Ostrogradski Hamiltonian approach for geodetic brane gravityRubén Cordero, Alberto Molgado, and Efraín Rojas Citation: AIP Conf. Proc. 1318, 216 (2010); doi: 10.1063/1.3531634 View online: http://dx.doi.org/10.1063/1.3531634 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1318&Issue=1 Published by the American Institute of Physics. Related ArticlesLink invariants for flows in higher dimensions J. Math. Phys. 51, 063506 (2010) Refining the shifted topological vertex J. Math. Phys. 50, 013509 (2009) Numerical Calabi–Yau metrics J. Math. Phys. 49, 032302 (2008) Normal ordering and boundary conditions in open bosonic strings J. Math. Phys. 46, 062302 (2005) Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Ostrogradski Hamiltonian approach for geodeticbrane gravity

Rubén Cordero∗, Alberto Molgado† and Efraín Rojas∗∗

∗Departamento de Física, Escuela Superior de Física y Matemáticas del IPN, Unidad AdolfoLópez Mateos, Edificio 9, 07738 México, Distrito Federal, México

†Unidad Académica de Física, Universidad Autónoma de Zacatecas, Calzada Solidaridad esq.Paseo a la Bufa s/n, 98062 Zacatecas, Zacatecas, México

∗∗Departamento de Física, Facultad de Física e Inteligencia Artificial, Universidad Veracruzana,91000 Xalapa, Veracruz, México

Abstract. We present an alternative Hamiltonian description of a branelike universe immersed ina flat background spacetime. This model is named geodetic brane gravity. We set up the Regge-Teitelboim model to describe our Universe where such field theory is originally thought as a secondorder derivative theory. We refer to an Ostrogradski Hamiltonian formalism to prepare the systemto its quantization. This approach comprize the manage of both first- and second-class constraintsand the counting of degrees of freedom follows accordingly.

Keywords: Hamiltonian formalism, second-order theories, constrained systemsPACS: 04.50.-h,04.60.Ds, 04.60.Kz,98.80.Jk

INTRODUCTIONWithin the framework of unified brane-like gravity [1] exist an effective model proposedlong time ago by Regge and Teitelboim (RT) motivated from first principles determiningthe dynamics of a string (particle) through its worldsheet (worldline) [2]. The coreof the idea behind the RT model is that gravitation can be described in a point- orstring-like fashion, as the worldvolume swept out by the motion of a three-dimensionalspacelike brane evolving in a higher dimensional bulk spacetime. When one address thisissue in a Minkowski spacetime, the model is named geodetic brane gravity and it hasbeen extensively studied by Davidson and Karasik (DK) [3, 4, 5]. The cosmology thatarise from the RT model is interesting in its own right because results an alternativeroute to understand better classical cosmology in extra dimensions as well as providesan interesting model to apply the canonical quantization methods. It is worthwhile tomention that RT model is a second order derivative model in the field variables, i.e.,the embedding functions but, one is able to identify a ’harmless’ surface term such thatit can be avoided at the beginning of its dynamical study. Thus, RT model becomestransformed in an ordinary first order field theory. At the Hamiltonian level it leads totroubles as was noticed first by Regge and Teitelboim because the scalar constraint is notwritten down in a closed form straightforwardly. In pursuing this endeavor, Davidsonand Karasik boldly tackled this problem successfully [3, 4]. In this work we chooseto proceed in another way around, that is, we reconsider the Hamiltonian approach forthe RT model in the light of the Ostrogradski formalism to constrained systems. In thistreatment it will be of a great interest to use the RT model straightforwardly for obtaining

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the quantum approach for brane cosmology.

GEODETIC BRANE GRAVITY

Consider a brane Σ of dimension d, evolving in a fixed Minkowski N dimensionalbackground spacetime with metric ηµν . Its trajectory, or worldvolume m of dimensiond + 1, is described by the embedding xµ = X µ(ξ a), where xµ are local coordinatesfor the background spacetime, ξ a local coordinates for m, and X µ the embeddingfunctions (µ,ν = 0,1, . . . ,N−1; a,b = 0,1, . . . ,d). We denote by eµ

a = ∂aX µ the tangentvectors to m. In this framework we introduce N − d − 1 unit normal vectors to theworldvolume, denoted by nµ

i (i = 1,2, . . . ,N− d− 1). These are defined implicitly byni · ea := ηµνnµ ieν

a = 0 and we choose to normalize them as ni ·n j = δi j.The RT model for Σ is given by

SRT [X ] =α

2

∫m

dd+1ξ√−gR−

∫m

dd+1ξ√−gΛ , (1)

where the constant α has dimensions [L](1−d), g denotes the determinant of the inducedmetric gab = ea ·eb. We have also included a cosmological constant term, Λ. The extrinsiccurvature of m is Kab

i =−ni ·Daeb, where Da = eµaDµ and Dµ is the covariant derivative

in the bulk spacetime. The mean extrinsic curvature is given by the trace Ki = gabKabi

where gab denotes the inverse of gab. The scalar curvature R of m can be obtainedeither directly from the induced metric gab, or, in terms of the extrinsic curvature, viathe contracted Gauss-Codazzi equation, R = KiKi−Ki

abKabi [6, 7].

The response of the action (1) to a deformation of the surface X → X + δX ischaracterized by a conserved stress tensor [8]

f a µ =−(

αG ab +Λgab)

eµb , (2)

where Gab = Rab − 12Rgab is the worldvolume Einstein tensor, with Rab being the

corresponding Ricci tensor. This quantity results physically relevant for the formulationof consistent conservation laws. Following the line of reasoning of [8], the classicalbrane trajectories can be obtained from the covariant conservation law, ∇a f a µ = 0,where ∇a is the covariant derivative compatible with gab. This yields

T ab Kabi = 0 , (3)

where T ab = α G ab +Λgab. In fact, T ab corresponds to the intrinsic stress tensor definedin the usual way by−2/

√−g(δSRT /δgab). Its conservation is supported by the Bianchy

identity. Notice that the equations of motion (3) are of second order in derivatives of theembedding functions because of the presence of the extrinsic curvature. Owing to thereparametrization invariance of the RT model, there are only N − d − 1 independentequations, along the normals; the remaining d + 1 tangential components are satisfiedidentically, as a consequence of the reparametrization invariance of the action (1). In thepresence of other possible matter sources with stress tensor T ab

m = (−2/√−g)δSm/δgab

where Sm is a matter action, there are not modifications in our approach. The equationsof motion (3) remain unchanged in form.

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MINISUPERSPACE MODEL

We turn now to specialize the RT model (1) to the case of a minisuperspace model.Consider a 3-brane Σ, evolving in a 5-dimensional Minkowski spacetime, ds2 =−dt2 +da2 + a2 dΩ2

3, where dΩ23 stands for the metric of a unit 3-sphere, i.e., dΩ2

3 = dχ2 +sin2

χdθ 2 + sin2χ sin2

θdφ 2. For the sake of simplicity, we choose the function sin2χ

in dΩ23 to consider a closed universe. If

xµ = X µ(ξ a) = (t(τ),a(τ),χ,θ ,φ) , (4)

is a parametric representation of the trajectory of Σ, we assure that the geometry of theworldvolume generated is that of the FRW case.

The basis adapted to the worldvolume is given by the four tangent vectors eµa (a =

0,1,2,3) together with the unit normal vector nµ = 1N (−a, t,0,0,0) , where the dot

stands for derivativation with respect to τ . For short in the notation we have introducedthe quantity, N =

√t2− a2, which coincides with the lapse function when we perform

an ADM decomposition of the action (1) [9, 10].The metric induced on the worldvolume is given by

ds2 = gabdξadξ

b =−N2dτ2 +a2dΩ

23 . (5)

For the Ricci scalar associated to the metric (5), it follows that

R =6t

a2N4

(aat−aat +N2t

). (6)

It is particularly remarkable the linear dependence in the accelerations of the dynamicalvariables t(τ) and a(τ), that the Ricci scalar possesses.

The Lagrangian density L =√−g(

α

2 R−Λ)

thus becomes

L =Θatα

N3

(aat−aat + t3− a2t

)− ΘNa3

3Λ , (7)

where Θ = 3sinθ sin2χ . Hence, the RT action (1) specialized to spherical configura-

tions, in terms of an arbitrary parameter τ , is

SRT = 6π2∫

dτ L(a, a, a, t, t) , (8)

where the Lagrangian function is given by 1

L(t, t,a, a, a) =atN3

(aat−aat +N2t

)−Na3H2 , (9)

1 It can be shown that this Lagrangian function can be brought into the total derivative of a sur-face term plus a function depending up to the velocities of the variables a and t, namely, L =atN3

(aat−aat + t3− a2t

)−Na3H2 =− aa2

N +aN(1−a2H2

)+ d

dτ

(a2 aN

).

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and we have introduced the quantity H2 = Λ/3α .The Einstein and the extrinsic curvature tensor to the worldvolume described by (5)

have the nonvanishing components

G ττ = − 3t2

a2N2 ,

G χχ = G θ

θ = G φφ = − t2

a2N4

[2at

ddτ

(at

)+N2

],

and

Kττ =

t2

N3d

dτ

(at

), Kχ

χ = Kθθ = Kφ

φ =t

aN. (10)

According to (3), there is only one equation of motion given by

ddτ

(at

)=−N2

at

(t2−3N2a2H2

3t2−N2a2H2

). (11)

This equation obviously only involves second derivatives of the field variables a and t.

OSTROGRADSKI HAMILTONIAN APPROACH

A deeper insight of the phase space structure of the theory defined by the Lagrangian(9) is achieved by an Ostrogradski procedure for higher order derivative systems2. Thehighest conjugate momenta to the velocities t, a are, respectively,

Pt =∂L∂ t

=−a2 a tN3 , (12)

Pa =∂L∂ a

=a2 t2

N3 , (13)

such that the highest momentum spacetime vector is Pµ = a2 tN3 (−a, t,0,0,0) . Though

this momentum has not a direct mechanical meaning it is useful to achieve a Legendretransformation [See Eq. (18) below]. Note that it is directed normal to the worldvolume.

The conjugate momenta to the position variables t,a are, respectively

pt =∂L∂ t− d

dτ

(∂L∂ t

)=

atN3

[a2 +a−2N2V (a)

]:=−Ω , (14)

pa =∂L∂ a− d

dτ

(∂L∂ a

)=−aa

N3

[a2 +a−2N2V (a)

]=(

at

)Ω , (15)

where we have introduced the standard potential in a minisuperspace approach, V (a) =a2(1−a2H2). The physical spacetime momentum vector can be expressed as pµ = Ω

t Xµ .

2 A complete description of the Hamiltonian formulation for branes whose action depends of the extrinsiccurvature of their worldvolume is provided in [9, 10].

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This vector is the projection of the conserved stress tensor (2) associated with theworldvolume, along the unit time like normal vector to Σ.

It is worthy mention that the momentum pa is obtained by two controbutions: onecoming from the ordinary theory and the other by a surface term, i. e., pa = pa + pa,with

pa =− aaN3 [a2 +2N2 +a−2N2V (a)] (16)

andpa =

2aaN

, (17)

where pa is the momentum conjugated to a(τ) when considering as the LagrangianLs = d

dτ(a2a/N).

The appropiate phase space of the system, Γ = t,a, t, a; pt , pa,Pt ,Pa, has been iden-tified explicitly. To continue build up the inherent Hamiltonian formalism in Γ, the Os-trogradski formalism yields the canonical Hamiltonian

H0 = p · X +P · X−L = paa+ pt t + JR , (18)

where we have defined

JR =− aN

[a2 +a−2N2V (a)

]=

N2

tΩ . (19)

This potential-like term results in an implicit function of the phase space variables in thecombination N3P2. Apparently this may look like an unnecessary complication to writeboth the physical momentum and H0 in terms of Ω but this quantity results importantbecause is nothing but the conserved bulk energy. Indeed, squaring the energy equation(14), results in the evolution equation

N2 + a2 = γ N2a2H2 (20)

where γ = γ(a) satisfies the cubic equation γ(γ−1)2 = Ω2/a8H6.

Constraint analysis

Since we are dealing with the Lagrangian (9), linear in the accelerations, we have twoprimary constraints

C1 = P · X = 0 , (21)

C2 = NP ·n− a2tN

= 0 . (22)

In fact, this is a generic feature of any brane Lagrangian linear in the accelerationspossesing reparametrization invariance.

Therefore, the Hamiltonian that generate time evolution of the fields is

H = H0 +λ1 C1 +λ2 C2 , (23)

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where λ1 and λ2 are Lagrange multipliers enforcing the primary constraints. For anycanonical variable z ∈ Γ, we have

z = z,H , (24)

on the constraint surface, where the corresponding Poisson bracket of any two functionsF(z) and G(z) in Γ is defined as

F,G=∂F∂ t

∂Gpt

+∂F∂a

∂Gpa

+∂F∂ t

∂GPt

+∂F∂ a

∂GPa− (F ←→ G) . (25)

With this symplectic structure, the constraints (21) and (22) are in involution, C1,C2=0. According to the Dirac-Bergmann program for constrained systems, C1 and C2 mustbe preserved by the evolution which demands the existence of the secondary constraints

C3 = H0 = p · X +N(

a3H2− 1a3 N2P2

)= 0 . (26)

C4 = p ·n = 0 . (27)

The vanishing of the canonical Hamiltonian is expected due to the reparametrizationinvariance of the RT model. It generates diffeomorphisms normal to the worldvolume.The secondary constraint (27) is characteristic of every linear brane model in accelera-tions. There are no other constraints. In fact, we are dealing with a theory with first- andsecond-class constraints which is a consequence of the gauge symmetry.

All the constraints should be divided into first- and second-class ones. For the RTmodel we have two first-class phase space constraints

F1 = C1 , (28)

F2 =N3

a2(3t2−N2a2H2)(t2−3N2a2H2)

C2 +C3 , (29)

and two second-class constraints. For our case, the selection of the second-class con-straints is straightforward. We choose them as

S1 = C2 , (30)S2 = C4 . (31)

Thus, we have the presence of 2 gauge transformations in the RT model. The countingof degrees of freedom is as follows: do f = [Total number of canonical variables− 2×first-class constraints−Number of second-class constraints]/2 = [8− 2× 2− 2]/2 = 1which agrees with the scale factor a.

Gauge-fixing

Following the Dirac’s program to extract the physical phase space for a constrainedsystem we need a gauge-fixing prescription which entails to introduce more constraints

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avoiding then the gauge freedom generated by the first-class constraints (29) and (28).To achieve this we will consider the usual cosmic gauge condition, namely

ϕ1 = N−1 =√

t2− a2−1≈ 0, (32)

and the evolution equation (20)

ϕ2 = N2 + a2− γ N2a2H2 = 0. (33)

From the geometric point of view, this set of gauge conditions is good enough since thematrix (

F ,ϕ1,2

) is not degenerate in the constraint surface.

In order to obtain quadratic constraints we implement the following canonical trans-formation to a new set of phase space variables:

N :=√

t2− a2, ΠN :=1N

(P · X) (34)

v := tanh−1(− pa

pt

), Πv := N(P ·n), (35)

together with the coordinate transformation Zµ := X µ −X µ ,vΠv while the momentapµ remains unaltered. Such transformation preserves the structure of the canonicalPoisson brackets

N,ΠN= 1 = v,Πv , Zµ , pν= δµ

ν , (36)

as expected.

Quadratic constraints

Taking into account the previous canonical transformation one can rewrite the first-and the second-class constraints as

F1 = NΠN = 0 , (37)

F2 = p · X +N(

a3H2 +1a3 N2

Π2N−

1a3 Π

2v

)= 0, (38)

and

S1 = Πv−a2tN

= 0 , (39)

S2 = pat− 2aaN

t = 0, (40)

respectively. One can develop further the constraint F2 by using the evolution equation(20) and the second-class constraint (40). A lenghty but straightforward computationyields

F2 = N

p2a−a2

[γ(γ−1)a4H4 +2(γa2H2−1)+(1− γ)a2H2 +

1a4 N2

Π2N

]×

[(γ−1)a2H2 +2]

(41)

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Remarkably, the use of the gauge condition (33) shifts the problem from the linear de-pendence in the momenta to a convenient quadratic expression for the physical momentapa.

CONCLUDING REMARKS

In this note we have presented an alternative Hamiltonian approach of the geodetic branegravity, treated as a second-order derivative theory. We have identified the phase spaceconstraints, some of them associated with the symmetry of reparametrization invarianceof the worldvolume. By using a canonical transformation we found a closed form for aquadratic constraint in the momenta. Finally, we would like to mention that our treatmentmakes feasible the quantization of the system.

ACKNOWLEDGMENTS

AM and ER acknowledge partial support from grants PROMEP-UV (CA Investig. yEnseñanza de la Física) and PROMEP-UAZ-PTC-086. RC and ER thank partial supportfrom SNI-México and CONACYT grant number J1-60621-I. RC thanks partial supportfrom grants COFAA-IPN 20100684 and EDI-IPN.

REFERENCES

1. A. Davidson and I. Gurwich, Phys. Rev. D74, 044023 (2006).2. T. Regge and C. Teitelboim in Proceedings of the Marcel Grossman Meeting, Trieste, Italy, (1975),

edited by R. Ruffini, North-Holland, Amsterdam, 1977.3. A. Davidson and D. Karasik, Mod. Phys. Lett. A13, 2187 (1998).4. D. Karasik and A Davidson, Phys. Rev. D67, 064012 (2003).5. A. Davidson, D. Karasik and Y. Lederer, Class. Quant. Grav. 16, 1349 (1999).6. R. Capovilla and J. Guven, Phys. Rev. D51, 6736 (1995).7. M. Spivak, Introduction to Differential Geometry Vols. I to 5, Publish or Perish, Boston MA, 1970.8. G. Arreaga, R. Capovilla and J. Guven, Annals of Phys. 279, 126 (2000).9. R. Capovilla, J. Guven and E. Rojas, Class. Quant. Grav. 21, 5563 (2004).10. R. Capovilla, A. Escalante, J. Guven and E. Rojas (2006)[arXiv:gr-qc/0603126].

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