New class of N-dimensional braneworlds

Gen Relativ Gravit (2014) 46:1697DOI 10.1007/s10714-014-1697-z

RESEARCH ARTICLE

New class of N-dimensional braneworlds

Merab Gogberashvili · Pavle Midodashvili ·Giorgi Tukhashvili

Received: 29 October 2013 / Accepted: 17 February 2014© Springer Science+Business Media New York 2014

Abstract The new class of the non-stationary solutions to the system ofN-dimensional equations for coupled gravitational and massless scalar field is found.The model represents a single (N-1)-brane in a space-time with one large (infinite)and (N-5) small (compact) space-like extra dimensions. In some particular cases themodel corresponds to the gravitational and scalar field standing waves bounded bythe brane. These braneworlds can be relevant in string and other higher dimensionalmodels.

Keywords Branes · Exact solutions · Standing waves

1 Introduction

Solutions to the Einstein equations in a higher dimensional space-times for the3-dimensional singular space-like surfaces with non-factorizable geometry, braneworlds[1–6], have attracted a lot of interest recently in high energy physics (see [7–10] forreviews). Most of these models were realized as static geometric configurations. How-ever, there have appeared several braneworld models that assumed time-dependentmetrics and fields [11–14].

M. Gogberashvili (B)Andronikashvili Institute of Physics, 6 Tamarashvili St., 0177 Tbilisi, Georgiae-mail: [email protected]

P. MidodashviliIlia State University, 3/5 Cholokashvili Ave., 0162 Tbilisi, Georgiae-mail: [email protected]

M. Gogberashvili · G. TukhashviliJavakhishvili State University, 3 Chavchavadze Ave., 0179 Tbilisi, Georgiae-mail: [email protected]

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1697 Page 2 of 12 M. Gogberashvili et al.

A key requirement for realizing the braneworld idea is that the various bulk fields belocalized on the brane. For reasons of economy and avoidance of charge universalityobstruction [15] one would like to have a universal gravitational trapping mechanismfor all fields. However, there are difficulties to realize such mechanism with expo-nentially warped space-times used in standard brane scenario [3–6]. In the existing(1+4)-dimensional models spin 0 and spin 2 fields are localized on the brane withthe decreasing warp factor [3–6], spin 1/2 field can be localized with the increas-ing warp factor [16], and spin 1 fields are not localized at all [17]. For the case of(1+5)-dimensions it was found that spin 0, spin 1 and spin 2 fields are localized on thebrane with the decreasing warp factor and spin 1/2 fields again are localized with theincreasing warp factor [18]. There exist also 6D models with non-exponential warpfactors that provide gravitational localization of all kind of bulk fields on the brane[19–23], however, these models require introduction of unnatural sources.

It turns out that proposed recently brane model [24–31], with standing waves in thebulk, can provide a natural alternative mechanism for universal gravitational trappingof zero modes of all kinds of matter fields. To clarify the mechanism of localization letus remind that standing electromagnetic waves, so-called optical lattices, can providetrapping of various particles by scattering, dipole and quadruple forces [32–35]. It isknown that the motion of test particles in the field of a gravitational wave is similarto the motion of charged particles in the field of an electromagnetic wave [36]. Thusstanding gravitational waves could also provide confinement of matter via quadrupleforces. Indeed, the equations of motion of the system of spinless particles in thequadruple approximation has the form [37–39]:

Dpμ

ds= Fμ = −1

6Jαβγ δ Dμ Rαβγ δ , (1)

where pμ is the total momentum of the matter field and Jαβγ δ is the quadruple momentof the stress-energy tensor for the matter field. The oscillating metric due to gravita-tional waves should induce a quadruple moment in the matter fields. If the inducedquadruple moment is out of phase with the gravitational wave the system energyincreases in comparison with the resonant case and the fields/particles will feel aquadruple force, Fμ, which ejects them out of the high curvature region, i.e. it wouldlocalize them at the nodes.

In this paper we introduce new general class of N-dimensional braneworlds thatcan provide universal gravitational trapping of all matter fields on the brane. They arerealized as standing wave solutions to the system of Einstein and Klein-Gordon equa-tions in N-dimensional space-time. Some particular cases of the introduced generalsolutions were considered in the papers: [40–43] (for N = 4), [24–29] (for N = 5with the ghost-like scalar field), [30,31] (for N = 5 with the real bulk scalar field)and [44–47] (for N = 6).

2 The model

Our setup consists of a single brane and non self interacting scalar field, φ, in N -dimensional space-time with the signature (+,−, . . . ,−). The action of the model

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New class of N-dimensional braneworlds Page 3 of 12 1697

has the form:

S =∫

d N x√|g|

(M N−2

2R + ε

2g AB∂Aφ∂Bφ + Lbrane

), (2)

where Lbrane is the brane Lagrangian and M is the fundamental scale, which is relatedto the N-dimensional Newton constant, G = 1/(8π M N−2). The sign coefficient ε infront of the Lagrangian of φ takes the values +1 and −1 for the real and phantom-like bulk scalar fields, respectively. Capital Latin indexes numerate N -dimensionalcoordinates, and we use the units where c = h = 1.

Variation of the action (2) with respect to gAB leads to the Einstein equations:

RAB − 1

2gAB R = 1

M N−2 (σAB + εTAB) , (3)

where the source terms are the energy-momentum tensors of the brane,

σ AB = M N−2δ(z)diag

[τt , τx1 , . . . , τx(N−3)

, τy, τz]

, (4)

with τN being brane tensions, and of the bulk scalar field,

TAB = ∂Aφ∂Bφ − 1

2gAB∂Cφ∂Cφ . (5)

The Einstein equations (3) can be rewritten in the form:

RAB = 1

M N−2

(σAB − 1

N − 2gABσ + ε∂Aφ∂Bφ

), (6)

where σ = g ABσAB .We consider the N -dimensional metric ansatz:

ds2 =(1+k|z|)aeS(

dt2 − dz2)−(1 + k|z|)b

[eV

N−3∑i=1

dx2i + e−(N−3)V dy2

], (7)

where the curvature scale k and the exponents a and b are some constants, and themetric functions S = S(t, |z|) and V = V (t, |z|) depend only on time, t , and on themodulus of the orthogonal to the brane coordinate, z. The determinant of (7) has theform: √|g| = eS(1 + k|z|) b(N−2)+2a

2 . (8)

By the metric (7) we want to describe geometry of the (N − 1)-brane placed at theorigin of the large space-like extra dimension z. Among the (N −2) spatial coordinatesof the brane, three of them, x1, x2 and y, denote the ordinary infinite dimensions ofour world, while the remaining ones, xi (i = 3, . . . , N − 5), can be assumed to becompact, curled up to the unobservable sizes for the present energies.

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The Klein-Gordon equation for the bulk scalar field, φ, in the background metric(7) has the form:

{∂2

z + b(N − 2)k sgn(z)

2(1 + k|z|) ∂z − ∂2t − (1 + k|z|)a−beS

[e−V

N−3∑i=1

∂2xi

+e(N−3)V ∂2y

] }φ = 0 . (9)

The non-zero components of N-dimensional Ricci tensor corresponding to (7) are:

Rtt = (ak + S′) δ(z) + 1

2

[S′′ + b(N − 2)k

2(1 + k|z|) S′ − S − (N − 2)(N − 3)

2V 2

]

+a[b(N − 2) − 2]k2

4(1 + k|z|)2 ,

Rtz = b(N − 2)k sgn(z)

4(1 + k|z|) S − (N − 2)(N − 3) sgn(z)

4V V ′ ,

Rx1x1 = · · · = Rx(N−3)x(N−3)= − (

bk + V ′) e−S+V δ(z) (10)

+ e−S+V

(1 + k|z|)a−b

{1

2

[V − V ′′ − b(N − 2)k

2(1 + k|z|) V ′]

− b[b(N − 2) − 2]k2

4(1 + k|z|)2

},

Ryy = − [bk − (N − 3)V ′] e−S−(N−3)V δ(z)

− e−S−(N−3)V

(1 + k|z|)a−b

{(N − 3)

2

[V − V ′′ − b(N − 2)k

2(1 + k|z|) V ′]

+b[b(N − 2) − 2]k2

4(1 + k|z|)2

},

Rzz = − {[a + b(N − 2)] k + S′} δ(z)

+1

2

[S − S′′ + b(N − 2)k

2(1 + k|z|) S′ − (N − 2)(N − 3)

2V ′2

]+ Dk2

4(1 + k|z|)2 ,

where overdots and primes denote the derivatives with respect to t and |z|, respectively,and to shorten the last expression we have introduced the constant:

D = a[b(N − 2) + 2] − b(N − 2)(b − 2) . (11)

The Einstein equations (6) can be split into the system of equations for metricfunctions:

1

2

[S′′ + b(N − 2)k

2(1 + k|z|) S′ − S − (N − 2)(N − 3)

2V 2

]+ a[b(N − 2) − 2]k2

4(1 + k|z|)2

= ε1

M N−2 ∂tφ2 ,

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b(N − 2)k sgn(z)

4(1 + k|z|) S − (N − 2)(N − 3) sgn(z)

4V V ′ = ε

sgn(z)

M N−2 ∂tφ∂zφ ,

e−S+V

(1 + k|z|)a−b

{1

2

[V − V ′′ − b(N − 2)k

2(1 + k|z|)V ′

]− b[b(N − 2) − 2]k2

4(1 + k|z|)2

}

= ε1

M N−2 ∂x1φ2 ,

. . . (12)

e−S+V

(1 + k|z|)a−b

{1

2

[V − V ′′ − b(N − 2)k

2(1 + k|z|) V ′]

− b[b(N − 2) − 2]k2

4(1 + k|z|)2

}

= ε1

M N−2 ∂x(N−3)φ2 ,

e−S−(N−3)V

(1 + k|z|)a−b

{(N − 3)

2

[V ′′ − V + b(N − 2)k

2(1 + k|z|) V ′]

− b[b(N − 2) − 2]k2

4(1 + k|z|)2

}

= ε1

M N−2 ∂yφ2 ,

1

2

[S − S′′ + b(N − 2)k

2(1 + k|z|) S′ − (N − 2)(N − 3)

2V ′2

]+ Dk2

4(1 + k|z|)2

= ε1

M N−2 ∂zφ2 ,

and for the brane energy-momentum tensor:

(ak + S′) δ(z) = 1

M N−2

(σt t − 1

N − 2gttσ

),

− (bk + V ′) e−S+V δ(z) = 1

M N−2

(σx1x1 − 1

N − 2gx1x1σ

),

. . . (13)

− (bk + V ′) e−S+V δ(z) = 1

M N−2

(σx(N−3)x(N−3)

− 1

N − 2gx(N−3)x(N−3)

σ

),

− [bk − (N − 3)V ′] e−S−(N−3)V δ(z) = 1

M N−2

(σyy − 1

N − 2gyyσ

),

− {[a + b(N − 2)]k + S′} δ(z) = 1

M N−2

(σzz − 1

N − 2gzzσ

).

In the following sections we present all nontrivial solutions to the system of equa-tions (9), (12) and (13) for our metric ansatz (7).

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3 Solutions with k = 0

In this section we assume k = 0 in (7) and consider the metric:

ds2 = eS(

dt2 − dz2)

−[

eVN−3∑i=1

dx2i + e−(N−3)V dy2

]. (14)

The Ricci tensor of this space-time has the δ-like singularity at z = 0, since the metricfunctions S and V are depending on of the modulus of z. To smooth this singularity weplace the brane at the origin of the extra coordinate z. Note that without modulus forz the metric (7) will correspond to the running wave solutions, considered in [48–50]for 4D case.

When k = 0 there exist three independent nontrivial solutions to the system (9),(12) and (13).

3.1 The oscillating brane

We start with the simplest case when the bulk scalar field and the metric function Vin (14) are equal to zero,

φ = 0 , (15)

V = 0 .

In this case the solution to the system (9) and (12) is:

S = [C1 sin(�t) + C2 cos(�t)] [C3 sin(�|z|) + C4 cos(�|z|)] , (16)

where Ci (i = 1, 2, 3, 4) and � are some constants. The solution corresponds to theoscillating brane at |z| = 0 in N -dimensional space-time.

Imposing on the metric function S the boundary condition on the brane:S||z|=0 = 0 , (17)

from the equations (13) for the brane tensions we find:

τt = 0 ,

τx1 = τx2 = · · · = τx(N−3)= −S′ ,

τy = −S′ , (18)

τz = 0 . (19)

3.2 Gravitational and phantom-like scalar field standing waves

Now we consider the case with the ghost-like scalar field, φ, when the metric functionS is absent in (14),

ε = −1 ,

S = 0 . (20)

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Then the equations (9) and (12) have the following solution:

V = [C1 sin(ωt) + C2 cos(ωt)] [C3 sin(ω|z|) + C4 cos(ω|z|)] ,

φ = 1

2

√M N−2(N − 2)(N − 3) [C1 sin(ωt) + C2 cos(ωt)] [C3 sin(ω|z|)

+C4 cos(ω|z|)] , (21)

where Ci (i = 1, 2, 3, 4) and ω are some constants. Imposing on the metric functionV the boundary condition:

V ||z|=0 = 0 , (22)

from (13) we find the brane tensions:

τt = 0 ,

τx1 = τx2 = · · · = τx(N−3)= V ′ ,

τy = −(N − 3)V ′ , (23)

τz = 0 .

3.3 Coupled gravitational and phantom-like scalar field waves

Now let us consider the case with the phantom-like scalar field,

ε = −1 , (24)

when the both metric functions S and V are presented in the metric (14). In this casethe system (9) and (12) has the standing wave solution of the form:

V = [C1 sin(ωt) + C2 cos(ωt)] [C3 sin(ω|z|) + C4 cos(ω|z|)] ,

φ = 1

2

√M N−2(N − 2)(N − 3) [C1 sin(ωt) + C2 cos(ωt)] [C3 sin(ω|z|)

+C4 cos(ω|z|)] , (25)

S = [C5 sin(�t) + C6 cos(�t)] [C7 sin(�|z|) + C8 cos(�|z|)] ,

with Ci (i = 1, 2, 3, . . . , 8), � and ω being some constants.Imposing on the metric functions S and V the boundary conditions on the brane:

S||z|=0 = 0 ,

V ||z|=0 = 0 , (26)

form (13) we find the brane tensions:

τt = 0 ,

τx1 = τx2 = · · · = τx(N−3)= −S′ + V ′ ,

τy = −S′ − (N − 3)V ′ , (27)

τz = 0 .

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From (25) it’s clear that there are two different frequencies associated with the metricfunctions S and V (� and ω, respectively), and that the oscillation frequency of thephantom-like bulk scalar field standing wave coincides with the frequency of V .

4 Solutions with k �= 0

In general case, when the metric curvature scale k in (7) is non-zero, the system ofEqs. (9) and (12) is self consistent only for the following values of the exponents aand b:

a = − N − 3

N − 2,

b = 2

N − 2. (28)

In this case the metric (7), together with the singularity at |z| = 0, where we placethe brane, and when k > 0, has the horizons at |z| = −1/k. At these points somecomponents of Ricci tensor get infinite values, while all gravitational invariants, forexample the Ricci scalar,

R = 2

(S′ + N − 1

N − 2k

)e−Sδ(z) + (1 + k|z|)(N−3)/(N−2) e−S (

S′′ − S)

, (29)

are finite. It resembles the situation with the Schwarzschild Black Hole, however, incontrast, the determinant of (7) becomes zero at |z| = −1/k. As a result nothing cancross these horizons and for the brane observer the extra space z is effectively finite.

For k �= 0 there also exist three independent nontrivial braneworld solutions.

4.1 The domain wall in N dimensions

First of all we want to mention the simplest case without the scalar field and metricfunctions:

φ = 0 ,

V = 0 , (30)

S = 0 ,

corresponding to the N -dimensional generalization of the known 4D [40–42] and 5D[30,31] domain wall solutions:

ds2 = (1 + k|z|)− N−3N−2

(dt2 − dz2

)− (1 + k|z|) 2

N−2

(N−3∑i=1

dx2i + dy2

). (31)

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In this case the brane tensions in (4) are:

τt = −2k ,

τx1 = · · · = τx(N−3)= τy = − N − 3

N − 2k , (32)

τz = 0 .

4.2 Gravitational and phantom-like scalar field standing waves bounded by the brane

Now we consider the case with the phantom-like scalar field and when the metricfunction S is zero:

ε = −1 ,

S = 0 . (33)

Then the metric (7) reduces to:

ds2 =(1+k|z|)a(

dt2 − dz2)−(1 + k|z|)b

[eV

N−3∑i=1

dx2i +e−(N−3)V dy2

], (34)

and the Eqs. (9) and (12) have the following solution:

V = A sin(ωt)J0(X) ,

φ = A

2

√M N−2(N − 2)(N − 3) sin(ωt)J0(X) , (35)

where A and ω are some constants, J0 is Bessel function of the first kind, and theargument X is defined as:

X = |ω||k| (1 + k|z|) . (36)

Imposing on the metric function V the boundary condition:

V ||z|=0 = 0 , (37)

the equations (13) for the brane tensions will give the solution:

τt = −2k,

τx1 = τx2 = · · · = τxN−3 = − N − 3

N − 2k + V ′ ,

τy = − N − 3

N − 2k − (N − 3)V ′ , (38)

τz = 0 .

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4.3 Standing wave braneworld with the real scalar field

Finally we introduce most realistic general solution to the Eqs. (9), (12) and (13) withthe real scalar field,

ε = +1 , (39)

and all parameters and functions in the metric (7) being non-zero. Nontrivial solutionto (9) and (12) in this case represents N -dimensional version of the standing wavebraneworld introduced in [30,31] and is done by:

V = [C1 sin(ωt) + C2 cos(ωt)] [C3 J0(X) + C4Y0(X)] ,

φ = 1

2

√M N−2(N − 2)(N − 3) [C1 cos(ωt) − C2 sin(ωt)] [C3 J0(X) + C4Y0(X)] ,

S = 1

2(N − 2)(N − 3)X2

{C2

3

[J0(X)2 + J1(X)2 − 1

XJ0(X)J1(X)

](40)

+C24

[Y0(X)2 + Y1(X)2 − 1

XY0(X)Y1(X)

]

+ C3C4

[2 [J0(X)Y0(X) + J1(X)Y1(X)] − 1

X[J0(X)Y1(X)

+J1(X)Y0(X)]]} + C5 ,

where Ci (i = 1, 2, 3, 4, 5) and ω are some constants, J0 (J1) and Y0 (Y1) are Besselfunctions of the first and the second kind, respectively, and X is defined by (36).

Imposing on the metric functions S and V the boundary conditions on the brane:

S||z|=0 = 0 ,

V ||z|=0 = 0 , (41)

the system of equations (13) for the brane tensions will have the following solution:

τt = −2k ,

τx1 = τx2 = · · · = τx(N−3)= − N − 3

N − 2k − S′ + V ′ ,

τy = − N − 3

N − 2k − S′ − (N − 3)V ′ , (42)

τz = 0 . (43)

5 Conclusions

In this paper we have presented the new class of solutions to the system of the Einsteinand Klein-Gordon equations in N-dimensional space-time. The model represents asingle (N-1)-brane in a space-time with one large (infinite) and (N-5) small (com-pact) space-like extra dimensions. We also have derived and explicitly solved thecorresponding junction conditions and have obtained analytical expressions for the

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tensions of the brane. Some cases of the general solutions, which correspond to thebraneworlds in 5D and 6D with gravitational and scalar field standing waves boundedby the brane, where already studied in [24–31,44–47].

Considered in this paper N-dimensional models can be relevant in string and otherhigher dimensional theories, since they can provide universal gravitational trappingof all kinds of matter fields on the brane, and realize the natural mechanisms ofdimensional reduction of multi-dimensional space-times and isotropization of initiallynon-symmetrically warped braneworlds [51,52].

Acknowledgments M.G. was partially supported by the grant of Shota Rustaveli National Science Foun-dation #DI/8/6 − 100/12. The research of P.M. was supported by Ilia State University (Georgia).

References

1. Arkani-Hamed, N., Dimopoulos, S., Dvali, G.: Phys. Lett. B 429, 263 (1998). arXiv:hep-ph/98033152. Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S., Dvali, G.: Phys. Lett. B 436, 257 (1998). arXiv:

hep-ph/98043983. Gogberashvili, M.: Int. J. Mod. Phys. D 11, 1635 (2002). arXiv:hep-ph/98122964. Gogberashvili, M.: Mod. Phys. Lett. A 14, 2025 (1999). arXiv:hep-ph/99043835. Randall, L., Sundrum, R.: Phys. Rev. Lett. 83, 3370 (1999). arXiv:hep-ph/99052216. Randall, L., Sundrum, R.: Phys. Rev. Lett. 83, 4690 (1999). arXiv:hep-th/99060647. Rubakov, V.A.: Phys. Usp. 44, 871 (2001); (Usp. Fiz. Nauk 171 (2001) 913)8. Langlois, D.: Prog. Theor. Phys. Suppl. 148, 181 (2003). arXiv:hep-th/02092619. Mannheim, P.D.: Brane-Localized Gravity. World Scientific, Singapore (2005)

10. Maartens, R., Koyama, K.: Living Rev. Rel. 13, 5 (2010). arXiv:1004.3962 [hep-th]11. Gutperle, M., Strominger, A.: JHEP 0204, 018 (2002). arXiv:hep-th/020221012. Kruczenski, M., Myers, R.C., Peet, A.W.: JHEP 0205, 039 (2002). arXiv:hep-th/020414413. Ivashchuk, V.D., Singleton, D.: JHEP 0410, 061 (2004). arXiv:hep-th/040722414. Burgess, C.P., Quevedo, F., Rabadan, R., Tasinato, G., Zavala, I.: JCAP 0402, 008 (2004). arXiv:

hep-th/031012215. Dubovsky, S.L., Rubakov, V.A., Tinyakov, P.G.: JHEP 0008, 041 (2000). arXiv:hep-ph/000717916. Bajc, B., Gabadadze, G.: Phys. Lett. B 474, 282 (2000). arXiv:hep-th/991223217. Pomarol, A.: Phys. Lett. B 486, 153 (2000). arXiv:hep-ph/991129418. Oda, I.: Phys. Rev. D 62, 126009 (2000). arXiv:hep-th/000801219. Gogberashvili, M., Midodashvili, P.: Phys. Lett. B 515, 447 (2001). arXiv:hep-ph/000529820. Gogberashvili, M., Midodashvili, P.: Europhys. Lett. 61, 308 (2003). arXiv:hep-th/011113221. Gogberashvili, M., Singleton, D.: Phys. Lett. B 582, 95 (2004). arXiv:hep-th/031004822. Gogberashvili, M., Singleton, D.: Phys. Rev. D 69, 026004 (2004). arXiv:hep-th/030524123. Gogberashvili, M., Midodashvili, P., Singleton, D.: JHEP 0708, 033 (2007). arXiv:0706.0676 [hep-th]24. Gogberashvili, M., Singleton, D.: Mod. Phys. Lett. A 25, 2131 (2010). arXiv:0904.2828 [hep-th]25. Gogberashvili, M., Midodashvili, P., Midodashvili, L.: Phys. Lett. B 702, 276 (2011). arXiv:1105.

1701 [hep-th]26. Gogberashvili, M., Midodashvili, P., Midodashvili, L.: Phys. Lett. B 707, 169 (2012). arXiv:1105.

1866 [hep-th]27. Gogberashvili, M., Midodashvili, P., Midodashvili, L.: Int. J. Mod. Phys. D 21, 1250081 (2012). arXiv:

1209.3815 [hep-th]28. Gogberashvili, M.: JHEP 09, 056 (2012). arXiv:1204.2448 [hep-th]29. Gogberashvili, M., Sakhelashvili, O., Tukhashvili, G.: Mod. Phys. Lett. A 28, 1350092 (2013). arXiv:

1304.6079 [hep-th]30. Gogberashvili, M., Midodashvili, P.: Adv. High Energy Phys. 2013, 873686 (2013). arXiv:1310.1911

[hep-th]31. Gogberashvili, M., Midodashvili, P.: Euro. Phys. Lett 104, 50002 (2013). arXiv:1312.6241 [hep-th]32. Phillips, W.D.: Rev. Mod. Phys. 70, 721 (1998)33. Metcalf, H.J., van der Straten, P.: Laser Cooling and Trapping. Springer, New York (1999)

123

http://arxiv.org/abs/arXiv:hep-ph/9803315






http://arxiv.org/abs/arXiv:hep-th/9906064


http://arxiv.org/abs/arXiv:1004.3962




























34. Moiseyev, N., Šindelka, M., Cederbaum, L.S.: Phys. Lett. A 362, 215 (2007)35. Moiseyev, N., Šindelka, M., Cederbaum, L.S.: Phys. Rev. A 74, 053420 (2006)36. Baskaran, D., Grishchuk, L.P.: Class. Quant. Grav. 21, 4041 (2004). arXiv:gr-qc/030905837. Dixon, W.G.: Nuovo Cim. 34, 318 (1964)38. Dixon, W.G.: Proc. R. Soc. Lond. A 314, 499 (1970)39. Dixon, W.G.: Gen. Relativ. Gravit. 4, 199 (1973)40. Taub, A.H.: Phys. Rev. 103, 454 (1956)41. Vilenkin, A.: Phys. Rev. D 23, 852 (1981)42. Ipser, J., Sikivie, P.: Phys. Rev. D 30 (1984)43. Gogberashvili, M., Myrzakul, S., Singleton, D.: Phys. Rev. D 80, 024040 (2009). arXiv:0904.1851

[gr-qc]44. Sousa, L.J.S., Silva, J.E.G., Almeida, C.A.S.: arXiv:1209.2727 [hep-th]45. Sousa, L.J.S., Silva, J.E.G., Almeida, C.A.S.: Phys. Rev. D 89 (2014); arXiv:1311.5848 [hep-th]46. Midodashvili, P.: Int. J. Theor. Phys. 53 (2014); arXiv:1211.0206 [hep-th]47. Sakhelashvili, O.: Int. J. Theor. Phys. 53 (2014); arXiv:1311.1030 [gr-qc]48. Yurtsever, U.: Phys. Rev. D 38, 1706 (1988)49. Feinstein, A., Ibañez, J.: Phys. Rev. D 39, 470 (1989)50. Griffiths, J.: Colliding Plane Waves in General Relativity. Oxford University Press, Oxford (1991);

Chapter 1051. Gogberashvili, M., Herrera-Aguilar, A., Malagón-Morejón, D., Mora-Luna, R.R., Nucamendi, U.:

Phys. Rev. D 87, 084059 (2013). arXiv:1201.4569 [hep-th]52. Gogberashvili, M., Herrera-Aguilar, A., Malagón-Morejón, D., Mora-Luna, R.R.: Phys. Lett. B 725,

208 (2013). arXiv:1202.1608 [hep-th]

123

http://arxiv.org/abs/arXiv:gr-qc/0309058








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