The seed of magnetic monopoles in the early inflationary universe from a 5D vacuum state

Physics Letters B 674 (2009) 143–145

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Physics Letters B www.elsevier.com/locate/physletb

The seed of magnetic monopoles in the early inflationary universe from a 5D vacuum state Jesús Martín Romero a , Mauricio Bellini a,b,∗ a b

Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, (7600) Mar del Plata, Argentina Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina

a r t i c l e

i n f o

Article history: Received 3 December 2008 Received in revised form 13 March 2009 Accepted 17 March 2009 Available online 21 March 2009 Editor: S. Dodelson

a b s t r a c t Starting from a 5D Riemann flat metric, we have induced an effective 4D Hermitian metric which has an antisymmetric part which is purely imaginary. We have worked an example in which both, non-metricity and cotorsion are zero. We obtained that the production of monopoles should be insignificant at the end of inflation and the tensor metric should come asymptotically diagonal and describing a nearly 4D de Sitter expansion. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The possibility that our world may be embedded in a (4 + d)dimensional universe with more than four large dimensions has attracted the attention of a great number of researches. One of these higher dimensional theories, where the cylinder condition of the Kaluza–Klein theory [1] is replaced by the conjecture that the ordinary matter and fields are confined to a 4D subspace usually referred to as a brane is the Randall and Sundrum model [2]. The original version of the KK theory assures, as a postulate, that the fifth dimension is compact. A few years ago, a non-compactified approach to KK gravity, known as Space–Time–Matter (STM) theory was proposed by Wesson and collaborators [3]. In this theory all classical physical quantities, such as matter density and pressure, are susceptible of a geometrical interpretation. Wesson’s proposal also assumes that the fundamental 5D space in which our usual spacetime is embedded, should be a solution of the classical 5D vacuum Einstein equations: R A B = 0.1 The mathematical basis of it is the Campbell’s theorem [4], which ensures an embedding of 4D general relativity with sources in a 5D theory whose field equations are apparently empty. That is, the Einstein equations G α β = −8π G T α β (we use c = h¯ = 1 units), are embedded perfectly in the Ricci-flat equations R A B = 0. The theory of magnetic monopoles was formulated many decades ago by Dirac in [5]. In his classical works, Dirac showed that the existence of a magnetic monopole would explain the

*

Corresponding author at: Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, (7600) Mar del Plata, Argentina. E-mail address: [email protected] (M. Bellini). 1 In our conventions capital Latin indices run from 0 to 4, greek indices run from 0 to 3 and latin indices run from 1 to 3. 0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.03.026

electric charge quantization. Also, a Lagrangian formulation describing the electromagnetic interaction mediated by topologically massive vector bosons-between charged, spin-1/2 fermions with an Abelian magnetic monopoles in a curved spacetime with nonminimally coupling and torsion potential, was presented in [6]. In the framework of inflation, magnetic monopole solutions of the Einstein–Yang–Mill–Higgs equations with a positive cosmological constant approach asymptotically the de Sitter spacetime background and exist only for a non-zero Higgs potential [7]. More recently, Maxwell equations with massive photons and magnetic monopoles were formulated using spacetime algebra [8]. From the quantum-theoretical standpoint monopoles and massive photons were discussed widely during the last decades [9]. To discuss massive photons one has to consider the Proca equations rather than the Maxwell ones. Further, if a magnetic charge (monopole) really exists, the Maxwell electrodynamics must be replaced by a generalized theory, with the dual field tensor having a nonvanishing divergence. Some years ago the Weyl–Dirac formalism was generalized in order to obtain a geometrically based general relativistic theory, possessing electric and magnetic currents and admitting massive photons [10]. In this Letter we are interested to study the evolution of primordial magnetic monopoles in the inflationary epoch by extending Gravitoelectromagnetic Inflation and using some ideas of Dirac [11] and Einstein [12] in the framework of the Induced Matter theory, where the extra dimension is space-like and non-compact. To make it we shall start from a 5D vacuum state, on which we define null all the external sources and magnetic monopoles [the absence of magnetic monopoles is a characteristic of any (N > 4) theory]. It means that the universe in higher dimensions will be considered as empty. If we consider the extra dimension as a complex function of other coordinates, it is possible to obtain an effective 4D Hermitian tensor metric which could be relevant to describe extended gravitation and electrody-

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J.M. Romero, M. Bellini / Physics Letters B 674 (2009) 143–145

namics, where 4D magnetic monopoles are taken into account. In this context, we shall study the production of magnetic monopoles in the early inflationary universe. 2. Formalism in a 5D vacuum





dS 2 = ψ 2 dN 2 − ψ 2 e 2N dr 2 + r 2 dΩ 2 − dψ 2 ,

0

(2)

= 0, so that 8π G G α β = −3g α β /ψ02 , Λ = 3/(8π G ψ02 ) being

1

L( A B , A C ; B ) = − Q BC Q BC ,

(3)

4

(4)

such that A B = ( A μ , ϕ ), A = ( A μ , −ϕ ), B

ϕ being the inflaton field. Since we are considering a 5D vacuum, one obtains absence of both, 5D gravitoelectric J B and gravitomagnetic K B currents  AB  Q ; A = 0,  †AB  Q = 0, ;A

(5) (6)

1 2

A BC D



|(5) g |

QC D .

(7)

3. Induced gravitoelectromagnetic equations from a 5D vacuum Now we consider the 4D embedding ψ ≡ ψ(r , N ) on the 5D metric (1), such that if we consider physical coordinates dψ = ∂ψ ∂ψ α β will be ∂ r dr + ∂ N dN. The induced operators Q

  Qα β = F α β + γ g α β A δ ;δ ,

(8)

 αβ  Q ;α = −4π J β ,  †α β  Q = −4π K β , ;α where (Q† )α β =

1√ α β μν Q . μν 2 | (4 ) g |

(9) (10)

Γ σ −Γ σ

In order to work an example with non-zero gravitomagnetic current we shall study an example where the fifth coordinate ψ ≡ ψ(r , N ) is a complex function of N and r. 4.1. An example with null cotorsion and non-metricity on a Hermitian metric We are interested to study the case where the resultant 4D effective metric is Hermitian

    2    ∂ψ 2    dN 2 − |ψ|2 e 2N +  ∂ψ  dr 2   ∂N ∂r  ∂ψ ∂ψ − (dN dr − dr dN ) − |ψ|2 e 2N r 2 dΩ 2 , ∂ N ∂r



(4)

dS 2 = |ψ|2 + 

such that |ψ|2 = ψψ ∗ and U A =

∂ψ ∂ψ ∗ =± , ∂r ∂r

dx A dS

(13)

are the penta-velocities such

∂ψ ∂ψ ∗ =∓ . ∂N ∂N

(14)

Hence, the metric (13) results to be represented by a Hermitian tensor with g α β = g β∗ α , where the asterisk denotes the complex conjugate. The determinant g = det | g α β | is non-zero and real. Furγ thermore, as in a real tensor metric, one has g α β g γ β = δβ , where γ

δβ is the Kronecker tensor. Here, the order of indices is important γ and, for example: g α β g β γ = δβ . Recently, Hermitian metrics has

λ κ g Γνμ + g λσ Γμσ [νκ ]    λ   + C λ + 1 g λσ ( Q νμσ + Q μνσ − Q σ μν ), =  μν μν  2

 αβ 1  α (ψ)  α (ψ) F + γ g ;β A (ψ) + g ;(ψ) A ;DD ;(ψ)

   λ  1 λσ    μν  = 2 g ( g μσ ,ν + g νσ ,μ − g μν ,σ ).

The induced gravitomagnetic current is given by

We denote with comma ordinary derivatives and with (;) covariant derivatives.

(16)

Finally, the non-zero gravitomagnetic current components on the effective 4D metric, are



KN = −

σ α N μν A σ T μν 4π |(4) g |



Kθ = (11)

(15)

where the non-metricity terms are due to Q νμσ ≡ − g μσ ;ν , torsion α = C α − C α and are related with cotorsion terms by 2T μν μν νμ

Kr = −

4π

 (ψ)  + g α (ψ) A ;DD ;(ψ) + A ;(ψ);β .

2

α β μν is the Ricci tensor density. Notice that when torsion

The induced gravitoelectric current

is Jα = −

(12)

, ;α

been subject of interest to describe cosmology [15]. The effective 4D connections are given by

where the dual tensor of Q A B is

(Q† ) A B =



|(4) g |

that g A B U A U B = 1. Furthermore, the conditions to obtain a Hermitian metric tensor are:

where Q A B is an operator given by

Q A B = F A B + γ g A B A ;DD ,

η

σ − Γ σ ) + γ g Γ Aρ A σ (Γνμ μν ρη μν

4. Induced gravitoelectromagnetic equations on a Hermitic metric

the cosmological constant. Physically, this metric removes the potentials of electromagnetic type and flattens the potential of scalar type, so that the fields A B in the action (2) must be considered as test ones. Here, (5) g 0 ≡ (5) g |[ N =0,ψ=ψ0 ,θ=π /2] is a constant. As in previous papers [14], we shall consider a Lagrangian density given by2 (5)

8π



α β μν

(1)

where (5) g = ψ 8 e 6N r 4 sin2 (θ) is the determinant of the covariant metric tensor g A B , on a Riemman spacetime. The metric (1) is Riemann-flat (R A BC D = 0) and is a particular case of the socalled canonical metric: dS 2 = ψ 2 g α β (xμ , ψ) dxα dxβ − dψ 2 , where ∂ gαβ ∂ψ

1

σ = μν νμ is zero and the metric tensor is diagonal, hence T νμ 2 gravitomagnetic current vanishes.

with the action

 (5)   R (5)  + L ( A , A ) , B C ;B  16π G



=− where

We consider the 5D Riemann flat metric [13]

   (5) g 4 I = d x dψ  (5) g



K β = Q†α β ;α

σ αr μν A σ T μν 4π |(4) g |



1

4π



|(4) g |





4π











σ 2 A σ T Nr − A σ T Nσ ϕ ;r − A σ T ϕσr ;N ;ϕ

1



(18)

, ;α

  η + γ g [Nr ] Γρη A ρ ;ϕ , Kϕ = −

(17)

, ;α

|(4) g |



(19)











σ 2 A σ T Nr − A σ T Nσ θ ;r − A σ T θσr ;N ;θ

  η + γ g [Nr ] Γρη A ρ ;θ .

(20)

J.M. Romero, M. Bellini / Physics Letters B 674 (2009) 143–145

Now we consider Eq. (15) with both, zero cotorsion and nonmetricity. In this case the last two terms in the right side are zero, and we obtain

   λ  λ κ g   Γνμ + g λσ Γμσ = [νκ ]  μν  ,

(21)

and the gravitomagnetic currents for this particular case results to be K N = K r = 0,











(22)



g [ Nr ],ρ γ g[rN ] ϕ θ + Aρ , + Kθ = − ρ θ ρϕ ( 4 ) g [ Nr ] 4π | g | ;ϕ       g [ Nr ],ρ γ g[Nr ] ϕ θ Kϕ = − + Aρ , + ρ θ ρϕ ( 4 ) g [ Nr ] 4π | g | ;θ

(23) (24)

where g [α β] denotes the antisymmetric components of the tensor

α

metric and β γ are the second kind Christoffel symbols. Notice that the existence of non-zero gravitomagnetic current components depends of the antisymmetry of the tensor metric. An interesting example, which is relevant for cosmology is the case in which the complex function ψ(r , N ) is given by



ψ( N , r ) = ψ0 e

−r

+ ie

−N

,

such that, at the end of inflation, the non-diagonal part of the metric tensor becomes negligible with respect to the diagonal one. If we take ψ0 = 1/ H (when H is the Hubble parameter, which in this case is a constant) and N = Ht, at the end of inflation the asymptotic tensor metric is diagonal g μν |t 1/ H

⎛

1

0

0

⎜ 0 − H −2 e 2Ht 0 ⎜ ⎜ ⎝0 0 − H −2 e 2Ht r 2 0

0

0

0 0 0

⎞ ⎟ ⎟ ⎟, ⎠

(25)

− H −2 e 2Ht r 2 sin2 (θ)

which describes a de Sitter expansion [16]. Notice that, at the end of inflation U ψ |t 1/ H → 0, and U t |t 1/ H → 1 for U θ = U φ = 0, so that observers are at this time in a nearly comoving frame: U r |t 1/ H → 0. 5. Final remarks In this Letter we have extended Gravitoelectromagnetic Inflation using some ideas of Dirac [11] and Einstein [12], in the framework of the Induced Matter theory, where the extra dimension is space-like, non-compact and complex. Starting from a 5D

145

Riemann flat metric (1), we have induced an effective 4D Hermitian metric which has an antisymmetric part g [μν ] which is purely imaginary. We have worked an example where the only ∂ψ ∂ψ non-diagonal components of the tensor metric are g Nr = i ∂ N ∂ r ∂ψ ∂ψ

and grN = −i ∂ r ∂ N in which both, non-metricity and cotorsion are zero. In this case the non-zero components of the gravitomagnetic current during inflation are K θ and K ϕ . However, at the end of inflation the production of monopoles should be insignificant and the tensor metric should describe a nearly 4D de Sitter expansion. The evolution of the universe from a de Sitter expansion to other Friedmann–Robertson–Walker cosmology was studied using different approaches [17]. Acknowledgements The authors acknowledge CONICET and UNMdP (Argentina) for financial support. References [1] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Math. Phys. K 1 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [2] L. Randall, R. Sundrum, Mod. Phys. Lett. A 13 (1998) 2807; L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690. [3] See, for instance, P.S. Wesson, Phys. Lett. B 276 (1992) 299; P.S. Wesson, Mod. Phys. Lett. A 7 (1992) 921; P.S. Wesson, J. Math. Phys. 33 (1992) 3883; J. Ponce de Leon, P.S. Wesson, J. Math. Phys. 34 (1993) 4080; J.M. Overduin, P.S. Wesson, Phys. Rep. 283 (1997) 303; J. Ponce de Leon, Class. Quantum Grav. 23 (2006) 3043; B. Mashhoon, P.S. Wesson, Gen. Relativ. Gravit. 39 (2007) 1403. [4] J.E. Campbell, A Course of Differential Geometry, Clarendon Press, Oxford, 1926. [5] P.A. Dirac, Proc. R. Soc. A 133 (1931) 60. [6] S.A. Ali, C. Cafaro, S. Capozziello, Ch. Corda, arXiv:0706.3130. [7] Y. Brihaye, B. Hartmann, E. Radu, C. Stelea, Nucl. Phys. B 763 (2007) 115. [8] C. Cafaro, S.A. Ali, Adv. Appl. Clifford Albegras 17 (2007) 23. ´ P. Senjanovic, ´ Phys. Rep. 157 (1988) 234; [9] M. Blagojevic, A.S. Golhaber, M.M. Nieto, Rev. Mod. Phys. 43 (1971) 277; B. Cabrera, W.P. Trower, Found. Phys. 13 (1983) 195. [10] M. Israelit, Gen. Relativ. Gravit. 29 (1997) 1597. [11] P.M. Dirac, Phys. Rev. 74 (1948) 817. [12] A. Einstein, Rev. Mod. Phys. 20 (1948) 35. [13] B. Mashhoon, H. Liu, P.S. Wesson, Phys. Lett. B 331 (1994) 305; D.S. Ledesma, M. Bellini, Phys. Lett. B 581 (2004) 1. [14] A. Raya, J.E. Madriz Aguilar, M. Bellini, Phys. Lett. B 638 (2006) 314; J.E. Madriz Aguilar, M. Bellini, Phys. Lett. 642 (2006) 302. [15] Ch. Mantz, T. Prokopec, arXiv:0804.0213. [16] J. Ponce de Leon, Gen. Relativ. Gravit. 20 (1988) 539; M. Bellini, Phys. Lett. B 609 (2005) 187; J.E. Madriz Aguilar, M. Bellini, Phys. Lett. B 619 (2005) 208. [17] S.S. Seahra, P.S. Wesson, Class. Quantum Grav. 19 (2002) 1139; J.E. Madriz Aguilar, M. Bellini, Eur. Phys. J. C 38 (2004) 367.