SOLUTIONS FOR MAXWELL-EQUATIONS’ SYSTEM IN A STATIC CONFORMAL SPACE-TIME

G. MURARIU1, GH. PUSCASU2

1Faculty of Sciences, University “Dunărea de Jos”, Galaţi, Domnească Street, No. 111, România 2Faculty of Computer Science, University “Dunărea de Jos”, Galaţi

E-mail: Gabriel.Murariu@ugal.ro

Received December 15, 2008

In this paper we have continued our studies from our previous works about the boson stars. The aim of this Letter is to study the ( )3,1 (1)SO U× gauge minimally coupled charged spinless filed to a time dependent and spherically symmetric space-time.

Supposing the private case of an electromagnetic field we have suggested on establish the concrete dependence on the space-temporal coordinates of the electric and magnetic fields. The calculations are performed with the Mathematica and Maple programs which have attached the GrTensor platform. In this paper we succeed in reaching general solutions for the Maxwell system equations in first order of approximation.

1. INTRODUCTION

The study of the boson stars (BS) systems can be drawn back more than 30 years ago. The starting point was due by the Kaup’s work [1] and Ruffini and Bonazzalo’s papers [2]. They succeed in founding asymptotical solutions of the Einstein-Klein-Gordon equations for spherically symmetric equilibrium. Jetzer and van der Bij extended the BS model to include the coupling with an U(1) gauge group [3]. All these models have demonstrated the same characteristic: new interactions tend to increase the critical values of mass and particle number, although the particular values are very model dependent.

It is important to mention here the computations performed by M.A. Dariescu and C. Dariescu which are of considerable interest in general relativity [4–7].

This result originates from the following specific feature of boson stars; the boson star is protected from gravitational collapse by the Heisenberg uncertainty principle, instead of Pauli’ s exclusion principle that applies to fermionic stars.

In this paper we have continued the studies from our previous works. The main focus of this contribution is to integrate the equations of the electromagnetic field in the static conformal space-time background.

Paper presented at the 4th National Conference on Applied Physics, September 25–26, 2008, Galaţi, Romania.

Rom. Journ. Phys., Vol. 55, Nos. 1–2, P. 47–52, Bucharest, 2010

G. Murariu, Gh. Puscasu 2

48

2. THEORY

The main focus of our paper is to succeed in reaching the first order solution of the Maxwell system equations in the static conformal background. Therefore we will concentrate on the special case, namely we will obtain the fields’ equations within a tetrad frame associated with this space-time configuration.

In a series of preceding works [4–7] it was described how can be performed the computation for the components of an orthonormal tetradic system in different space-time configurations. In this way, still it was developed specific software package in order to succeed in reaching these particular frames [5, 8].

Let us consider a spherically symmetric configuration describe by a static conformal metric tensor type, expressed in Schwarzchild coordinates as

( ) ( ) ( )( )

( )

2 22 2 2 2 2 2 2

2 2 2

( ) sin

( )

t t

t

ds e a r dr e r d r d

e b r dt

θ θ ϕΞ +Σ Ξ +Σ

Ξ +Σ

= + + −

− (1)

For this space-time configuration, the pseudo-orthonormal tetradic frame { } 1,4a ae

=, with the correspondent metric tensor abη of minkowskian kind

[ ]1 1 1 1ab diagη = −

is related to the dual base given by the following relations:

( )1 ( )te a r drω Ξ +Σ= ( )2 te rdω θΞ +Σ=

( )3 ( ) sinte a r r dω θ ϕΞ +Σ= ( )4 ( )te b r dtω Ξ +Σ=

Considering a charged boson of mass 0m coupled to the electromagnetic field, described by the (3,1) (1)SO U× gauge invariance Lagrangean density [1, 2, 4–7] of the form

20, ,

14

ab aba b abL m F Fη= Φ Φ + ΦΦ + (2)

where we used ( ) ( )|. aa e= and

, |a a aieAΦ =Φ − Φ , , |a a aieAΦ =Φ + Φ (3)

In this way, the Maxwell tensor : :ab b a a bF A A= − (4)

is expressed in the terms of the Levi-Civita covariant derivative of the four-potential { } 1,4a a

A=

, i.e.

; |c

a b a b c abA A A= − Γ (5)

3 Maxwell-equations’ system in a static conformal space-time 49

Working in the minimally symmetric ansatz 1 1( , )A A r t= , 4 4 ( , )A A r t= , ( , )r tΦ =Φ [4–7], the single non-vanishing Maxwell tensor component is

( )

14 41

1, 4 4, 1( ) '( ) ( ) ( )( ) ( )

t rt

F F

a r A b r A b r A a r Ae

a r b r− Ξ +Σ

= − =

− − + Ξ = − (6)

The Klein - Gordon equation can be derived using (2) by varying with respect to scalar fields,

Φ+Φ=Φ−Φ cc

c|c AAeieAm 22

0 2 and it’s hermitic conjugated. (7)

and admit the explicit form

( )

( ) ( ) ( )

2 2 2

,2 2

2 2 2 2, , , 0

2 221 , 4 , 1 4

2 ( ) '( ) ( ) '( ) ( )( ) ( ) ( )

2 ( ) ( ) ( )

1 12( ) ( )

t

r

t rr tt

tr t

e b r a r b r b r b ra r b r r a r

a r b r a r m

iee A A e A Aa r b r

− Ξ +Σ

− Ξ +Σ

Φ − + −

− Φ Ξ + Φ − Φ − Φ =

= Φ − Φ + Φ −

(8)

and its’ h.c. The resulting Maxwell system equations

( ) ( ): | |ab ab

c b b b bF ie ieA ieAη = − Φ Φ − Φ − Φ − Φ Φ (9)

on the particular ansatz, can be written as

( )

( )

( )

22

4 4,2 2

21 1, 4 1,

24 4, 4, 1

2 2

4,2 2

( ) ' '( ) ( ) '( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) '( ) ( ) '( ) 2

'( ) ( ) ( ) ( )2 ( ) '( ) ( ) 2

( )2 (( ) ( )

t

r

t t

r rr

t

r

e a r b r b r a rA b r Aa r b r a r a r

a r b ra r b r A A b r b r A Ar

b r b r a r b rA b r b r A b r A Ar r

e b r A aa r b r r

− Ξ +Σ

− Ξ +Σ

− − +

+ Ξ + + + −

Ξ − + + + +

+ −

]( )

( )

1, 4

21, , , 4

) ( ) ( ) "( )

( ) ( ) 2( )

r

t

rt t t

r b r A b r b r A

ea r b r A ie e Ab r

− Ξ +Σ

Ξ + −

= ΦΦ −Φ Φ + ΦΦ

(10)

and respectively

G. Murariu, Gh. Puscasu 4

50

( )

( )

( )( )

( )( )

( )

22

1, 4 4,2 2

22

4, 4, 1 1,2

2, , 1

2 ( ) ( ) '( )( ) ( )

( ) ( )( )

2( )

t

t t

t

r rt tt

t

r r

e a r A a r b r A Aa r b r

ea r b r A A A Ab r

eie e Aa r

− Ξ +Σ

− Ξ +Σ

− Ξ +Σ

Ξ + −Ξ − +

+ −Ξ − + Ξ + =

= − ΦΦ −Φ Φ − ΦΦ

(11)

while the necessary Lorentz condition can be read as

( )

1, 4 1 4, 12 ( )( ) 3 ( ) ( ) '( ) 0

( ) ( )

t

r te b rb r A a r A A a r A b r A

a r b r r

− Ξ +Σ − Ξ + − + = (12)

For a completeness discussion and for the sake of the results, should be built up the energy-momentum tensor

: : : :c

ab a b b a ac b abT F F Lη= Φ Φ +Φ Φ + − , (13)

in order to succeed in deriving the Einstein equation

ab abG kT= , (14) which adopt the explicit expression

( )

( )

( ) ( )

22 2 2 2 2

2 2 2

22;1 ;1 ;4 ;4 0 14

( ) ( ) ( ) 2 '( ) ( )( ) ( )

12

te a r b r r b r rb r b ra r b r r

m Fκ

− Ξ +Σ

− + Ξ − − =

= Φ Φ +Φ Φ − ΦΦ −

(15)

and

( )

( )

( )

( ) ( )

23 2

3 2

2

22;1 ;1 ;4 ;4 0 14

( ) ( ) '( ) ( ) ( ) "( )( ) ( )

'( ) ( ) ( ) '( )

12

te a r r a r b r b r rb r b rra r b r

a r b r rb r b r

m Fκ

− Ξ +Σ − Ξ − +

+ + =

= − Φ Φ −Φ Φ + ΦΦ −

(16)

respectively

( )

( ) ( )

( ) ( )

23 2 2 2 2

2 3 2

22;1 ;1 ;4 ;4 0 14

( ) 3 ( ) ( ) 2 '( ) ( )( ) ( )

12

te a r r b r b r ra r a rr a r b r

m Fκ

− Ξ +Σ

Ξ + + − =

= Φ Φ +Φ Φ + ΦΦ +

(17)

5 Maxwell-equations’ system in a static conformal space-time 51

The aim of this paper is to compute the first order solutions for the Klein-Gordon-Maxwell system equations and using them to compute the electric charge density of the boson system.

2. KLEIN-GORDON-MAXWELL FIRST ORDER SOLUTIONS

For solving, we start with the physically reasonable assumptions that can be neglected the feedback of gravity in the first order approximation [1, 2, 4, 5]. This estimation is due in order to succeed in ahead calculate approximately the reaction on the curved manifold’ functions. The equation for potential Φ (imposing 0Ξ = ,

( ) ( ) 1a r b r= = and 0Σ = ) can be written as:

2, , , 0

2 0rr t tt mr

Φ − Φ −Φ − Φ = and h.c. (18)

with the solutions

( )i t krN er

ω −Φ = and ( )i t krN er

ω− −Φ = (19)

Using the same conditions, from Maxwell equations it can be read, from (10) and (11)

2

1, 1, 1 1,2 2

1 1 22 2rr r tt

NA A A A ek

r r r+ − − = − (20)

and respectively

2

4, 4, 4, 2

2 2rr r tt

NA A A e

r rω+ − = (21)

The solution for the first Maxwell equation has the form

2

21 4

ek NA ek N

r= + (22)

while from the second Maxwell’s equation, can be found the general solution of the progressive wave form [11–13]

( ) ( ) ( )2 2 1 24

2( , ) 2 log 2

F t r F t rA r t e N r e N

rω ω

− + += − + (23)

Using these results, the electric charge density

( )

( ) ( )4 , , 42 2 ( )

( )

tt

t tej ie ieb r e Ab r

− Ξ +ΣΞ +Σ = ΦΦ −Φ Φ + ΦΦ (24)

has the explicit expression

2 22

24 22 4

( )e N N

j Nb r r r

κ = − +

(25)

G. Murariu, Gh. Puscasu 6

52

Acknowlegdments. In this paper we succeed in reaching general solutions for the Maxwell system equations in first order of approximation [4, 14]. This estimation is due in order to succeed in ahead calculate approximately the response on the curved manifold’ functions [4, 6]. In this way we get a sensitive generalization of our previous computations for the electromagnetic field. Future works will contain such response‘calculation. A thoughtful analysis of symmetries of such solutions should be developed [9]. The author warmly thanks to all for useful suggestions and advices which significantly improved of present paper.

REFERENCES

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Chaos, Solitons & Fractals, 28, 1–7, (2006). 7. M.A. Dariescu, C. Dariescu, G. Murariu, “Gravitoelectromagnetically induced transitions in

charged boson nebulae”, European Phys. Letters, 74, 978–984, (2006). 8. B.Ciobanu, I. Radinschi, Rom. Journal Phys., 53, 405–416, (2008). 9. M. Visinescu, Rom. Journ. Phys., 53, 1213–1219, (2008). 10. G. Murariu, AIP Conference Proceedings vol. 895, 333–336, (2007). 11. K.W. Howard and P. Candelas, Phys. Rev. Lett., 53, 403 (1984). 12. K.W. Howard, Phys. Rev., D 30, 2532 (1984). 13. G. Zet, Journal Phys., 50, 63–75, (2005). 14. B. Ciobanu, I. Radinschi, Rom. Journal Phys., 53, 29–35, (2008).