SISOM 2007 and Homagial Session of the Commission of Acoustics, Bucharest 29-31 May

DISCRETE SYMMETRIES OF THE FIELD EQUATIONS IN CURVED SPACE - TIME

Gabriel Murariu*, Marina Aura Dariescu**, Ciprian Dariescu**

Faculty of Sciences, University “Dunarea de Jos” of Galati, Gabriel.Murariu@ugal.ro Faculty of Physics, University “Alexandru.I. Cuza” of Iasi

The aim of this paper is to study the discrete symmetries for all the elements and fields equations on spherically symmetric curved space-time configuration. The specific results are considered for a consecrated space –time configuration.

Key words: Maxwell equation; Einstein equation; Klein-Gordon equation;

1. INTRODUCTION

The study of boson stars can be traced back to the work of Kaup and Ruffini and Bonazzalo more than 30 years ago. They found asymptotically at, spherically symmetric equilibrium solutions of the Einstein-Klein-Gordon equations.

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 the Pauli exclusion principle that applies to fermionic stars..

Jetzer and van der Bij extended the boson star model to include the coupling with a U (1) gauge group. 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. The discrete symmetries transformations problem was less discussed in the literature.

The aim of this paper is to present an MAPLE package due to built the field equation and to reveal the influence of the discrete symmetries on the geometrically and fields equation system.

2. FIELDS EQUATIONS

For a curved space-time, the line element can be expressed using the metric tensor as [1 , 2] μνg

2ds g dx dxμ νμν= , (1)

Considering a charged boson of mass coupled to the electromagnetic field, described by the

gauge invariance Lagrangean density of the form

0m

(3,1) (1)SO U×

20, ,

14

ab aba b abL m= η Φ Φ + ΦΦ + F F , (2)

the Maxwell tensor

: :ab b a a bF A A= − , (3)

can be used to derive the electromagnetic field equation [3, 4].

Gabriel MURARIU Marina Aura DARIESCU, Ciprian DARIESCU 95

( ) ( ): | |ab abc b b b bF ie ieA ieA⎡ ⎤= − η Φ Φ − Φ − Φ − Φ Φ⎣ ⎦ , (4)

For the scalar field, the Klein - Gordon equation can be derived using (2) by varying with respect to

different fields, 2 2

0 |2 c cc cm ieA e A AΦ − Φ = Φ + Φ , (5)

and it’s hermitic conjugated.

The Einstein system equations can be written as

ab abG kT= , (6)

where the energy-momentum tensor is expressed as

: : : :c

ab a b b a ac b abT F=Φ Φ +Φ Φ + −ηF L , (7)

3. PACKAGE PROCEDURES OVERVIEW

In order to build an efficient tool for the analysis of the system (4--7), using a set of procedures and routines, let us describe the structure and the main features of the procedures of the formalism. Actually, two major parts of the programs can be detected: the first one is regarding the tensorial structure of the objects needed in the algorithm, while the second one contains line-commands specific to each version.

The first part of the program starts after initializing the main used package, with a set of definitions for the entire set of necessary objects. For example, using the MAPLE platform, one can write (for the particular case of using polar coordinates)

>coord:=[r,theta,phi,t]; >Mg:=array(1..4,1..4,symmetric, [(1,1)=g11(coord[1],coord[2],coord[3],coord[4])), …………………………. (4,4)=g44(coord[1],coord[2],coord[3],coord[4])]) >g:=create([-1,-1],op(Mg));

Now, using the MAPLE platform, one is able to compute the first and second order derivatives of the metric tensor, the Christoffel symbols, the Riemann and Ricci tensors, and finally, the Einstein tensor components, as it follows:

> D1g:=d1metric(g,coord); > Cf1:=Christoffel1(D1g); > g_inv:=invert(g,'det_g'); > Cf2:=Christoffel2(g_inv,Cf1); …………………… > Einstein1:=lin_com(1,Rc,1/2*Rsc,g);

The definition of the gauge-covariant derivatives of the complex scalar field is an important step in building the system lagrangian and in writing down the KGM coupled equations. In order to get a coherent structure, the terms are defined as tensors. The gauge fields can be defined in a direct manner and, using the tensor structure of definition, it can be easily used in building the gauge derivatives of the scalar field.

> P:=create([],Phi(coord[1],coord[2],coord[3],coord[4])); > P_conjugate:=create([],conjugate(Phi(coord[1],…………….coord[4]))); > P_bar_a:=partial_diff(P,coord); > P_bar_a_conjugate:=partial_diff(P_conjugate,coord); > A_gauge:=create([-1],array(1..4, ………………………. (4)=A[4](coord[1],coord[2],coord[3],coord[4])]));

Dyscrete symmetries of the field equations in curved space - time 96

> ..... > P_bar_a_2:=prod(A_gauge,P); > P_coma_a:=lin_com(1,P_bar_a,-I*E,P_bar_a_2); > P_bar_a_conjugate:=prod(P_conjugate,A_gauge); > P_coma_a_conjugate:=lin_com(1,P_bar_a_conjugate,I*E,P_bar_a_2);

Further, using a tensor for the interaction term in the Klein--Gordon equations, the evolution equation for the complex scalar field reads: ( )tr,Φ

> prod(A_gauge,P_bar_a_conjugate); > J_Klein_Gordon1:=prod(A_gauge_up,P_bar_a,[1,1]); > J_Klein_Gordon_temp:=prod(A_gauge,P); > J_Klein_Gordon2:=prod(A_gauge_up,J_Klein_Gordon_temp,[1,1]); > J_Klein_Gordon:=lin_com(2*I*E,J_Klein_Gordon1,E^2,J_Klein_Gordon2); > J_Klein_GordonM:=J_Klein_Gordon[compts]; > Eq_Klein_Gordon:=Klein_GordonM=J_Klein_GordonM;

The next step is devoted to the Maxwell tensor, defined in (3), and to the corresponding Maxwell equations (4). The currents and the interaction term are put in the same tensorial structure [5, 6].

> Fab1:=cov_diff(A_gauge, coord,Cf2); > Fba2:=permute_indices(Fab1,[2,1]); > Fab:=lin_com(1,Fab1,-1,Fba2); > JMaxwell2:=prod(P,P_coma_a_conjugate); > JMaxwell3:=lin_com(1,JMaxwell1,-1,JMaxwell2); > JMaxwell4:=prod(g_inv,JMaxwell3,[2,1]); > JMaxwell:=lin_com(-I*E,JMaxwell4);

Putting together all the objects defined in previous steps of the algorithm, the Einstein equations (6) explicitly are [7, 8]:

> EinsteinM:=Einstein1[compts]; > J_EinsteinM:=J_Einstein[compts]; > for i from 1 to 4 do : for j from 1 to 4 do EqEinstein[i,j]:=EinsteinM[i,j]=J_EinsteinM[i,j] end do : end do;

Finally, one has to impose specific ansatz conditions, in order to obtain a set of simpler equations, easier to manipulate and of course, much interesting for didactic reasons.

In the last part of this short presentation, will be briefly discuss only procedures used in order to obtain the result of applying the discrete symmetry transformation of parity, time reversal and charge conjugation. To succeed in this purpose, it was build procedures to compute, at every algorithm step, all the considered elements used in order to write down the field equation.

4. SPECIFIC RESULTS

For a space – time with the line element of the form

( ) ( ) ( )2 22 2 2 2 2 2 2sinf hds e dr r d d e dt= + θ + θ ϕ − , (8)

The corresponding Christoffel coefficients are

111 ( , )F r t

r∂

Γ =∂

14 411 1 ( , )F r t

t∂

Γ = Γ =∂

221 2F r tre−Γ = − ( , ) ( )33

21 2 ( , ) sinF r tre−Γ = − θ

[ ]44

2 ( , ) ( , )1 ( , ) H r t F r tH r t er

−∂Γ =

∂ 12 21

2 2 1r

Γ = Γ =

( ) (332 sin cosΓ = − θ θ) 13 31

3 3 1r

Γ = Γ =

(9)

Gabriel MURARIU Marina Aura DARIESCU, Ciprian DARIESCU 97

( )( )23 32

3 3 sincos

θΓ = Γ =

θ [ ]

112 ( , ) ( , )4 ( , ) H r t F r tF r t e

t− −∂

Γ =∂

14 414 4 ( , )H r t

r∂

Γ = Γ =∂

444 ( , )H r t

t∂

Γ =∂

,

Under a parity transformation, the differences between the original and derived values, are

2 ( , )114

( , )2 f r tf r t et

∂ΔΓ = −

∂ 2 ( , )

141( , )2 f r tf r t et

∂ΔΓ =

∂

2 ( , )444

( , )2 h r th r t er

∂ΓΔΓ = −

∂ 2 ( , )

411( , )2 f r tf r t et

∂ΔΓ =

∂

(10)

Using this package, the connection coefficients are

22 121 2 (1 F r te

r−Γ = −Γ = − , ) 41 11

1 4 (( , ) , )H r tF r t et

−∂Γ = Γ =

∂

44 141 4 (( , ) F r tH r t e

r−∂

Γ = Γ =∂

, ) 33 131 3 (1 F r te

r−Γ = −Γ = − , )

( )( )33 23

2 3 sin1cosr

θΓ = −Γ = −

θ

(11)

The scalar and electromagnetic fields equations, in the spherically symmetric ansatz of the form 1 1( , )A A r t= , , , 2 0A = 3 0A = 4 4 ( , )A A r t= , ( , )r tΦ =Φ , can be read as

{ }( ) ( )

2, , , ,

22, , , , 0

2 221 , 4 , 1 4

2

2 2

frr r r r

htt t r t

f hr t

e h fr

e f f m

ieA e ieA e e A A

−

−

− −

⎧ ⎫⎡ ⎤Φ + − + Φ −⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤− Φ + − Φ − Φ =⎣ ⎦

⎡ ⎤= Φ − Φ + −⎣ ⎦

(12)

respectively ( ) ( )( )

4, , 4 1, , 1 ,

, , 12

h f hr r t t t

fr r

e e A h A e A f A

ie e ieA

− − −

−

⎡ ⎤+ − +⎣ ⎦

⎡ ⎤= ΦΦ −Φ Φ − ΦΦ⎣ ⎦

= (13)

and

( ) ( ) ( )

( ) ( )

4, , 4 1, , 1 4, , 4,

1, , 1 , , 4

2

2 2

ff f h f

r r t t r rr

fh h

r t t t

ee e A h A e A f A e A h Ar

e e A f A ie e ieAr

−− − − −

−− −

⎡ ⎤ ⎡+ − + + +⎣ ⎦ ⎣

⎡ ⎤⎡ ⎤− + = ΦΦ −Φ Φ − ΦΦ⎣ ⎦ ⎣ ⎦

⎤⎦ (14)

The necessary Lorentz condition is of the form

( )1, , 1 4, , 42 0f h f

r r t te A h A e e A f Ar

− − −⎡ ⎤⎛ ⎞ ⎡ ⎤+ + + + =⎜ ⎟⎢ ⎥ ⎣ ⎦⎝ ⎠⎣ ⎦ (15)

Under a combined time reversal –charge conjugation -parity transformation, the local fields

equations are invariant, like in the minkowskian configuration.

AKNOWLEGMENTS

This work is supported by the Grant Type A CNCSIS Code 1160/2006.

Dyscrete symmetries of the field equations in curved space - time 98

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