla119871
119892119871119879ℎ119879119899
minus 12120597119899
ℎ = minus 120596 (119863 minus 1)119863 minus 2 + 120596119889
120597119899
ℎ00
= 0 (15)
Because 120596 = 0 to satisfy this equation we must demand that
120597119899
ℎ00
= 21198882120597119899
120593 (119883) = 0 (16)
Hence the gravitational potential is the function of thespatial coordinates of the externalour space 120593(119883) = 120593(119909)Obviously it takes place only if the rest-mass density alsodepends only on the spatial coordinates of the external space120588(119883) = 120588(119909) In other words the gravitating masses should
be uniformly smeared over the internal space In short weusually call such internal spaces ldquosmearedrdquo extra dimensions
This conclusion has the following important effect Sup-pose that we have solved for the considered particle amultidi-mensional quantumSchrodinger equation and found itswavefunction Ψ(119883) In general this function depends on allspatial coordinates 119883 = (119909 119910) and we can expand it inappropriate eigenfunctions of the compact internal spacethat is in the Kaluza-Klein (KK) modes The ground statecorresponds to the absence of these particles In this state thewave function may depend only on the coordinates 119909 of theexternal spaceThe classical rest-mass density is proportionalto the probability density |Ψ|2 Therefore the demand thatthe rest-mass density depends only on the coordinates of theexternal space means that the particle can be only in theground quantum state and KK excitations are absent Thislooks very unnatural from the viewpoint of quantum andstatistical physics
3 Metric Components
Now we want to get the expression for the metrics up to 11198882terms Since 1199090 = 119888119905 the component 119892
00
should be calculatedup to 119874(11198884)
11989200
asymp 12057800
+ ℎ00
+ 11989100
(17)
where ℎ00
sim 119874(11198882) and 11989100
sim 119874(11198884) For the rest metriccoefficients we still have (6) 119892
119872
119873
asymp 119892
119872
119873
+ ℎ
119872
119873
ℎ
119872
119873
sim119874(11198882) We recall that we consider the steady-state case1198920
119872
= 0According to the results of the previous section the
gravitating masses are uniformly smeared over the internalspace Therefore the rest-mass density (5) now reads
120588 = 1119881119889
119873
sum119901=1
119898119901
120575 (x minus x119901
)
radicminus119892(4)= 1119881119889
9848583
radicminus119892(4) (18)
where 119881119889
= intM119889119889119889119910radic|119892(119889)| is the volume of the internal
space and
9848583
(119909) equiv119873
sum119901=1
119898119901
120575 (x minus x119901
) (19)
Here x is a three-dimensional radius vector in the externalspace Additionally it is clear that all prefactors 120585
1
1205852
and 1205853
in (A10) are also functions of the external spatial coordinates119909 For convenience we redetermine the function 120585
3
1205853
(119909) equiv120572(119909) Then ℎ
119898119899
(119909 119910) = 120572(119909)119892119898119899
(119910) where 120572(119909) sim 119874(11198882)To get the metric correction terms we should solve the
Einstein equation (7) First we consider the 119874(11198882) termsℎ119872119873
These terms satisfy the system (8)ndash(10) where we mustperform the following replacements 120588 rarr 984858
3
119889
where 119889
=intM119889119889119889119910radic|119892(119889)| is the volume of the unperturbed internal
4 Advances in High Energy Physics
space and minusnabla
119871
nabla119871 rarr Δ = 120575120583]1205972120597119909120583120597119909] Introducing thedefinition ℎ
00
equiv 21205931198882 we can easily get from this system
120593 = minus119866119873
119873
sum119901=1
119898119901
10038161003816100381610038161003816x minus x11990110038161003816100381610038161003816 (20)
where
119866119873
= 119878119863 (119863 minus 2 + 120596119889)2120587119889
(119863 minus 1)119866D (21)
is theNewtonian gravitational constant and120593(119909) is the nonre-lativistic gravitational potential created by 119873 masses At thesame time
ℎ120583 ] =
1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
120575120583 ]
ℎ119898119899
(119909 119910) = 120572 (119909) 119892119898119899
(119910)
120572 = minus120596 (119863 minus 1) + 1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
(22)
Now we intend to determine the 11989100
sim 119874(11198884) correc-tion term Obviously to do it we need to solve only the 00-component of the Einstein equation (7) Keeping in mind theprefactor 11198884 in the right hand side of (7) it is clear thatthe energy-momentum tensor component 119879
00
and the trace119879 must be determined up to 119874(1) Therefore similar to thepaper [11] we get
11987900
asymp9848583
1198882
119889
[1 + 3119863 minus 4 + 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
]
119879 asymp9848583
1198882
119889
[1 + 119863 minus 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
] (1 minus 120596119889)
(23)
To get the Ricci-tensor component 11987700
up to 119874(11198884) wecan use (B2) and Equations (229) (238) and (239) in [15]and arrive at the following formula
11987700
asymp 12Δ(11989100
minus 211988841205932) + 1
1198882Δ120593
+ 119863 minus 1119863 minus 2 + 120596119889
21198884120593Δ120593
(24)
where Δ = 120575120583]1205972120597119909120583120597119909] is the 3-dimensional Laplace oper-ator Therefore the 00-component of the Einstein equationreads
12Δ(11989100
minus 211988841205932) = 1
1198884120593Δ120593
=4120587119866119873
1198884119873
sum119901=1
119898119901
1205931015840 (x119901
) 120575 (x minus x119901
) (25)
where we used the equation Δ120593 = 4120587119866119873
sum119873119901=1
119898119901
120575(x minus x119901
)The function 1205931015840(x
119901
) is the potential of the gravitational field
at a point with the radius vector x119901
produced by all particlesexcept the 119901th
1205931015840 (x119901
) = sum119902 = 119901
120593119902
(x119901
minus x119902
)
Δ120593119901
(x minus x119901
) = 4120587119866119873
119898119901
120575 (x minus x119901
) (26)
Therefore the solution of (25) is
11989100
(x) = 211988841205932 (x) + 2
1198884119873
sum119901=1
120593119901
(x minus x119901
) 1205931015840 (x119901
) (27)
Obviously in the case of the toroidal internal spacethis section reproduces the results of our paper [11] Thefound metric correction terms enable us to calculate thegravitational tests (the deflection of light the time-delay ofthe radar echoes and the perihelion advance) in full analogywith the paper [18]
4 Conclusion
In our paper we have studied the multidimensional KKmodel with the Ricci-flat internal space Such models areof special interest because Calabi-Yau manifolds which arewidely used in the superstring theory [10] belong to this classIn the absence of matter sources the background manifold isa direct product of the four-dimensional Minkowski space-time and the Ricci-flat internal space Then we perturbedthis background metrics by a system of gravitating massesfor example astrophysical objects such as our Sun It is wellknown that pressure inside these objects is much less thanthe energy density Therefore we supposed that gravitatingmasses are pressureless in the external space Howeverthey may have relativistic pressure in the internal spaceMoreover the presence of such pressure is the necessarycondition for agreement with the gravitational tests (thedeflection of light the time-delay of the radar echoes and theperihelion advance) [16ndash18] We clearly demonstrated thatmetric perturbations do not depend on the coordinates of theinternal space and gravitating masses are uniformly smearedover the internal space This means first that KK modescorresponding to the metric fluctuations are absent andsecond the particles corresponding to the gravitating massesshould be only in the ground quantum state with respectto the internal space Of course it looks very unnaturalAccording to statistical physics any nonzero temperatureshould result in fluctuations that is in KK modes On theother hand the conclusion about the absence of KK particlesis consistent with the results of [19 20] where a mechanismfor the symmetries formation of the internal spaces wasproposed According to this mechanism due to the entropydecreasing in the compact subspace its metric undergoesthe process of symmetrization during some time after itsquantum nucleation A high symmetry of internal spacesactually means suppression of the extra degrees of freedom
We also obtained the formulae for the metric correctionterms For the 00-component we got it up to 119874(11198884) Wefound other components up to 119874(11198882) This enables us to
Advances in High Energy Physics 5
calculate the gravitational tests the deflection of light thetime-delay of the radar echoes and the perihelion advanceIt also gives a possibility to construct the Lagrange functionfor the considered many-body system (see eg [11])
Appendices
A Ricci Tensor in the Weak-FieldApproximation
In our paper we use the signature of the metrics (+ minus minusminus ) and determine the Ricci-tensor components as follows
119877119872119873
equiv 119877119878119872119878119873
= 120597119871
Γ119871119872119873
minus 120597119873
Γ119871119872119871
+ Γ119871119872119873
Γ119878119871119878
minus Γ119878119872119871
Γ119871119873119878
(A1)
where the Christoffel symbols are
Γ119871119872119873
= 12119892119871119879 (minus120597
119879
119892119872119873
+ 120597119872
119892119873119879
+ 120597119873
119892119872119879
)
119872119873 119871 = 0 1 2 119863(A2)
The metrics 119892(D)119872119873
equiv 119892119872119873
is determined on (D = 1 + 119863)-dimensional manifoldMD
Now we suppose that the metric coefficients 119892119872119873
cor-respond to the perturbed metrics and in the weak-fieldapproximation they can be decomposed as
119892119872119873
asymp 119892119872119873
+ ℎ119872119873
119892119872119873 asymp 119892119872119873 minus ℎ119872119873 (A3)
where 119892119872119873
is the background unperturbed metrics and themetric perturbations ℎ
119872119873
sim 119874(11198882) (119888 is the speed oflight) As usual ℎ119873
119872
equiv 119892119873119879ℎ119879119872
Here and after the ldquohatsrdquodenote unperturbed quantities Then up to linear terms ℎthe Christoffel symbols read
Γ119871119872119873
asymp Γ119871119872119873
+ 12119892119871119879 [minus120597
119879
ℎ119872119873
+ (120597119872
ℎ119873119879
minus Γ119875119872119873
ℎ119875119879
)
+ (120597119873
ℎ119872119879
minus Γ119875119873119872
ℎ119875119879
)]
= Γ119871119872119873
+ 12119892119871119879 [minusnabla
119879
ℎ119872119873
+ nabla119872
ℎ119873119879
+ nabla119873
ℎ119872119879
]
(A4)
and for the Ricci tensor we get
119877119872119873
asymp 119872119873
+ 12(minusnabla119871
nabla119871ℎ119872119873
+ 119876119872119873
) (A5)
where we introduced a new tensor
119876119872119873
equiv nabla119871
nabla119872
ℎ119871119873
+ nabla119871
nabla119873
ℎ119871119872
minus nabla119873
nabla119872
ℎ119871119871
= [nabla119872
(nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
) + nabla119873
(nabla119871
ℎ119871119872
minus 12120597119872
ℎ119871119871
)]
minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A6)
To get the last expression we used the relation (918) in[12]Thegauge conditions (see eg the paragraph 105 in [12])
nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
= 0 (A7)
simplify considerably this expression
119876119872119873
= minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A8)
Now we suppose that the manifold MD is a productmanifoldMD =M4timesM119889 with a block-diagonal backgroundmetrics of the form (1) However in this appendix we do notassume that the internal manifoldM
119889
is a Ricci-flat one Forthe metrics (1) and an arbitrary internal space M
119889
the onlynonzero components of the Riemann tensor
119871119873119872119875
[119892(D)]and the Ricci tensor
119873119872
[119892(D)] are
119897119899119898119901
[119892(D)] = 119897119899119898119901
[119892(119889)]
119898119899
[119892(D)] = 119898119899
[119892(119889)] (A9)
Typically the matter which perturbs the backgroundmetrics is taken in the form of a perfect fluid with isotropicpressure In the case of the product manifold the pressureis isotropic in each factor manifold (see eg the energy-momentum tensor (3)) Obviously such perturbation doesnot change the topology of the model for example thetopologies of the factor manifolds in the case of the productmanifold Additionally it preserves also the block-diagonalstructure of the metric tensor In the case of a steady-statemodel (our case) the nondiagonal perturbations ℎ
0
119872
( =1 2 3 119863) are also absentTherefore themetric correctionterms are conformal to the background metrics
ℎ00
= 1205851
119892(D)00
= 1205851
12057800
ℎ120583] = 1205852119892
(D)
120583]= minus1205852
120575120583] 120583 ] = 1 2 3
ℎ119898119899
= 1205853
119892(D)119898119899
= 1205853
119892(119889)119898119899
(A10)
where 120585123
are some scalar functions of all spatial coordinates119883 It can be easily verified that under these conditions (it isworth noting that as it follows from (A8)119876
119872119873
is also equalto zero when the following condition holds 119871
119873119875119872
= minus119871119872119875119873
resulting in the Ricci-flat space
119872119873
= 0 (but not vice versa))the tensor119876
119872119873
in (A8) is equal to zero119876119872119873
= 0Thereforeup to 119874(11198882) terms we get
119877119872119873
[119892(D)] asymp 119872119873
[119892(D)] minus 12nabla119871
nabla119871ℎ119872119873
= 119872119873
[119892(D)] minus 12nabla
119871
nabla119871ℎ119872119873
+ 119900 ( 11198882)
(A11)
where = 1 2 119863 and we took into account that1198830 = 119888119905
B Ricci Tensor for a Block-Diagonal Metrics
In this appendix we consider a block-diagonal metrics
[119892(D)119872119873
(119909 119910)] = [119892(1198891)120583] (119909)] oplus [119892
(1198892)
119898119899
(119909 119910)]
= [119892(1198891)120583] (119909)] oplus [119890
120572(119909)119892(1198892)119898119899
(119910)] (B1)
6 Advances in High Energy Physics
Then for the Ricci-tensor components we get the follow-ing expressions
119877120583] [119892119872119873] = 119877120583] [119892
(1198891)
120582120591
] minus 11988922[nabla(1198891)120583
(120597]120572) +12(120597120583
120572) (120597]120572)](B2)
119877119898] [119892119872119873] = 0 (B3)
119877119898119899
[119892119872119873
] = 119877119898119899
[119892(1198892)119896119897
] minus 12119890120572(119909)119892(1198892)
119898119899
119892(1198891)120582120591
times [nabla(1198891)120582
(120597120591
120572) + 11988922(120597120582
120572) (120597120591
120572)] (B4)
Acknowledgments
This work was supported by the ldquoCosmomicrophysics-2rdquoprogramme of the Physics and Astronomy Division of theNational Academy of Sciences of Ukraine The work ofMaxim Eingorn was supported by NSF CREST Award HRD-0833184 and NASA Grant NNX09AV07A
References
[1] T Kaluza ldquoOn the problem of unity in physicsrdquo Sitzungsberichteder Koniglich Preussischen Akademie der Wissenschaften zuBerlin vol 1921 pp 966ndash972 1921
[2] O Klein ldquoQuantum theory and five-dimensional theory ofrelativityrdquoZeitschrift fur Physik vol 37 no 12 pp 895ndash906 1926
[3] NArkani-Hamed SDimopoulos andGDvali ldquoThehierarchyproblem and new dimensions at a millimeterrdquo Physics Letters Bvol 429 no 3-4 pp 263ndash272 1998
[4] I Antoniadis N Arkani-Hamed S Dimopoulos and G DvalildquoNew dimensions at a millimeter to a fermi and superstrings ata TeVrdquo Physics Letters B vol 436 no 3-4 pp 257ndash263 1998
[5] A Salam and J Strathdee ldquoOn Kaluza-Klein theoryrdquo Annals ofPhysics vol 141 no 2 pp 316ndash352 1982
[6] P van Nieuwenhuizen ldquoThe complete mass spectrum of 119889 = 11supergravity compactified on S
4
and a general mass formula forarbitrary cosets M
4
rdquo Classical and Quantum Gravity vol 2 no1 pp 1ndash20 1985
[7] H J Kim L J Romans and P van Nieuwenhuizen ldquoThe massspectrum of chiral 119873 = 2119863 = 10 supergravity on 1198785rdquo PhysicalReview D vol 32 no 2 pp 389ndash399 1985
[8] K Hinterbichler J Levin and C Zukowski ldquoKaluza-Kleintowers on generalmanifoldsrdquo httparxivorgabs13106353
[9] A R Brown and A Dahlen ldquoStability and spectrum of com-pactifications on product manifoldsrdquo httparxivorgabs13106360
[10] M B Green J H Schwarz and E Witten Superstring TheoryLoop Amplitudes Anomalies and Phenomenology vol 2 ofCambridge Monographs on Mathematical Physics CambridgeUniversity Press Cambridge UK 1987
[11] A Chopovsky M Eingorn and A Zhuk ldquoMany-body prob-lem in Kaluza-Klein models with toroidal compactificationrdquoaccepted in European Physical Journal C
[12] L D Landau and E M Lifshitz ldquoThe classical theory of fieldsrdquovol 2 of Course of Theoretical Physics Pergamon Press OxfordUK 4th edition 2000
[13] M Eingorn and A Zhuk ldquoMultidimensional gravity in non-relativistic limitrdquo Physical Review D vol 80 no 12 Article ID124037 5 pages 2009
[14] M Eingorn andA Zhuk ldquoNon-relativistic limit ofmultidimen-sional gravity exact solutions and applicationsrdquo Classical andQuantum Gravity vol 27 no 5 Article ID 055002 p 17 2010
[15] M Eingorn and A Zhuk ldquoClassical tests of multidimensionalgravity negative resultrdquo Classical and Quantum Gravity vol 27no 20 Article ID 205014 p 18 2010
[16] M Eingorn and A Zhuk ldquoRemarks on gravitational interactionin Kaluza-Klein modelsrdquo Physics Letters B vol 713 no 3 pp154ndash159 2012
[17] M Eingorn and A Zhuk ldquoKaluza-Klein models can we con-struct a viable examplerdquo Physical Review D vol 83 no 4 Arti-cle ID 044005 11 pages 2011
[18] M Eingorn O R De Medeiros L C B Crispino and A ZhukldquoLatent solitons black strings black branes and equations ofstate in Kaluza-Klein modelsrdquo Physical Review D vol 84 no 2Article ID 024031 8 pages 2011
[19] A A Kirillov A A Korotkevich and S G Rubin ldquoEmergenceof symmetriesrdquo Physics Letters B vol 718 no 2 pp 237ndash2402012
[20] K A Bronnikov and S G Rubin Black Holes Cosmology andExtra Dimensions World Scientific Singapore 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Advances in High Energy Physics
space and minusnabla
119871
nabla119871 rarr Δ = 120575120583]1205972120597119909120583120597119909] Introducing thedefinition ℎ
00
equiv 21205931198882 we can easily get from this system
120593 = minus119866119873
119873
sum119901=1
119898119901
10038161003816100381610038161003816x minus x11990110038161003816100381610038161003816 (20)
where
119866119873
= 119878119863 (119863 minus 2 + 120596119889)2120587119889
(119863 minus 1)119866D (21)
is theNewtonian gravitational constant and120593(119909) is the nonre-lativistic gravitational potential created by 119873 masses At thesame time
ℎ120583 ] =
1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
120575120583 ]
ℎ119898119899
(119909 119910) = 120572 (119909) 119892119898119899
(119910)
120572 = minus120596 (119863 minus 1) + 1 minus 120596119889119863 minus 2 + 120596119889
ℎ00
(22)
Now we intend to determine the 11989100
sim 119874(11198884) correc-tion term Obviously to do it we need to solve only the 00-component of the Einstein equation (7) Keeping in mind theprefactor 11198884 in the right hand side of (7) it is clear thatthe energy-momentum tensor component 119879
00
and the trace119879 must be determined up to 119874(1) Therefore similar to thepaper [11] we get
11987900
asymp9848583
1198882
119889
[1 + 3119863 minus 4 + 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
]
119879 asymp9848583
1198882
119889
[1 + 119863 minus 1205961198892 (119863 minus 2 + 120596119889)
ℎ00
] (1 minus 120596119889)
(23)
To get the Ricci-tensor component 11987700
up to 119874(11198884) wecan use (B2) and Equations (229) (238) and (239) in [15]and arrive at the following formula
11987700
asymp 12Δ(11989100
minus 211988841205932) + 1
1198882Δ120593
+ 119863 minus 1119863 minus 2 + 120596119889
21198884120593Δ120593
(24)
where Δ = 120575120583]1205972120597119909120583120597119909] is the 3-dimensional Laplace oper-ator Therefore the 00-component of the Einstein equationreads
12Δ(11989100
minus 211988841205932) = 1
1198884120593Δ120593
=4120587119866119873
1198884119873
sum119901=1
119898119901
1205931015840 (x119901
) 120575 (x minus x119901
) (25)
where we used the equation Δ120593 = 4120587119866119873
sum119873119901=1
119898119901
120575(x minus x119901
)The function 1205931015840(x
119901
) is the potential of the gravitational field
at a point with the radius vector x119901
produced by all particlesexcept the 119901th
1205931015840 (x119901
) = sum119902 = 119901
120593119902
(x119901
minus x119902
)
Δ120593119901
(x minus x119901
) = 4120587119866119873
119898119901
120575 (x minus x119901
) (26)
Therefore the solution of (25) is
11989100
(x) = 211988841205932 (x) + 2
1198884119873
sum119901=1
120593119901
(x minus x119901
) 1205931015840 (x119901
) (27)
Obviously in the case of the toroidal internal spacethis section reproduces the results of our paper [11] Thefound metric correction terms enable us to calculate thegravitational tests (the deflection of light the time-delay ofthe radar echoes and the perihelion advance) in full analogywith the paper [18]
4 Conclusion
In our paper we have studied the multidimensional KKmodel with the Ricci-flat internal space Such models areof special interest because Calabi-Yau manifolds which arewidely used in the superstring theory [10] belong to this classIn the absence of matter sources the background manifold isa direct product of the four-dimensional Minkowski space-time and the Ricci-flat internal space Then we perturbedthis background metrics by a system of gravitating massesfor example astrophysical objects such as our Sun It is wellknown that pressure inside these objects is much less thanthe energy density Therefore we supposed that gravitatingmasses are pressureless in the external space Howeverthey may have relativistic pressure in the internal spaceMoreover the presence of such pressure is the necessarycondition for agreement with the gravitational tests (thedeflection of light the time-delay of the radar echoes and theperihelion advance) [16ndash18] We clearly demonstrated thatmetric perturbations do not depend on the coordinates of theinternal space and gravitating masses are uniformly smearedover the internal space This means first that KK modescorresponding to the metric fluctuations are absent andsecond the particles corresponding to the gravitating massesshould be only in the ground quantum state with respectto the internal space Of course it looks very unnaturalAccording to statistical physics any nonzero temperatureshould result in fluctuations that is in KK modes On theother hand the conclusion about the absence of KK particlesis consistent with the results of [19 20] where a mechanismfor the symmetries formation of the internal spaces wasproposed According to this mechanism due to the entropydecreasing in the compact subspace its metric undergoesthe process of symmetrization during some time after itsquantum nucleation A high symmetry of internal spacesactually means suppression of the extra degrees of freedom
We also obtained the formulae for the metric correctionterms For the 00-component we got it up to 119874(11198884) Wefound other components up to 119874(11198882) This enables us to
Advances in High Energy Physics 5
calculate the gravitational tests the deflection of light thetime-delay of the radar echoes and the perihelion advanceIt also gives a possibility to construct the Lagrange functionfor the considered many-body system (see eg [11])
Appendices
A Ricci Tensor in the Weak-FieldApproximation
In our paper we use the signature of the metrics (+ minus minusminus ) and determine the Ricci-tensor components as follows
119877119872119873
equiv 119877119878119872119878119873
= 120597119871
Γ119871119872119873
minus 120597119873
Γ119871119872119871
+ Γ119871119872119873
Γ119878119871119878
minus Γ119878119872119871
Γ119871119873119878
(A1)
where the Christoffel symbols are
Γ119871119872119873
= 12119892119871119879 (minus120597
119879
119892119872119873
+ 120597119872
119892119873119879
+ 120597119873
119892119872119879
)
119872119873 119871 = 0 1 2 119863(A2)
The metrics 119892(D)119872119873
equiv 119892119872119873
is determined on (D = 1 + 119863)-dimensional manifoldMD
Now we suppose that the metric coefficients 119892119872119873
cor-respond to the perturbed metrics and in the weak-fieldapproximation they can be decomposed as
119892119872119873
asymp 119892119872119873
+ ℎ119872119873
119892119872119873 asymp 119892119872119873 minus ℎ119872119873 (A3)
where 119892119872119873
is the background unperturbed metrics and themetric perturbations ℎ
119872119873
sim 119874(11198882) (119888 is the speed oflight) As usual ℎ119873
119872
equiv 119892119873119879ℎ119879119872
Here and after the ldquohatsrdquodenote unperturbed quantities Then up to linear terms ℎthe Christoffel symbols read
Γ119871119872119873
asymp Γ119871119872119873
+ 12119892119871119879 [minus120597
119879
ℎ119872119873
+ (120597119872
ℎ119873119879
minus Γ119875119872119873
ℎ119875119879
)
+ (120597119873
ℎ119872119879
minus Γ119875119873119872
ℎ119875119879
)]
= Γ119871119872119873
+ 12119892119871119879 [minusnabla
119879
ℎ119872119873
+ nabla119872
ℎ119873119879
+ nabla119873
ℎ119872119879
]
(A4)
and for the Ricci tensor we get
119877119872119873
asymp 119872119873
+ 12(minusnabla119871
nabla119871ℎ119872119873
+ 119876119872119873
) (A5)
where we introduced a new tensor
119876119872119873
equiv nabla119871
nabla119872
ℎ119871119873
+ nabla119871
nabla119873
ℎ119871119872
minus nabla119873
nabla119872
ℎ119871119871
= [nabla119872
(nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
) + nabla119873
(nabla119871
ℎ119871119872
minus 12120597119872
ℎ119871119871
)]
minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A6)
To get the last expression we used the relation (918) in[12]Thegauge conditions (see eg the paragraph 105 in [12])
nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
= 0 (A7)
simplify considerably this expression
119876119872119873
= minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A8)
Now we suppose that the manifold MD is a productmanifoldMD =M4timesM119889 with a block-diagonal backgroundmetrics of the form (1) However in this appendix we do notassume that the internal manifoldM
119889
is a Ricci-flat one Forthe metrics (1) and an arbitrary internal space M
119889
the onlynonzero components of the Riemann tensor
119871119873119872119875
[119892(D)]and the Ricci tensor
119873119872
[119892(D)] are
119897119899119898119901
[119892(D)] = 119897119899119898119901
[119892(119889)]
119898119899
[119892(D)] = 119898119899
[119892(119889)] (A9)
Typically the matter which perturbs the backgroundmetrics is taken in the form of a perfect fluid with isotropicpressure In the case of the product manifold the pressureis isotropic in each factor manifold (see eg the energy-momentum tensor (3)) Obviously such perturbation doesnot change the topology of the model for example thetopologies of the factor manifolds in the case of the productmanifold Additionally it preserves also the block-diagonalstructure of the metric tensor In the case of a steady-statemodel (our case) the nondiagonal perturbations ℎ
0
119872
( =1 2 3 119863) are also absentTherefore themetric correctionterms are conformal to the background metrics
ℎ00
= 1205851
119892(D)00
= 1205851
12057800
ℎ120583] = 1205852119892
(D)
120583]= minus1205852
120575120583] 120583 ] = 1 2 3
ℎ119898119899
= 1205853
119892(D)119898119899
= 1205853
119892(119889)119898119899
(A10)
where 120585123
are some scalar functions of all spatial coordinates119883 It can be easily verified that under these conditions (it isworth noting that as it follows from (A8)119876
119872119873
is also equalto zero when the following condition holds 119871
119873119875119872
= minus119871119872119875119873
resulting in the Ricci-flat space
119872119873
= 0 (but not vice versa))the tensor119876
119872119873
in (A8) is equal to zero119876119872119873
= 0Thereforeup to 119874(11198882) terms we get
119877119872119873
[119892(D)] asymp 119872119873
[119892(D)] minus 12nabla119871
nabla119871ℎ119872119873
= 119872119873
[119892(D)] minus 12nabla
119871
nabla119871ℎ119872119873
+ 119900 ( 11198882)
(A11)
where = 1 2 119863 and we took into account that1198830 = 119888119905
B Ricci Tensor for a Block-Diagonal Metrics
In this appendix we consider a block-diagonal metrics
[119892(D)119872119873
(119909 119910)] = [119892(1198891)120583] (119909)] oplus [119892
(1198892)
119898119899
(119909 119910)]
= [119892(1198891)120583] (119909)] oplus [119890
120572(119909)119892(1198892)119898119899
(119910)] (B1)
6 Advances in High Energy Physics
Then for the Ricci-tensor components we get the follow-ing expressions
119877120583] [119892119872119873] = 119877120583] [119892
(1198891)
120582120591
] minus 11988922[nabla(1198891)120583
(120597]120572) +12(120597120583
120572) (120597]120572)](B2)
119877119898] [119892119872119873] = 0 (B3)
119877119898119899
[119892119872119873
] = 119877119898119899
[119892(1198892)119896119897
] minus 12119890120572(119909)119892(1198892)
119898119899
119892(1198891)120582120591
times [nabla(1198891)120582
(120597120591
120572) + 11988922(120597120582
120572) (120597120591
120572)] (B4)
Acknowledgments
This work was supported by the ldquoCosmomicrophysics-2rdquoprogramme of the Physics and Astronomy Division of theNational Academy of Sciences of Ukraine The work ofMaxim Eingorn was supported by NSF CREST Award HRD-0833184 and NASA Grant NNX09AV07A
References
[1] T Kaluza ldquoOn the problem of unity in physicsrdquo Sitzungsberichteder Koniglich Preussischen Akademie der Wissenschaften zuBerlin vol 1921 pp 966ndash972 1921
[2] O Klein ldquoQuantum theory and five-dimensional theory ofrelativityrdquoZeitschrift fur Physik vol 37 no 12 pp 895ndash906 1926
[3] NArkani-Hamed SDimopoulos andGDvali ldquoThehierarchyproblem and new dimensions at a millimeterrdquo Physics Letters Bvol 429 no 3-4 pp 263ndash272 1998
[4] I Antoniadis N Arkani-Hamed S Dimopoulos and G DvalildquoNew dimensions at a millimeter to a fermi and superstrings ata TeVrdquo Physics Letters B vol 436 no 3-4 pp 257ndash263 1998
[5] A Salam and J Strathdee ldquoOn Kaluza-Klein theoryrdquo Annals ofPhysics vol 141 no 2 pp 316ndash352 1982
[6] P van Nieuwenhuizen ldquoThe complete mass spectrum of 119889 = 11supergravity compactified on S
4
and a general mass formula forarbitrary cosets M
4
rdquo Classical and Quantum Gravity vol 2 no1 pp 1ndash20 1985
[7] H J Kim L J Romans and P van Nieuwenhuizen ldquoThe massspectrum of chiral 119873 = 2119863 = 10 supergravity on 1198785rdquo PhysicalReview D vol 32 no 2 pp 389ndash399 1985
[8] K Hinterbichler J Levin and C Zukowski ldquoKaluza-Kleintowers on generalmanifoldsrdquo httparxivorgabs13106353
[9] A R Brown and A Dahlen ldquoStability and spectrum of com-pactifications on product manifoldsrdquo httparxivorgabs13106360
[10] M B Green J H Schwarz and E Witten Superstring TheoryLoop Amplitudes Anomalies and Phenomenology vol 2 ofCambridge Monographs on Mathematical Physics CambridgeUniversity Press Cambridge UK 1987
[11] A Chopovsky M Eingorn and A Zhuk ldquoMany-body prob-lem in Kaluza-Klein models with toroidal compactificationrdquoaccepted in European Physical Journal C
[12] L D Landau and E M Lifshitz ldquoThe classical theory of fieldsrdquovol 2 of Course of Theoretical Physics Pergamon Press OxfordUK 4th edition 2000
[13] M Eingorn and A Zhuk ldquoMultidimensional gravity in non-relativistic limitrdquo Physical Review D vol 80 no 12 Article ID124037 5 pages 2009
[14] M Eingorn andA Zhuk ldquoNon-relativistic limit ofmultidimen-sional gravity exact solutions and applicationsrdquo Classical andQuantum Gravity vol 27 no 5 Article ID 055002 p 17 2010
[15] M Eingorn and A Zhuk ldquoClassical tests of multidimensionalgravity negative resultrdquo Classical and Quantum Gravity vol 27no 20 Article ID 205014 p 18 2010
[16] M Eingorn and A Zhuk ldquoRemarks on gravitational interactionin Kaluza-Klein modelsrdquo Physics Letters B vol 713 no 3 pp154ndash159 2012
[17] M Eingorn and A Zhuk ldquoKaluza-Klein models can we con-struct a viable examplerdquo Physical Review D vol 83 no 4 Arti-cle ID 044005 11 pages 2011
[18] M Eingorn O R De Medeiros L C B Crispino and A ZhukldquoLatent solitons black strings black branes and equations ofstate in Kaluza-Klein modelsrdquo Physical Review D vol 84 no 2Article ID 024031 8 pages 2011
[19] A A Kirillov A A Korotkevich and S G Rubin ldquoEmergenceof symmetriesrdquo Physics Letters B vol 718 no 2 pp 237ndash2402012
[20] K A Bronnikov and S G Rubin Black Holes Cosmology andExtra Dimensions World Scientific Singapore 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 5
calculate the gravitational tests the deflection of light thetime-delay of the radar echoes and the perihelion advanceIt also gives a possibility to construct the Lagrange functionfor the considered many-body system (see eg [11])
Appendices
A Ricci Tensor in the Weak-FieldApproximation
In our paper we use the signature of the metrics (+ minus minusminus ) and determine the Ricci-tensor components as follows
119877119872119873
equiv 119877119878119872119878119873
= 120597119871
Γ119871119872119873
minus 120597119873
Γ119871119872119871
+ Γ119871119872119873
Γ119878119871119878
minus Γ119878119872119871
Γ119871119873119878
(A1)
where the Christoffel symbols are
Γ119871119872119873
= 12119892119871119879 (minus120597
119879
119892119872119873
+ 120597119872
119892119873119879
+ 120597119873
119892119872119879
)
119872119873 119871 = 0 1 2 119863(A2)
The metrics 119892(D)119872119873
equiv 119892119872119873
is determined on (D = 1 + 119863)-dimensional manifoldMD
Now we suppose that the metric coefficients 119892119872119873
cor-respond to the perturbed metrics and in the weak-fieldapproximation they can be decomposed as
119892119872119873
asymp 119892119872119873
+ ℎ119872119873
119892119872119873 asymp 119892119872119873 minus ℎ119872119873 (A3)
where 119892119872119873
is the background unperturbed metrics and themetric perturbations ℎ
119872119873
sim 119874(11198882) (119888 is the speed oflight) As usual ℎ119873
119872
equiv 119892119873119879ℎ119879119872
Here and after the ldquohatsrdquodenote unperturbed quantities Then up to linear terms ℎthe Christoffel symbols read
Γ119871119872119873
asymp Γ119871119872119873
+ 12119892119871119879 [minus120597
119879
ℎ119872119873
+ (120597119872
ℎ119873119879
minus Γ119875119872119873
ℎ119875119879
)
+ (120597119873
ℎ119872119879
minus Γ119875119873119872
ℎ119875119879
)]
= Γ119871119872119873
+ 12119892119871119879 [minusnabla
119879
ℎ119872119873
+ nabla119872
ℎ119873119879
+ nabla119873
ℎ119872119879
]
(A4)
and for the Ricci tensor we get
119877119872119873
asymp 119872119873
+ 12(minusnabla119871
nabla119871ℎ119872119873
+ 119876119872119873
) (A5)
where we introduced a new tensor
119876119872119873
equiv nabla119871
nabla119872
ℎ119871119873
+ nabla119871
nabla119873
ℎ119871119872
minus nabla119873
nabla119872
ℎ119871119871
= [nabla119872
(nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
) + nabla119873
(nabla119871
ℎ119871119872
minus 12120597119872
ℎ119871119871
)]
minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A6)
To get the last expression we used the relation (918) in[12]Thegauge conditions (see eg the paragraph 105 in [12])
nabla119871
ℎ119871119873
minus 12120597119873
ℎ119871119871
= 0 (A7)
simplify considerably this expression
119876119872119873
= minus (119871119873119875119872
+ 119871119872119875119873
) ℎ119875119871
+ 119875119872
ℎ119875119873
+ 119875119873
ℎ119875119872
(A8)
Now we suppose that the manifold MD is a productmanifoldMD =M4timesM119889 with a block-diagonal backgroundmetrics of the form (1) However in this appendix we do notassume that the internal manifoldM
119889
is a Ricci-flat one Forthe metrics (1) and an arbitrary internal space M
119889
the onlynonzero components of the Riemann tensor
119871119873119872119875
[119892(D)]and the Ricci tensor
119873119872
[119892(D)] are
119897119899119898119901
[119892(D)] = 119897119899119898119901
[119892(119889)]
119898119899
[119892(D)] = 119898119899
[119892(119889)] (A9)
Typically the matter which perturbs the backgroundmetrics is taken in the form of a perfect fluid with isotropicpressure In the case of the product manifold the pressureis isotropic in each factor manifold (see eg the energy-momentum tensor (3)) Obviously such perturbation doesnot change the topology of the model for example thetopologies of the factor manifolds in the case of the productmanifold Additionally it preserves also the block-diagonalstructure of the metric tensor In the case of a steady-statemodel (our case) the nondiagonal perturbations ℎ
0
119872
( =1 2 3 119863) are also absentTherefore themetric correctionterms are conformal to the background metrics
ℎ00
= 1205851
119892(D)00
= 1205851
12057800
ℎ120583] = 1205852119892
(D)
120583]= minus1205852
120575120583] 120583 ] = 1 2 3
ℎ119898119899
= 1205853
119892(D)119898119899
= 1205853
119892(119889)119898119899
(A10)
where 120585123
are some scalar functions of all spatial coordinates119883 It can be easily verified that under these conditions (it isworth noting that as it follows from (A8)119876
119872119873
is also equalto zero when the following condition holds 119871
119873119875119872
= minus119871119872119875119873
resulting in the Ricci-flat space
119872119873
= 0 (but not vice versa))the tensor119876
119872119873
in (A8) is equal to zero119876119872119873
= 0Thereforeup to 119874(11198882) terms we get
119877119872119873
[119892(D)] asymp 119872119873
[119892(D)] minus 12nabla119871
nabla119871ℎ119872119873
= 119872119873
[119892(D)] minus 12nabla
119871
nabla119871ℎ119872119873
+ 119900 ( 11198882)
(A11)
where = 1 2 119863 and we took into account that1198830 = 119888119905
B Ricci Tensor for a Block-Diagonal Metrics
In this appendix we consider a block-diagonal metrics
[119892(D)119872119873
(119909 119910)] = [119892(1198891)120583] (119909)] oplus [119892
(1198892)
119898119899
(119909 119910)]
= [119892(1198891)120583] (119909)] oplus [119890
120572(119909)119892(1198892)119898119899
(119910)] (B1)
6 Advances in High Energy Physics
Then for the Ricci-tensor components we get the follow-ing expressions
119877120583] [119892119872119873] = 119877120583] [119892
(1198891)
120582120591
] minus 11988922[nabla(1198891)120583
(120597]120572) +12(120597120583
120572) (120597]120572)](B2)
119877119898] [119892119872119873] = 0 (B3)
119877119898119899
[119892119872119873
] = 119877119898119899
[119892(1198892)119896119897
] minus 12119890120572(119909)119892(1198892)
119898119899
119892(1198891)120582120591
times [nabla(1198891)120582
(120597120591
120572) + 11988922(120597120582
120572) (120597120591
120572)] (B4)
Acknowledgments
This work was supported by the ldquoCosmomicrophysics-2rdquoprogramme of the Physics and Astronomy Division of theNational Academy of Sciences of Ukraine The work ofMaxim Eingorn was supported by NSF CREST Award HRD-0833184 and NASA Grant NNX09AV07A
References
[1] T Kaluza ldquoOn the problem of unity in physicsrdquo Sitzungsberichteder Koniglich Preussischen Akademie der Wissenschaften zuBerlin vol 1921 pp 966ndash972 1921
[2] O Klein ldquoQuantum theory and five-dimensional theory ofrelativityrdquoZeitschrift fur Physik vol 37 no 12 pp 895ndash906 1926
[3] NArkani-Hamed SDimopoulos andGDvali ldquoThehierarchyproblem and new dimensions at a millimeterrdquo Physics Letters Bvol 429 no 3-4 pp 263ndash272 1998
[4] I Antoniadis N Arkani-Hamed S Dimopoulos and G DvalildquoNew dimensions at a millimeter to a fermi and superstrings ata TeVrdquo Physics Letters B vol 436 no 3-4 pp 257ndash263 1998
[5] A Salam and J Strathdee ldquoOn Kaluza-Klein theoryrdquo Annals ofPhysics vol 141 no 2 pp 316ndash352 1982
[6] P van Nieuwenhuizen ldquoThe complete mass spectrum of 119889 = 11supergravity compactified on S
4
and a general mass formula forarbitrary cosets M
4
rdquo Classical and Quantum Gravity vol 2 no1 pp 1ndash20 1985
[7] H J Kim L J Romans and P van Nieuwenhuizen ldquoThe massspectrum of chiral 119873 = 2119863 = 10 supergravity on 1198785rdquo PhysicalReview D vol 32 no 2 pp 389ndash399 1985
[8] K Hinterbichler J Levin and C Zukowski ldquoKaluza-Kleintowers on generalmanifoldsrdquo httparxivorgabs13106353
[9] A R Brown and A Dahlen ldquoStability and spectrum of com-pactifications on product manifoldsrdquo httparxivorgabs13106360
[10] M B Green J H Schwarz and E Witten Superstring TheoryLoop Amplitudes Anomalies and Phenomenology vol 2 ofCambridge Monographs on Mathematical Physics CambridgeUniversity Press Cambridge UK 1987
[11] A Chopovsky M Eingorn and A Zhuk ldquoMany-body prob-lem in Kaluza-Klein models with toroidal compactificationrdquoaccepted in European Physical Journal C
[12] L D Landau and E M Lifshitz ldquoThe classical theory of fieldsrdquovol 2 of Course of Theoretical Physics Pergamon Press OxfordUK 4th edition 2000
[13] M Eingorn and A Zhuk ldquoMultidimensional gravity in non-relativistic limitrdquo Physical Review D vol 80 no 12 Article ID124037 5 pages 2009
[14] M Eingorn andA Zhuk ldquoNon-relativistic limit ofmultidimen-sional gravity exact solutions and applicationsrdquo Classical andQuantum Gravity vol 27 no 5 Article ID 055002 p 17 2010
[15] M Eingorn and A Zhuk ldquoClassical tests of multidimensionalgravity negative resultrdquo Classical and Quantum Gravity vol 27no 20 Article ID 205014 p 18 2010
[16] M Eingorn and A Zhuk ldquoRemarks on gravitational interactionin Kaluza-Klein modelsrdquo Physics Letters B vol 713 no 3 pp154ndash159 2012
[17] M Eingorn and A Zhuk ldquoKaluza-Klein models can we con-struct a viable examplerdquo Physical Review D vol 83 no 4 Arti-cle ID 044005 11 pages 2011
[18] M Eingorn O R De Medeiros L C B Crispino and A ZhukldquoLatent solitons black strings black branes and equations ofstate in Kaluza-Klein modelsrdquo Physical Review D vol 84 no 2Article ID 024031 8 pages 2011
[19] A A Kirillov A A Korotkevich and S G Rubin ldquoEmergenceof symmetriesrdquo Physics Letters B vol 718 no 2 pp 237ndash2402012
[20] K A Bronnikov and S G Rubin Black Holes Cosmology andExtra Dimensions World Scientific Singapore 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Advances in High Energy Physics
Then for the Ricci-tensor components we get the follow-ing expressions
119877120583] [119892119872119873] = 119877120583] [119892
(1198891)
120582120591
] minus 11988922[nabla(1198891)120583
(120597]120572) +12(120597120583
120572) (120597]120572)](B2)
119877119898] [119892119872119873] = 0 (B3)
119877119898119899
[119892119872119873
] = 119877119898119899
[119892(1198892)119896119897
] minus 12119890120572(119909)119892(1198892)
119898119899
119892(1198891)120582120591
times [nabla(1198891)120582
(120597120591
120572) + 11988922(120597120582
120572) (120597120591
120572)] (B4)
Acknowledgments
This work was supported by the ldquoCosmomicrophysics-2rdquoprogramme of the Physics and Astronomy Division of theNational Academy of Sciences of Ukraine The work ofMaxim Eingorn was supported by NSF CREST Award HRD-0833184 and NASA Grant NNX09AV07A
References
[1] T Kaluza ldquoOn the problem of unity in physicsrdquo Sitzungsberichteder Koniglich Preussischen Akademie der Wissenschaften zuBerlin vol 1921 pp 966ndash972 1921
[2] O Klein ldquoQuantum theory and five-dimensional theory ofrelativityrdquoZeitschrift fur Physik vol 37 no 12 pp 895ndash906 1926
[3] NArkani-Hamed SDimopoulos andGDvali ldquoThehierarchyproblem and new dimensions at a millimeterrdquo Physics Letters Bvol 429 no 3-4 pp 263ndash272 1998
[4] I Antoniadis N Arkani-Hamed S Dimopoulos and G DvalildquoNew dimensions at a millimeter to a fermi and superstrings ata TeVrdquo Physics Letters B vol 436 no 3-4 pp 257ndash263 1998
[5] A Salam and J Strathdee ldquoOn Kaluza-Klein theoryrdquo Annals ofPhysics vol 141 no 2 pp 316ndash352 1982
[6] P van Nieuwenhuizen ldquoThe complete mass spectrum of 119889 = 11supergravity compactified on S
4
and a general mass formula forarbitrary cosets M
4
rdquo Classical and Quantum Gravity vol 2 no1 pp 1ndash20 1985
[7] H J Kim L J Romans and P van Nieuwenhuizen ldquoThe massspectrum of chiral 119873 = 2119863 = 10 supergravity on 1198785rdquo PhysicalReview D vol 32 no 2 pp 389ndash399 1985
[8] K Hinterbichler J Levin and C Zukowski ldquoKaluza-Kleintowers on generalmanifoldsrdquo httparxivorgabs13106353
[9] A R Brown and A Dahlen ldquoStability and spectrum of com-pactifications on product manifoldsrdquo httparxivorgabs13106360
[10] M B Green J H Schwarz and E Witten Superstring TheoryLoop Amplitudes Anomalies and Phenomenology vol 2 ofCambridge Monographs on Mathematical Physics CambridgeUniversity Press Cambridge UK 1987
[11] A Chopovsky M Eingorn and A Zhuk ldquoMany-body prob-lem in Kaluza-Klein models with toroidal compactificationrdquoaccepted in European Physical Journal C
[12] L D Landau and E M Lifshitz ldquoThe classical theory of fieldsrdquovol 2 of Course of Theoretical Physics Pergamon Press OxfordUK 4th edition 2000
[13] M Eingorn and A Zhuk ldquoMultidimensional gravity in non-relativistic limitrdquo Physical Review D vol 80 no 12 Article ID124037 5 pages 2009
[14] M Eingorn andA Zhuk ldquoNon-relativistic limit ofmultidimen-sional gravity exact solutions and applicationsrdquo Classical andQuantum Gravity vol 27 no 5 Article ID 055002 p 17 2010
[15] M Eingorn and A Zhuk ldquoClassical tests of multidimensionalgravity negative resultrdquo Classical and Quantum Gravity vol 27no 20 Article ID 205014 p 18 2010
[16] M Eingorn and A Zhuk ldquoRemarks on gravitational interactionin Kaluza-Klein modelsrdquo Physics Letters B vol 713 no 3 pp154ndash159 2012
[17] M Eingorn and A Zhuk ldquoKaluza-Klein models can we con-struct a viable examplerdquo Physical Review D vol 83 no 4 Arti-cle ID 044005 11 pages 2011
[18] M Eingorn O R De Medeiros L C B Crispino and A ZhukldquoLatent solitons black strings black branes and equations ofstate in Kaluza-Klein modelsrdquo Physical Review D vol 84 no 2Article ID 024031 8 pages 2011
[19] A A Kirillov A A Korotkevich and S G Rubin ldquoEmergenceof symmetriesrdquo Physics Letters B vol 718 no 2 pp 237ndash2402012
[20] K A Bronnikov and S G Rubin Black Holes Cosmology andExtra Dimensions World Scientific Singapore 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
