Research Article Modified Hamiltonian Formalism for Regge ...downloads.hindawi.com/archive/2014/606727.pdf · Research Article Modified Hamiltonian Formalism for Regge-Teitelboim

Research ArticleModified Hamiltonian Formalism forRegge-Teitelboim Cosmology

Pinaki Patra Md Raju Gargi Manna and Jyoti Prasad Saha

Department of Physics University of Kalyani Kalyani 741235 India

Correspondence should be addressed to Pinaki Patra monkjugmailcom

Received 22 July 2014 Revised 17 November 2014 Accepted 8 December 2014 Published 28 December 2014

Academic Editor Ashok Chatterjee

Copyright copy 2014 Pinaki Patra et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Ostrogradski approach for the Hamiltonian formalism of higher derivative theory is not satisfactory because the Lagrangiancannot be viewed as a function on the tangent bundle to coordinate manifold In this paper we have used an alternative approachwhich leads directly to the Lagrangian which being a function on the tangent manifold gives correct equation of motion no newcoordinate variables need to be added This approach can be used directly to the singular (in Ostrogradski sense) Lagrangian Wehave used this method for the Regge-Teitelboim (RT) minisuperspace cosmological model We have obtained the Hamiltonian ofthe dynamical equation of the scale factor of RT model

1 Introduction

It is fairly well known that adding higher derivatives termin Lagrangian may improve the theory in some respectslike ultraviolet behavior [1 2] gravity renormalization [3] oreven making modified gravity asymptotically free [4] Alsohigher-derivative Lagrangians appear to be a useful tool todescribe some interesting models like relativistic particleswith rigidity curvature and torsion [5ndash10]

The starting of the study of higher derivative theorywas started long back Classical dynamics of a test particlersquosmotion with higher-order time derivatives of the coordinateswas first described in 1850 by Ostrogradski [11] and is knownas Ostrogradskirsquos formalism which has been extensivelystudied by several authors for its wide applicability [12ndash30]

An interesting occurrence of higher derivative terms inthe action appears in general relativity In some cases suchterms are isolated as surface terms and dropped Howeverin the case of gravity the surface term is never ignorablefor example the requirement of the Gibbons-Hawking termin the action This is more so in the brane world scenariowhere the universe is viewed as a hypersurface immersedin a bulk A classic model is due to Regge and Teitelboim(RT) [31] where gravitation is described as the world volumeswept out by the motion of a three-dimensional brane in

a higher dimensional Minkowski spacetime Hamiltoniananalysis of the model and its quantization was furtherexplored in [32ndash35] Unlike the Einstein gravity in the RTmodel the independent fields are the embedding functionsrather than the metric In the RT model second derivativesof the fields appear in the action and like general relativitythese higher derivative terms may be clubbed in a surfaceterm In the usual formulation this surface term is dropped[35] thereby reducing the original model to a first-ordertheory However this makes the Hamiltonian formulationof the model problematic [35] These problems are bypassedby introducing an auxiliary field [35] On the other handrecently it has been pointed out that no such auxiliary fieldis needed if one includes the surface term in the RT modelcontaining higher derivative terms [34] Obviously thereforethe Hamiltonian formulation of this model is far from closedThe analysis of the RTmodel in the ambit of higher derivativetheory [34] was done from the Ostrogradski approach andthis work was based on the minisuperspace model followingfrom the RT theory The minisuperspace model carries thereparametrization invariance of the original RT gravity whichappears as gauge invariance in the Hamiltonian analysis

However the main disadvantage of the Ostrogradskiapproach is that the Hamiltonian being a linear functionof some momenta is necessarily unbounded from below In

Hindawi Publishing CorporationPhysics Research InternationalVolume 2014 Article ID 606727 6 pageshttpdxdoiorg1011552014606727

2 Physics Research International

general this cannot be cured by trying to devise an alternativecanonical formalism In fact any Hamiltonian is an integralof motion while it is by far not obvious that a genericsystem described by higher derivative Lagrangians possessesglobally defined integrals of motion except the one related totime translation invariance Moreover the instability of theOstrogradski Hamiltonian is not related to finite domains inphase space which implies that it will survive in the standardquantization procedure (ie it cannot be cured by the uncer-tainty principle) The Ostrogradski approach also has someother disadvantages There is no straightforward transitionfrom the Lagrangian to the Hamiltonian formalism

Recently Andrzejewski et al [36] have proposed a mod-ified formalism for the higher derivative theory which cancure some of the drawbacks of Ostrogradski formalism Basicidea is the same as that of Ostrogradski But the advantageof this approach is that the Legendre transformation can beperformed in a straightforward way though the Hamiltonianof the modified formalism is directly connected to theHamiltonian obtained in Ostrogradski formalism through acanonical transformation

In this paper we have used the modified formalismproposed byAndrzejewski et al [36] for theRegge-Teitelboim(RT) minisuperspace cosmological model

2 Regge-Teitelboim Cosmological Model

The Regge-Teitelboim cosmological model has been studiedin [37] We include this section for the completeness of ourpaper and we used their notation The RT model considers a119889-dimensional brane Σ which evolves in an 119873-dimensionalbulk spacetime with fixed Minkowski metric 120578120583] The worldvolume swept out by the brane is a119889+1-dimensionalmanifold119898 defined by the embedding 119909

120583= 119883120583120577(119886) where 119909

120583 arethe local coordinates of the background spacetime and 120577(119886)

are local coordinates for119898 The theory is given by the actionfunctional

119878 [119883] = int119898

119889119889+1

120577radicminus119892(120573

2R minus Λ) (1)

where 120573 has the dimension [119871]1minus119889 and 119892 is the determinant ofthe induced metric 119892119886119887 Λ denotes the cosmological constantand 119877 is the Ricci scalar As has been already stated above wewill be confined to the minisuperspace cosmological modelfollowing from the RT model

The standard procedure in cosmology is to assume thaton the large scale the universe is homogeneous and isotropicThese special symmetries enable the four-dimensional worldvolume representing the evolving universe to be embeddedin a five-dimensional Minkowski spacetime

1198891199042= minus119889119905

2+ 1198891198862+ 1198862119889Ω2

3 (2)

where 119889Ω23is the metric for unit 3 sphere To ensure the FRW

case we take the following parametric representation for thebrane

119909120583= 119883120583(120577119886) = (119905 (120591) 119886 (120591) 120594 120579 120601) (3)

where 119886(120591) is known as the scale factor

After ADM decomposition [38 39] with space-like unitnormals (119873 = radic 1199052 minus 1198862 is the lapse function)

119899120583 =1

119873(minus 119886 119905 0 0 0) (4)

the induced metric on the world volume is given by

1198891199042= minus119873

21198891205912+ 1198862119889Ω2

3 (5)

Now one can compute the Ricci scalar which is given by

R =6 119905

11988621198734(119886 119886 119905 minus 119886 119886 119905 + 119873

2 119905) (6)

With these functions we can easily construct the Lagrangiandensity as

L = radicminus119892(120573

2R minus Λ) (7)

The Lagrangian in terms of arbitrary parameter 120591 can bewritten as [34]

L (119886 119886 119886 119905 119905) =119886 119905

1198733(119886 119886 119905 minus 119886 119886 119905 + 119873

2 119905) minus 11987311988631198672 (8)

where

1198672=

Λ

3120573 (9)

119867 is called the Hubble parameterVarying the action with respect to 119886(120591) we get the corres-

ponding Euler-Lagrange equation

119889

119889120591(

119886

119905) = minus

1198732( 1199052minus 3119873211988621198672)

119886 119905 (3 1199052 minus 119873211988621198672) (10)

Please note that the Lagrangian contains higher derivativeterms of field 119886 However we can write it as [34]

119871 = minus119886 1198862

119873+ 119886119873(1 minus 119886

21198672) +

119889

119889120591(1198862119886

119873) (11)

If we neglect the boundary term the resulting Lagrangianbecomes the usual first-order one However the Hamiltoniananalysis is facilitated if we retain the higher derivative termThus our Hamiltonian analysis will proceed from the aboveequation containing higher derivative term Note that thehigher-order model was also considered in [34] where theHamiltonian analysis was performed following the Ostro-gradski approach We on the contrary will follow the equi-valent first-order approach of Andrzejewski et al [36]

3 Hamiltonian Formalism forRegge-Teitelboim Cosmological Model

Our concerned Lagrangian as mentioned in the previous sec-tion is

119871 (119886 119886 119886 119905 119905) =119886 119905

1198733(119886 119886 119905 minus 119886 119886 119905 + 119873

2 119905) minus 11987311988631198672 (12)

Physics Research International 3

where

119873 = radic 119905 minus 1198862 (13)

and the Hubble parameter which we are considering to be aconstant in the present discussion is

1198672=

Λ

3120573 (14)

We set for the notational convenience

119886 = 1199021

1 119905 = 119902

2

1 119886 = 119902

1

1

119905 = 1199022

1 119886 = 119902

1

2 119905 = 119902

2

2

(15)

Our Lagrangian is singular in Ostrogradski sense because

det (119882120583]) = det( 1205972119871

120597 119902120583120597 119902]) = det

[[[[

[

1205972119871

120597 119886120597 119886

1205972119871

120597 119886120597 119905

1205972119871

120597 119905120597 119886

1205972119871

120597 119905120597 119905

]]]]

]

= det [0 0

0 0] = 0

(16)

Now we define 119865(11990211 1199022

1 1199021

1 1199022

1 1199021

3 1199022

3) such that

det( 1205972119865

120597 119902120583

1120597119902]3

) =

[[[[

[

1205972119865

120597 1199021112059711990213

1205972119865

120597 1199021112059711990223

1205972119865

120597 1199022112059711990213

1205972119865

120597 1199022112059711990223

]]]]

]

= 0 (17)

One possible choice is

119865 (1199021

1 1199022

1 1199021

1 1199022

1 1199021

3 1199022

3) = 120572 ( 119902

1

11199021

3+ 1199022

11199022

3) + 120573 ( 119902

1

11199022

3+ 1199022

11199021

3)

Δ = 1205722minus 1205732

= 0

(18)

Now using the suggestions prescribed in [36] we define

L (1199021

1 1199022

1 1199021

1 1199022

1 1199021

2 1199022

2 1199021

3 1199022

3 1199021

3 1199022

3)

= 119871 +120597119865

120597119902120583

1

119902120583

1+

120597119865

120597119902120583

3

119902120583

3+

120597119865

120597 119902120583

1

119902120583

2

=1199021

11199022

1

1198733(1199021

11199021

21199022

1minus 1199021

11199021

11199022

2+ 11987321199022

1) minus 119873(119902

1

1)3

1198672

+ (120572 1199021

1+ 120573 1199022

1) 1199021

3+ (120573 119902

1

1+ 120572 1199022

1) 1199022

3+ 1199021

2(1205721199021

3+ 1205731199022

3)

+ (1205721199022

3+ 1205731199021

3) 1199022

2

(19)

The conjugatemomenta corresponding to the coordinates aregiven by 119901119894119895 = 120597L120597 119902

119895

119894 In particular

11990111 =3

1198735(1199021

1)2

1199021

11199022

1(1199021

21199022

1minus 1199021

11199022

2) +

1199021

11199022

1

1198733( 1199021

11199022

1minus 1199021

11199022

2)

minus1198672

1198731199021

1(1199021

1)3

+ 120572 1199021

3+ 120573 1199022

3

(20)

11990112 =3 (1199021

11199022

1)2

1198735( 1199021

11199022

2minus 1199021

21199022

1)

+1199021

1

1198733(21199021

11199021

21199022

1minus 1199021

11199021

11199022

2minus ( 1199022

1)3

)

+1199021

1

119873(2 1199022

1minus 1198672(1199021

1)2

1199022

1) + (120572 119902

2

3+ 120573 1199021

3)

(21)

11990121 = 0 (22)

11990122 = 0 (23)

11990131 =120597119865

12059711990213

= 120572 1199021

1+ 120573 1199022

1 (24)

11990132 =120597119865

12059711990223

= 120573 1199021

1+ 120572 1199022

1 (25)

Equations (22) and (23) provide primary constraint Equation(17) enables us to solve 119902rsquos in terms of momenta These areexplicitly given by

1199021

1=

1

Δ11987531 119902

2

1=

1

Δ11987532

1199021

3=1205729848581 minus 1205739848582

Δ 119902

2

3=1205729848582 minus 1205739848581

Δ

(26)

where

11987531 = 12057211990131 minus 12057311990132

11987532 = (12057211990132 minus 12057311990131)

9848581 = 11990111 +1

119873Δ1198672(1199021

1)3

11987531 minus1199021

1

1198733Δ11987532 [

1

Δ21198753111987532 minus 119902

1

11199022

2]

minus3 (1199021

1)2

1198735Δ31198753111987532 [119902

1

211987532 minus 119902

2

211987531]

9848582 = 11990112 +(1199021

111987532)2

Δ(1199021

211987532 minus 119902

2

211987531) +

1199021

1

1198733Δ

times [1199021

1119875311199022

2+

1

Δ21198753

32minus 21199021

11199021

211987532]

1199021

1

119873Δ

times [1198672(1199021

1)2

minus 2] 11987532

(27)


Therefore119873 reduces to

119873 =radic(11990132)2minus (11990131)

2

Δ

(28)

Now the Dirac Hamiltonian (H) is given by

H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2+ 1198881205831199012120583 (29)

where 119888120583 are two Lagrange multipliers enforcing the ldquopri-

maryrdquo constraints

Φ1120583 equiv 1199012120583 asymp 0 (30)

The Hamiltonian explicitly reads

H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2

minus1

1198733Δ2[(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

+ 119888111990121 + 119888

211990122

(31)

And the Hamilton equations of motion are given by

1199021

1=

1

Δ11987531

11 =1

119873Δ2(11987532)2+

21199021

1

1198733Δ2(1199021

211987532 minus 119902

2

211987531) 11987532

minus 31198672119873(1199021

1)2

1199022

1=

1

Δ11987532 12 = 0 119902

1

2= 1198881

21 =1

1198733Δ2(1199021

111987532)2

+ 1205721199021

3+ 1205731199022

3

1199022

2= 1198882

22 = minus1

1198733Δ2(1199021

1)2

1198753111987532 + 1205721199022

3+ 1205731199021

3

1199021

3=

1

Δ(12057211990111 minus 12057311990112) minus

1

1198733Δ31199021

111990131 (11987532)

2

+2120573

119873Δ21199021

111987532 minus

311990131

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[1205731199022

211987531 minus (120572119902

2

2+ 2120573119902

1

2) 11987532]

minus1

119873Δ119867211990131 (119902

1

1)3

31 = 1205721199021

2+ 1205731199022

2

1199022

3=

1

Δ(12057211990112 minus 12057311990111) +

1

1198733Δ31199021

111990132 (11987532)

2

minus2120572

119873Δ21199021

111987532 +

311990132

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[minus1205721199022

211987531 + (2120572119902

1

2+ 1205731199022

2) 11987532]

+1

119873Δ119867211990132 (119902

1

1)3

32 = 1205721199022

2+ 1205731199021

2

(32)

The secondary constraints read

0 asymp Φ2120583 =120597119871 (1199021 1199021 1199022)

120597119902120583

2

+120597119865 (1199021 1199021 1199023)

120597 119902120583

1

(33)

that is

0 asymp Φ21 =1

1198733(1199021

11199022

1)2

+ 1205721199021

3+ 1205731199022

3

0 asymp Φ22 = minus1

1198733(1199021

1)2

1199021

11199022

1+ 1205721199022

3+ 1205731199021

3

(34)

To determine 119888120583 usually one uses the stability condition ofthe secondary constraints

0 asymp Φ2120583H (35)

But for our system under consideration 119882 in (16) has rank0 This implies the existence of 2 linearly independent nullvectors so one cannot obtain the values of 119888120583 in this caseHowever if we are interested in the dynamical equations for119886 and 119905 we can use the Hamiltonian

H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2 (36)

which is explicitly given by

H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2minus

1

1198733Δ2

times [(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

(37)

If we promote it to quantization it can easily be seen thatHis hermitian

4 Conclusions

We have obtained the Hamiltonian structure for the scalefactor of the RT model In the modified formalism used inthis paper the Legendre transformation can be performed in


a straightforwardway Summarizing we have foundmodifiedHamiltonian formulations of RT gravity which is equivalentto the Ostrogradski formalism in the sense that they arerelated to the latter by a canonical transformation

The stability condition of constraint for the modifiedformalism proposed in [36] fails to determine the lagrangemultipliers for the model discussed in this paper In thatsense one can conclude that the modified formalism pro-posed in [36] is not always superior to the usual Ostrogradskiformalism used in literature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their careful reading of the paper and for their valuablecomments whichmade this paper in the present form PinakiPatra is grateful forCSIRGovernment of India for fellowshipsupport Gargi Manna is grateful for DST Government ofIndia for DST-INSPIRE scholarship and Jyoti Prasad Sahais grateful for DST-PURSE for financial support

References

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[2] A Pais and G E Uhlenbeck ldquoOn field theories with non-loca-lized actionrdquo Physical Review Letters vol 79 pp 145ndash165 1950

[3] K S Stelle ldquoRenormalization of higher-derivative quantumgravityrdquo Physical Review D vol 16 no 4 pp 953ndash969 1977

[4] E S Fradkin and A A Tseytlin ldquoRenormalizable asymptoti-cally free quantum theory of gravityrdquoNuclear Physics B vol 201no 3 pp 469ndash481 1982

[5] Y A Kuznetsov andM S Plyushchay ldquoThemodel of the relativ-istic particle with curvature and torsionrdquoNuclear Physics B vol389 no 1 pp 181ndash205 1993

[6] R D Pisarski ldquoField theory of paths with a curvature-depend-ent termrdquo Physical Review D Particles and Fields Third Seriesvol 34 no 2 pp 670ndash673 1986

[7] V V Nesterenko ldquoSingular Lagrangians with higher deriva-tivesrdquo Journal of Physics A Mathematical and General vol 22no 10 pp 1673ndash1687 1989

[8] M S Plyushchay ldquoMassive relativistic point particle with rigi-dityrdquo International Journal of Modern Physics A vol 4 p 38511989

[9] M S Plyushchay ldquoThe model of the relativistic particle withtorsionrdquo Nuclear Physics B vol 362 no 1-2 pp 54ndash72 1991

[10] R Banerjee P Mukherjee and B Paul ldquoGauge symmetry andW-algebra in higher derivative systemsrdquo Journal of High EnergyPhysics vol 2011 article 85 2011

[11] M Ostrogradski Memoires de lrsquoAcademie Imperiale des Sci-ences de Saint-Petersbourg Series 4 1850

[12] B Podolsky ldquoA generalized electrodynamics I NonquantumrdquoPhysical Review vol 62 pp 68ndash71 1942

[13] B Podolsky and C Kikuchi ldquoA generalized electrodynamicspart IImdashquantumrdquo Physical Review vol 65 p 228 1944

[14] B Podolsky and C Kikuchi ldquoAuxiliary conditions and electro-static interaction in generalized quantum electrodynamicsrdquo vol67 pp 184ndash192 1945

[15] J Iliopoulos and B Zumino ldquoBroken supergauge symmetry andrenormalizationrdquo Nuclear Physics B vol 76 no 2 pp 310ndash3321974

[16] D A Eliezer and R P Woodard ldquoThe problem of nonlocalityin string theoryrdquo Nuclear Physics B vol 325 no 2 pp 389ndash4691989

[17] I P Neupane ldquoConsistency of higher derivative gravity in thebrane backgroundrdquo Journal of High Energy Physics vol 2000no 9 article 040 2000

[18] S Nojiri S D Odintsov and S Ogushi ldquoCosmological andblack hole brane-world universes in higher derivative gravityrdquoPhysical Review D vol 65 Article ID 023521 2001

[19] C-S Chu J Lukierski andW J Zakrzewski ldquoHermitian analy-ticity IRUV mixing and unitarity of noncommutative fieldtheoriesrdquoNuclear Physics B vol 632 no 1ndash3 pp 219ndash239 2002

[20] S M Carroll M Hoffman and M Trodden ldquoCan the darkenergy equation-of-state parameter w be less than minus1rdquo PhysicalReview D vol 68 Article ID 023509 2003

[21] A Anisimov E Babichev and A Vikman ldquoB-inflationrdquo Journalof Cosmology and Astroparticle Physics vol 2005 no 06 2005

[22] R PWoodard ldquoAvoiding dark energy with 1Rmodifications ofgravityrdquo inThe Invisible Universe DarkMatter andDark Energyvol 720 of Lecture Notes in Physics pp 403ndash433 SpringerBerlin Germany 2007

[23] R Cordero A Molgado and E Rojas ldquoOstrogradski approachfor the Regge-Teitelboim type cosmologyrdquo Physical Review DndashParticles Fields Gravitation and Cosmology vol 79 no 2Article ID 024024 10 pages 2009

[24] R Andringa E A Bergshoeff M De Roo O Hohm E Sezginand P K Townsend ldquoMassive 3D supergravityrdquo Classical andQuantum Gravity vol 27 no 2 Article ID 025010 2010

[25] E A Bergshoeff O Hohm J Rosseel E Sezgin and P KTownsend ldquoMore on massive 3D supergravityrdquo Classical andQuantum Gravity vol 28 no 1 Article ID 015002 2011

[26] F S Gama M Gomes J R Nascimento A Yu Petrov and AJ da Silva ldquoHigher-derivative supersymmetric gauge theoryrdquoPhysical Review D vol 84 Article ID 045001 2011

[27] P Mukherjee and B Paul ldquoGauge invariances of higher deriva-tive Maxwell-Chern-Simons field theorymdasha new Hamiltonianapproachrdquo Physical Review D vol 85 no 4 Article ID 0450282012

[28] A Escalante J Guven E Rojas and R Capovilla ldquoHamiltoniandynamics of extended objects Regge-Teitelboim modelrdquo Inter-national Journal of Theoretical Physics vol 48 pp 2486ndash24982009

[29] R Cordero M Cruz A Molgado and E Rojas ldquoQuantummodified Regge-Teitelboim cosmologyrdquo General Relativity andGravitation vol 46 no 7 p 1761 2014 Erratum in GeneralRelativity and Gravitation vol 46 no 8 p 1770 2014

[30] R Cordero A Molgado and E Rojas ldquoQuantum charged rigidmembranerdquo Classical and Quantum Gravity vol 28 no 6Article ID 065010 2011

[31] T Regge and C Teitelboim in Proceedings of the Marcel Gross-manMeeting R Ruffini Ed North-Holland Trieste Italy 1977

[32] A Davidson D Karasik and Y Lederer ldquoWavefunction of abrane-like universerdquo Classical Quantum Gravity vol 16 no 4pp 1349ndash1356 1999


[33] A Davidson D Karasik and Y Lederer ldquoGeodesic evolutionand nucleation of a de Sitter branerdquo Physical Review D vol 72no 6 Article ID 064011 5 pages 2005

[34] R Cordero A Molgado and E Rojas ldquoOstrogradski approachfor the Regge-Teitelboim type cosmologyrdquo Physical Review Dvol 79 no 2 Article ID 024024 2009

[35] D Karasik and A Davidson ldquoGeodetic brane gravityrdquo PhysicalReview D vol 67 no 6 2003

[36] K Andrzejewski J Gonera P MacHalski and P MaslankaldquoModified Hamiltonian formalism for higher-derivative the-oriesrdquo Physical Review D-Particles Fields Gravitation andCosmology vol 82 no 4 Article ID 045008 2010

[37] R Banerjee PMukherjee and B Paul ldquoNewHamiltonian anal-ysis of Regge-Teitelboim minisuperspace cosmologyrdquo PhysicalReview DmdashParticles Fields Gravitation and Cosmology vol 89no 4 Article ID 043508 2014

[38] R Arnowitt S Deser and C W Misner ldquoRepublication of thedynamics of general relativityrdquo General Relativity and Gravi-tation vol 40 no 9 pp 1997ndash2027 2008

[39] S Deser F A E Pirani and D C Robinson ldquoNew embeddingmodel of general relativityrdquo Physical ReviewD vol 14 no 12 pp3301ndash3303 1976

Submit your manuscripts athttpwwwhindawicom

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general this cannot be cured by trying to devise an alternativecanonical formalism In fact any Hamiltonian is an integralof motion while it is by far not obvious that a genericsystem described by higher derivative Lagrangians possessesglobally defined integrals of motion except the one related totime translation invariance Moreover the instability of theOstrogradski Hamiltonian is not related to finite domains inphase space which implies that it will survive in the standardquantization procedure (ie it cannot be cured by the uncer-tainty principle) The Ostrogradski approach also has someother disadvantages There is no straightforward transitionfrom the Lagrangian to the Hamiltonian formalism

Recently Andrzejewski et al [36] have proposed a mod-ified formalism for the higher derivative theory which cancure some of the drawbacks of Ostrogradski formalism Basicidea is the same as that of Ostrogradski But the advantageof this approach is that the Legendre transformation can beperformed in a straightforward way though the Hamiltonianof the modified formalism is directly connected to theHamiltonian obtained in Ostrogradski formalism through acanonical transformation

In this paper we have used the modified formalismproposed byAndrzejewski et al [36] for theRegge-Teitelboim(RT) minisuperspace cosmological model

2 Regge-Teitelboim Cosmological Model

The Regge-Teitelboim cosmological model has been studiedin [37] We include this section for the completeness of ourpaper and we used their notation The RT model considers a119889-dimensional brane Σ which evolves in an 119873-dimensionalbulk spacetime with fixed Minkowski metric 120578120583] The worldvolume swept out by the brane is a119889+1-dimensionalmanifold119898 defined by the embedding 119909

120583= 119883120583120577(119886) where 119909

120583 arethe local coordinates of the background spacetime and 120577(119886)

are local coordinates for119898 The theory is given by the actionfunctional

119878 [119883] = int119898

119889119889+1

120577radicminus119892(120573

2R minus Λ) (1)

where 120573 has the dimension [119871]1minus119889 and 119892 is the determinant ofthe induced metric 119892119886119887 Λ denotes the cosmological constantand 119877 is the Ricci scalar As has been already stated above wewill be confined to the minisuperspace cosmological modelfollowing from the RT model

The standard procedure in cosmology is to assume thaton the large scale the universe is homogeneous and isotropicThese special symmetries enable the four-dimensional worldvolume representing the evolving universe to be embeddedin a five-dimensional Minkowski spacetime

1198891199042= minus119889119905

2+ 1198891198862+ 1198862119889Ω2

3 (2)

where 119889Ω23is the metric for unit 3 sphere To ensure the FRW

case we take the following parametric representation for thebrane

119909120583= 119883120583(120577119886) = (119905 (120591) 119886 (120591) 120594 120579 120601) (3)

where 119886(120591) is known as the scale factor

After ADM decomposition [38 39] with space-like unitnormals (119873 = radic 1199052 minus 1198862 is the lapse function)

119899120583 =1

119873(minus 119886 119905 0 0 0) (4)

the induced metric on the world volume is given by

1198891199042= minus119873

21198891205912+ 1198862119889Ω2

3 (5)

Now one can compute the Ricci scalar which is given by

R =6 119905

11988621198734(119886 119886 119905 minus 119886 119886 119905 + 119873

2 119905) (6)

With these functions we can easily construct the Lagrangiandensity as

L = radicminus119892(120573

2R minus Λ) (7)

The Lagrangian in terms of arbitrary parameter 120591 can bewritten as [34]

L (119886 119886 119886 119905 119905) =119886 119905

1198733(119886 119886 119905 minus 119886 119886 119905 + 119873

2 119905) minus 11987311988631198672 (8)

where

1198672=

Λ

3120573 (9)

119867 is called the Hubble parameterVarying the action with respect to 119886(120591) we get the corres-

ponding Euler-Lagrange equation

119889

119889120591(

119886

119905) = minus

1198732( 1199052minus 3119873211988621198672)

119886 119905 (3 1199052 minus 119873211988621198672) (10)

Please note that the Lagrangian contains higher derivativeterms of field 119886 However we can write it as [34]

119871 = minus119886 1198862

119873+ 119886119873(1 minus 119886

21198672) +

119889

119889120591(1198862119886

119873) (11)

If we neglect the boundary term the resulting Lagrangianbecomes the usual first-order one However the Hamiltoniananalysis is facilitated if we retain the higher derivative termThus our Hamiltonian analysis will proceed from the aboveequation containing higher derivative term Note that thehigher-order model was also considered in [34] where theHamiltonian analysis was performed following the Ostro-gradski approach We on the contrary will follow the equi-valent first-order approach of Andrzejewski et al [36]

3 Hamiltonian Formalism forRegge-Teitelboim Cosmological Model

Our concerned Lagrangian as mentioned in the previous sec-tion is

119871 (119886 119886 119886 119905 119905) =119886 119905

1198733(119886 119886 119905 minus 119886 119886 119905 + 119873

2 119905) minus 11987311988631198672 (12)


where

119873 = radic 119905 minus 1198862 (13)


1198672=

Λ

3120573 (14)


119886 = 1199021

1 119905 = 119902

2

1 119886 = 119902

1

1

119905 = 1199022

1 119886 = 119902

1

2 119905 = 119902

2

2

(15)


det (119882120583]) = det( 1205972119871

120597 119902120583120597 119902]) = det

[[[[

[

1205972119871

120597 119886120597 119886

1205972119871

120597 119886120597 119905

1205972119871

120597 119905120597 119886

1205972119871

120597 119905120597 119905

]]]]

]

= det [0 0

0 0] = 0

(16)

Now we define 119865(11990211 1199022

1 1199021

1 1199022

1 1199021

3 1199022

3) such that

det( 1205972119865

120597 119902120583

1120597119902]3

) =

[[[[

[

1205972119865

120597 1199021112059711990213

1205972119865

120597 1199021112059711990223

1205972119865

120597 1199022112059711990213

1205972119865

120597 1199022112059711990223

]]]]

]

= 0 (17)


119865 (1199021

1 1199022

1 1199021

1 1199022

1 1199021

3 1199022

3) = 120572 ( 119902

1

11199021

3+ 1199022

11199022

3) + 120573 ( 119902

1

11199022

3+ 1199022

11199021

3)

Δ = 1205722minus 1205732

= 0

(18)


L (1199021

1 1199022

1 1199021

1 1199022

1 1199021

2 1199022

2 1199021

3 1199022

3 1199021

3 1199022

3)

= 119871 +120597119865

120597119902120583

1

119902120583

1+

120597119865

120597119902120583

3

119902120583

3+

120597119865

120597 119902120583

1

119902120583

2

=1199021

11199022

1

1198733(1199021

11199021

21199022

1minus 1199021

11199021

11199022

2+ 11987321199022

1) minus 119873(119902

1

1)3

1198672

+ (120572 1199021

1+ 120573 1199022

1) 1199021

3+ (120573 119902

1

1+ 120572 1199022

1) 1199022

3+ 1199021

2(1205721199021

3+ 1205731199022

3)

+ (1205721199022

3+ 1205731199021

3) 1199022

2

(19)


119895


11990111 =3

1198735(1199021

1)2

1199021

11199022

1(1199021

21199022

1minus 1199021

11199022

2) +

1199021

11199022

1

1198733( 1199021

11199022

1minus 1199021

11199022

2)

minus1198672

1198731199021

1(1199021

1)3

+ 120572 1199021

3+ 120573 1199022

3

(20)

11990112 =3 (1199021

11199022

1)2

1198735( 1199021

11199022

2minus 1199021

21199022

1)

+1199021

1

1198733(21199021

11199021

21199022

1minus 1199021

11199021

11199022

2minus ( 1199022

1)3

)

+1199021

1

119873(2 1199022

1minus 1198672(1199021

1)2

1199022

1) + (120572 119902

2

3+ 120573 1199021

3)

(21)

11990121 = 0 (22)

11990122 = 0 (23)

11990131 =120597119865

12059711990213

= 120572 1199021

1+ 120573 1199022

1 (24)

11990132 =120597119865

12059711990223

= 120573 1199021

1+ 120572 1199022

1 (25)


1199021

1=

1

Δ11987531 119902

2

1=

1

Δ11987532

1199021

3=1205729848581 minus 1205739848582

Δ 119902

2

3=1205729848582 minus 1205739848581

Δ

(26)

where

11987531 = 12057211990131 minus 12057311990132

11987532 = (12057211990132 minus 12057311990131)

9848581 = 11990111 +1

119873Δ1198672(1199021

1)3

11987531 minus1199021

1

1198733Δ11987532 [

1

Δ21198753111987532 minus 119902

1

11199022

2]

minus3 (1199021

1)2

1198735Δ31198753111987532 [119902

1

211987532 minus 119902

2

211987531]

9848582 = 11990112 +(1199021

111987532)2

Δ(1199021

211987532 minus 119902

2

211987531) +

1199021

1

1198733Δ

times [1199021

1119875311199022

2+

1

Δ21198753

32minus 21199021

11199021

211987532]

1199021

1

119873Δ

times [1198672(1199021

1)2

minus 2] 11987532

(27)



119873 =radic(11990132)2minus (11990131)

2

Δ

(28)


H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2+ 1198881205831199012120583 (29)



Φ1120583 equiv 1199012120583 asymp 0 (30)


H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2

minus1

1198733Δ2[(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

+ 119888111990121 + 119888

211990122

(31)


1199021

1=

1

Δ11987531

11 =1

119873Δ2(11987532)2+

21199021

1

1198733Δ2(1199021

211987532 minus 119902

2

211987531) 11987532

minus 31198672119873(1199021

1)2

1199022

1=

1

Δ11987532 12 = 0 119902

1

2= 1198881

21 =1

1198733Δ2(1199021

111987532)2

+ 1205721199021

3+ 1205731199022

3

1199022

2= 1198882

22 = minus1

1198733Δ2(1199021

1)2

1198753111987532 + 1205721199022

3+ 1205731199021

3

1199021

3=

1

Δ(12057211990111 minus 12057311990112) minus

1

1198733Δ31199021

111990131 (11987532)

2

+2120573

119873Δ21199021

111987532 minus

311990131

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[1205731199022

211987531 minus (120572119902

2

2+ 2120573119902

1

2) 11987532]

minus1

119873Δ119867211990131 (119902

1

1)3

31 = 1205721199021

2+ 1205731199022

2

1199022

3=

1

Δ(12057211990112 minus 12057311990111) +

1

1198733Δ31199021

111990132 (11987532)

2

minus2120572

119873Δ21199021

111987532 +

311990132

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[minus1205721199022

211987531 + (2120572119902

1

2+ 1205731199022

2) 11987532]

+1

119873Δ119867211990132 (119902

1

1)3

32 = 1205721199022

2+ 1205731199021

2

(32)


0 asymp Φ2120583 =120597119871 (1199021 1199021 1199022)

120597119902120583

2

+120597119865 (1199021 1199021 1199023)

120597 119902120583

1

(33)

that is

0 asymp Φ21 =1

1198733(1199021

11199022

1)2

+ 1205721199021

3+ 1205731199022

3


1198733(1199021

1)2

1199021

11199022

1+ 1205721199022

3+ 1205731199021

3

(34)


0 asymp Φ2120583H (35)


H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2 (36)


H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2minus

1

1198733Δ2

times [(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

(37)


4 Conclusions







Acknowledgments


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where

119873 = radic 119905 minus 1198862 (13)


1198672=

Λ

3120573 (14)


119886 = 1199021

1 119905 = 119902

2

1 119886 = 119902

1

1

119905 = 1199022

1 119886 = 119902

1

2 119905 = 119902

2

2

(15)


det (119882120583]) = det( 1205972119871

120597 119902120583120597 119902]) = det

[[[[

[

1205972119871

120597 119886120597 119886

1205972119871

120597 119886120597 119905

1205972119871

120597 119905120597 119886

1205972119871

120597 119905120597 119905

]]]]

]

= det [0 0

0 0] = 0

(16)

Now we define 119865(11990211 1199022

1 1199021

1 1199022

1 1199021

3 1199022

3) such that

det( 1205972119865

120597 119902120583

1120597119902]3

) =

[[[[

[

1205972119865

120597 1199021112059711990213

1205972119865

120597 1199021112059711990223

1205972119865

120597 1199022112059711990213

1205972119865

120597 1199022112059711990223

]]]]

]

= 0 (17)


119865 (1199021

1 1199022

1 1199021

1 1199022

1 1199021

3 1199022

3) = 120572 ( 119902

1

11199021

3+ 1199022

11199022

3) + 120573 ( 119902

1

11199022

3+ 1199022

11199021

3)

Δ = 1205722minus 1205732

= 0

(18)


L (1199021

1 1199022

1 1199021

1 1199022

1 1199021

2 1199022

2 1199021

3 1199022

3 1199021

3 1199022

3)

= 119871 +120597119865

120597119902120583

1

119902120583

1+

120597119865

120597119902120583

3

119902120583

3+

120597119865

120597 119902120583

1

119902120583

2

=1199021

11199022

1

1198733(1199021

11199021

21199022

1minus 1199021

11199021

11199022

2+ 11987321199022

1) minus 119873(119902

1

1)3

1198672

+ (120572 1199021

1+ 120573 1199022

1) 1199021

3+ (120573 119902

1

1+ 120572 1199022

1) 1199022

3+ 1199021

2(1205721199021

3+ 1205731199022

3)

+ (1205721199022

3+ 1205731199021

3) 1199022

2

(19)


119895


11990111 =3

1198735(1199021

1)2

1199021

11199022

1(1199021

21199022

1minus 1199021

11199022

2) +

1199021

11199022

1

1198733( 1199021

11199022

1minus 1199021

11199022

2)

minus1198672

1198731199021

1(1199021

1)3

+ 120572 1199021

3+ 120573 1199022

3

(20)

11990112 =3 (1199021

11199022

1)2

1198735( 1199021

11199022

2minus 1199021

21199022

1)

+1199021

1

1198733(21199021

11199021

21199022

1minus 1199021

11199021

11199022

2minus ( 1199022

1)3

)

+1199021

1

119873(2 1199022

1minus 1198672(1199021

1)2

1199022

1) + (120572 119902

2

3+ 120573 1199021

3)

(21)

11990121 = 0 (22)

11990122 = 0 (23)

11990131 =120597119865

12059711990213

= 120572 1199021

1+ 120573 1199022

1 (24)

11990132 =120597119865

12059711990223

= 120573 1199021

1+ 120572 1199022

1 (25)


1199021

1=

1

Δ11987531 119902

2

1=

1

Δ11987532

1199021

3=1205729848581 minus 1205739848582

Δ 119902

2

3=1205729848582 minus 1205739848581

Δ

(26)

where

11987531 = 12057211990131 minus 12057311990132

11987532 = (12057211990132 minus 12057311990131)

9848581 = 11990111 +1

119873Δ1198672(1199021

1)3

11987531 minus1199021

1

1198733Δ11987532 [

1

Δ21198753111987532 minus 119902

1

11199022

2]

minus3 (1199021

1)2

1198735Δ31198753111987532 [119902

1

211987532 minus 119902

2

211987531]

9848582 = 11990112 +(1199021

111987532)2

Δ(1199021

211987532 minus 119902

2

211987531) +

1199021

1

1198733Δ

times [1199021

1119875311199022

2+

1

Δ21198753

32minus 21199021

11199021

211987532]

1199021

1

119873Δ

times [1198672(1199021

1)2

minus 2] 11987532

(27)



119873 =radic(11990132)2minus (11990131)

2

Δ

(28)


H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2+ 1198881205831199012120583 (29)



Φ1120583 equiv 1199012120583 asymp 0 (30)


H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2

minus1

1198733Δ2[(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

+ 119888111990121 + 119888

211990122

(31)


1199021

1=

1

Δ11987531

11 =1

119873Δ2(11987532)2+

21199021

1

1198733Δ2(1199021

211987532 minus 119902

2

211987531) 11987532

minus 31198672119873(1199021

1)2

1199022

1=

1

Δ11987532 12 = 0 119902

1

2= 1198881

21 =1

1198733Δ2(1199021

111987532)2

+ 1205721199021

3+ 1205731199022

3

1199022

2= 1198882

22 = minus1

1198733Δ2(1199021

1)2

1198753111987532 + 1205721199022

3+ 1205731199021

3

1199021

3=

1

Δ(12057211990111 minus 12057311990112) minus

1

1198733Δ31199021

111990131 (11987532)

2

+2120573

119873Δ21199021

111987532 minus

311990131

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[1205731199022

211987531 minus (120572119902

2

2+ 2120573119902

1

2) 11987532]

minus1

119873Δ119867211990131 (119902

1

1)3

31 = 1205721199021

2+ 1205731199022

2

1199022

3=

1

Δ(12057211990112 minus 12057311990111) +

1

1198733Δ31199021

111990132 (11987532)

2

minus2120572

119873Δ21199021

111987532 +

311990132

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[minus1205721199022

211987531 + (2120572119902

1

2+ 1205731199022

2) 11987532]

+1

119873Δ119867211990132 (119902

1

1)3

32 = 1205721199022

2+ 1205731199021

2

(32)


0 asymp Φ2120583 =120597119871 (1199021 1199021 1199022)

120597119902120583

2

+120597119865 (1199021 1199021 1199023)

120597 119902120583

1

(33)

that is

0 asymp Φ21 =1

1198733(1199021

11199022

1)2

+ 1205721199021

3+ 1205731199022

3


1198733(1199021

1)2

1199021

11199022

1+ 1205721199022

3+ 1205731199021

3

(34)


0 asymp Φ2120583H (35)


H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2 (36)


H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2minus

1

1198733Δ2

times [(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

(37)


4 Conclusions







Acknowledgments


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119873 =radic(11990132)2minus (11990131)

2

Δ

(28)


H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2+ 1198881205831199012120583 (29)



Φ1120583 equiv 1199012120583 asymp 0 (30)


H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2

minus1

1198733Δ2[(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

+ 119888111990121 + 119888

211990122

(31)


1199021

1=

1

Δ11987531

11 =1

119873Δ2(11987532)2+

21199021

1

1198733Δ2(1199021

211987532 minus 119902

2

211987531) 11987532

minus 31198672119873(1199021

1)2

1199022

1=

1

Δ11987532 12 = 0 119902

1

2= 1198881

21 =1

1198733Δ2(1199021

111987532)2

+ 1205721199021

3+ 1205731199022

3

1199022

2= 1198882

22 = minus1

1198733Δ2(1199021

1)2

1198753111987532 + 1205721199022

3+ 1205731199021

3

1199021

3=

1

Δ(12057211990111 minus 12057311990112) minus

1

1198733Δ31199021

111990131 (11987532)

2

+2120573

119873Δ21199021

111987532 minus

311990131

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[1205731199022

211987531 minus (120572119902

2

2+ 2120573119902

1

2) 11987532]

minus1

119873Δ119867211990131 (119902

1

1)3

31 = 1205721199021

2+ 1205731199022

2

1199022

3=

1

Δ(12057211990112 minus 12057311990111) +

1

1198733Δ31199021

111990132 (11987532)

2

minus2120572

119873Δ21199021

111987532 +

311990132

1198735Δ3(1199021

1)2

11987532 [1199021

211987532 minus 119902

2

211987531]

minus1

1198733Δ2(1199021

1)2

[minus1205721199022

211987531 + (2120572119902

1

2+ 1205731199022

2) 11987532]

+1

119873Δ119867211990132 (119902

1

1)3

32 = 1205721199022

2+ 1205731199021

2

(32)


0 asymp Φ2120583 =120597119871 (1199021 1199021 1199022)

120597119902120583

2

+120597119865 (1199021 1199021 1199023)

120597 119902120583

1

(33)

that is

0 asymp Φ21 =1

1198733(1199021

11199022

1)2

+ 1205721199021

3+ 1205731199022

3


1198733(1199021

1)2

1199021

11199022

1+ 1205721199022

3+ 1205731199021

3

(34)


0 asymp Φ2120583H (35)


H = 1199011120583 119902120583

1minus 119871 minus

120597119865

120597119902120583

1

119902120583

1minus

120597119865

120597 119902120583

1

119902120583

2 (36)


H =1

Δ(1199011111987531 + 1199011211987532) minus

1

119873Δ21199021

1(11987532)2minus

1

1198733Δ2

times [(1199021

1)2

1199021

2(11987532)2minus (1199021

1)2

1199022

21198753111987532]

minus 120572 (1199021

31199021

2+ 1199022

31199022

2) minus 120573 (119902

2

31199021

2+ 1199021

31199022

2) + 119867

2119873(1199021

1)3

(37)


4 Conclusions







Acknowledgments


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








Acknowledgments


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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