# A Note On Proper Conformal Vector Fields In Cylindrically

## Transcript Of A Note On Proper Conformal Vector Fields In Cylindrically

U.P.B. Sci. Bull., Series A, Vol. 70, No. 2, 2008

ISSN 1223-7027

A NOTE ON PROPER CONFORMAL VECTOR FIELDS IN CYLINDRICALLY SYMMETRIC STATIC SPACE-TIMES

Ghulam SHABBIR, 1 Shaukat IQBAL2

A study of proper conformal vector field in non conformally flat cylindrically symmetric static space-times is given by using the direct integration technique. Using the above mentioned technique we have shown that a very special class of the above space-time admits proper conformal vector fields.

Keywords: direct integration technique; conformal vector fields

1. Introduction

The aim of this paper is to find the existence of conformal vector fields in

the non conformally flat cylindrically symmetric static space-times. The

conformal vector field which preserves the metric structure up to a conformal

factor carries significant interest in Einstein’s theory of general relativity. It is

therefore important to study this symmetry. Different approaches [1, 3-8] were

adopted to study conformal vector fields. In this paper a direct integration

technique is used to study conformal vector fields in the non conformally flat

cylindrically symmetric static space-times. Throughout M represents a four dimensional, connected, Hausdorff space-time manifold with Lorentz metric g of

signature (-, +, +, +). The curvature tensor associated with gab , through the Levi-

Civita connection, is denoted in component form by

R

a bcd

,

the Weyl tensor

components are C abcd , and the Ricci tensor components are Rab = Rc acb . The

usual covariant, partial and Lie derivatives are denoted by a semicolon, a comma and the symbol L, respectively. Round and square brackets denote the usual

symmetrization and skew-symmetrization, respectively. The space-time M will

1 Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology Topi Swabi, NWFP, PAKISTAN, Email: [email protected] 2 Faculty of Computer Science and Engineering GIK Institute of Engineering Sciences and Technology Topi Swabi, NWFP, PAKISTAN

24

Ghulam Shabbir, Shaukat Iqbal

be assumed non conformally flat in the sense that the Weyl tensor does not vanish over any non empty open subset of M .

The covariant derivative of any vector field X on M can be

decomposed as

X =1h +F

(1)

a;b 2 ab

ab

where hab (= hba ) = LX gab and Fab (= −Fba ) are symmetric and skew symmetric

tensors on M , respectively. A vector field X is called conformal vector field if

the local diffeomorphisms ψ t (for appropriate t ) associated with X preserves the

metric structure up to a conformal factor i.e. ψ t* g = φ g, where φ is a nowhere

zero positive function on some open subset of M and ψ t* is a pullback map on

some open subset of M [3]. This is equivalent to the condition that hab = φ g ab ,

or, equivalently, if

g ab,c X c

+

g

cb

X

c ,a

+

g

ac

X

c ,b

=φ

g ab ,

(2)

where φ :U → R is a smooth conformal function on some subset of M , then X

is a called conformal vector field. If φ is constant on M , X is homothetic

(proper homothetic if φ ≠ 0 ) while if φ = 0, then it is Killing [1]. If the vector

field X is conformal, but not homothetic, then it is called proper conformal. It follows from [3] that for a conformal vector field X , the bivector F and the function φ satisfy (putting φa = φ,a )

Fab;c = Rabcd X d − 2φ[a gb]c ,

(3)

φ = −1 L X c −φ L + R F c,

(4)

a;b

2 ab;c

ab

c(a b)

where L = R − 1 R g .

ab

ab 6 ab

2. Main Results

Consider a cylindrically symmetric static space-time in the usual coordinate system (t, r,θ , z) with line element [2]

ds 2 = −eV (r )dt 2 + dr 2 + eU (r )dθ 2 + eW (r )dz 2 .

(5)

A note on proper conformal vector fields in cylindrically symmetric static space-times 25

The possible Segre type of the above space-time is {1,111}, or one of its

degeneracies. The above space-time (5) admits three linearly independent killing

vector fields, which are ∂ , ∂ , ∂ . (6) ∂t ∂θ ∂z

A vector field X is said to be a conformal vector field if it satisfies the

equation (2). One can write (2) explicitly using (5), and we have

V ′(r)X 1

+

2

X

0 ,0

=φ

(7)

X ,10 − eV (r ) X ,01 = 0 (8)

eU (r ) X ,20 − eV (r ) X ,02 = 0 (9)

eW (r ) X ,30 − eV (r ) X ,03 = 0

(10)

2

X

1 ,1

=φ

(11)

eU (r ) X ,21 + X ,12 = 0

(12)

eW (r ) X ,31 + X ,13 = 0

(13)

U ′(r)X 1

+

2

X

2 ,2

=φ

(14)

eW (r ) X ,32 + eU (r ) X ,23 = 0

(15)

W ′(r)X 1

+

2

X

3 ,3

=

φ,

(16)

where φ = φ(t, r,θ , z). Equations (11), (8), (12) and (13) give

26

Ghulam Shabbir, Shaukat Iqbal

∫ ∫ ∫ X 0 =

e−V (r ) ⎜⎛ 1

φ dr ⎟⎞dr + A1(t,θ , z) e−V (r)dr + A2 (t,θ , z)

t

t

⎫ ⎪

⎝2

⎠

⎪

∫ X 1 = 1 φ dr +A1(t,θ , z)

⎪ ⎪

2

⎪⎬, (17)

X 2 = − e−U (r)⎜⎛ 1 φ dr ⎟⎞dr − A1 (t,θ , z) e−U (r)dr + A3 (t,θ , z)⎪

∫ ∫ ∫ ⎝ 2 θ ⎠

θ

⎪

⎪

X 3 = − e−W (r)⎜⎛ 1 φ dr ⎟⎞dr − A1 (t,θ , z) e−W (r)dr + A4 (t,θ , z)⎪

∫ ∫ ∫ ⎝ 2 z ⎠

z

⎪⎭

where A1(t,θ , z), A2 (t,θ , z), A3 (t,θ , z) and A4 (t,θ , z) are functions of integration. In order to determine A1(t,θ , z), A2 (t,θ , z), A3 (t,θ , z) and A4 (t,θ , z) we need to

integrate the remaining six equations. To avoid details, here we will present only results, when the above space-time (5) admits proper conformal vector fields. It follows after some tedious and lengthy calculations that there exist two cases when the above space-time (5) admits proper conformal Killing vector fields which are:

Case (1)

In this case the space-time (5) becomes ds 2 = dr 2 + M 2 (r)(−e −2d7 N (r) dt 2 + e −2d11 N (r) dθ 2 + e −2d14 N (r) dz 2 ),

(18)

where M (r) = 12 ∫φ(r)dr + d9 and N (r) = ∫ M1(r) dr. The conformal

vector fields in this case are

X 0 = d7t + d8 , X 1 = M (r), X 2 = d11θ + d12 , X 3 = d14 z + d15 ,

(19)

where d7 , d8 , d9 , d11 , d12 , d14 , d15 ∈ IR(d7 ≠ 0, d11 ≠ 0, d14 ≠ 0, d7 ≠ d11 , d7 ≠ d14 ,

d ≠ d ) and the conformal factor is φ(r) = 2 dM . One can write the above

11

14

dr

equation (19), after subtracting Killing vector fields (which are given in equation

(6)) as

X = (d7t, M (r), d11θ , d14 z).

(20)

Here, the above space-time (18) admits four independent conformal vector

fields (see equation (19)) in which one is proper conformal which is given in

A note on proper conformal vector fields in cylindrically symmetric static space-times 27

equation (20) and three are independent Killing vector fields which are given in equation (6).

Case (2)

This case is the sub case of case (1). In this case we consider

d11 = d14 and d7 ≠ d11 and the space-time (18) becomes

ds 2 = dr 2 + M 2 (r)(−e −2d7 N (r) dt 2 + e −2d11 N (r) (dθ 2 + dz 2 )).

(21)

The above space-time (21) admits four independent Killing vector fields

which are

∂ , ∂ , ∂ , z ∂ −θ ∂ .

(22)

∂t ∂θ ∂z ∂θ ∂z

The conformal vector fields in this case are

X 0 = d7t + d8 , X 1 = M (r), X 2 = d11θ + d16 z + d12 , X 3 = d11z − d16θ + d15 , (23)

where d7 , d8 , d11, d12 , d15 , d16 ∈ IR(d7 ≠ 0, d11 ≠ 0, d7 ≠ d11 ) and the conformal

factor is φ(r) = 2 dM . One can write the above equation (23), after subtracting dr

Killing vector fields (see equation (22)) as

X = (d7t, M (r), d11θ , d11z).

(24)

The above space-time (21) admits five independent conformal vector fields in

which four independent Killing vector fields which are given in equation (22) and

one proper conformal vector field which is given in equation (24). The cases when d7 = d11, d7 ≠ d14 and d7 = d14 , d11 ≠ d14 are exactly the same.

R E F E R E N C E S

[1] G. S. Hall, conformal vector fields and conformal type collineations in space-times, General Relativity and Gravitation, 32 (2000) 933-941.

[2] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselears and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press, 2003.

[3] G. S. Hall, symmetries and curvature structure in general relativity, World Scientific, 2004. [4] A. D. Catalano, Closed conformal vector fields on pseudo Riemannian manifolds, International

journal of Mathematics and Mathematical Sciences, 2006 (2006) 1-8. [5] M. Steller, conformal vector fields on space-times, Ann. Global Anal. Geometry, 29 (2006)

293-317.

28

Ghulam Shabbir, Shaukat Iqbal

[6] B. Biadabad, conformal vector fields on tangent bundle of Finsler manifolds, Balkan J. of Geom. and its application, 11 (2006) 28-35.

[7] D. S. Kim, Y. H. Kim and H. S. Park, Einstein spaces and conformal vector fields, Journal of the Korean Mathematics, 43 (2006) 133-145.

[8] F. M. Sim and C. Tezer, conformal vector fields with respect to their Sasaki metric tensor, J. Geometry, 84 (2005) 133-151.

ISSN 1223-7027

A NOTE ON PROPER CONFORMAL VECTOR FIELDS IN CYLINDRICALLY SYMMETRIC STATIC SPACE-TIMES

Ghulam SHABBIR, 1 Shaukat IQBAL2

A study of proper conformal vector field in non conformally flat cylindrically symmetric static space-times is given by using the direct integration technique. Using the above mentioned technique we have shown that a very special class of the above space-time admits proper conformal vector fields.

Keywords: direct integration technique; conformal vector fields

1. Introduction

The aim of this paper is to find the existence of conformal vector fields in

the non conformally flat cylindrically symmetric static space-times. The

conformal vector field which preserves the metric structure up to a conformal

factor carries significant interest in Einstein’s theory of general relativity. It is

therefore important to study this symmetry. Different approaches [1, 3-8] were

adopted to study conformal vector fields. In this paper a direct integration

technique is used to study conformal vector fields in the non conformally flat

cylindrically symmetric static space-times. Throughout M represents a four dimensional, connected, Hausdorff space-time manifold with Lorentz metric g of

signature (-, +, +, +). The curvature tensor associated with gab , through the Levi-

Civita connection, is denoted in component form by

R

a bcd

,

the Weyl tensor

components are C abcd , and the Ricci tensor components are Rab = Rc acb . The

usual covariant, partial and Lie derivatives are denoted by a semicolon, a comma and the symbol L, respectively. Round and square brackets denote the usual

symmetrization and skew-symmetrization, respectively. The space-time M will

1 Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology Topi Swabi, NWFP, PAKISTAN, Email: [email protected] 2 Faculty of Computer Science and Engineering GIK Institute of Engineering Sciences and Technology Topi Swabi, NWFP, PAKISTAN

24

Ghulam Shabbir, Shaukat Iqbal

be assumed non conformally flat in the sense that the Weyl tensor does not vanish over any non empty open subset of M .

The covariant derivative of any vector field X on M can be

decomposed as

X =1h +F

(1)

a;b 2 ab

ab

where hab (= hba ) = LX gab and Fab (= −Fba ) are symmetric and skew symmetric

tensors on M , respectively. A vector field X is called conformal vector field if

the local diffeomorphisms ψ t (for appropriate t ) associated with X preserves the

metric structure up to a conformal factor i.e. ψ t* g = φ g, where φ is a nowhere

zero positive function on some open subset of M and ψ t* is a pullback map on

some open subset of M [3]. This is equivalent to the condition that hab = φ g ab ,

or, equivalently, if

g ab,c X c

+

g

cb

X

c ,a

+

g

ac

X

c ,b

=φ

g ab ,

(2)

where φ :U → R is a smooth conformal function on some subset of M , then X

is a called conformal vector field. If φ is constant on M , X is homothetic

(proper homothetic if φ ≠ 0 ) while if φ = 0, then it is Killing [1]. If the vector

field X is conformal, but not homothetic, then it is called proper conformal. It follows from [3] that for a conformal vector field X , the bivector F and the function φ satisfy (putting φa = φ,a )

Fab;c = Rabcd X d − 2φ[a gb]c ,

(3)

φ = −1 L X c −φ L + R F c,

(4)

a;b

2 ab;c

ab

c(a b)

where L = R − 1 R g .

ab

ab 6 ab

2. Main Results

Consider a cylindrically symmetric static space-time in the usual coordinate system (t, r,θ , z) with line element [2]

ds 2 = −eV (r )dt 2 + dr 2 + eU (r )dθ 2 + eW (r )dz 2 .

(5)

A note on proper conformal vector fields in cylindrically symmetric static space-times 25

The possible Segre type of the above space-time is {1,111}, or one of its

degeneracies. The above space-time (5) admits three linearly independent killing

vector fields, which are ∂ , ∂ , ∂ . (6) ∂t ∂θ ∂z

A vector field X is said to be a conformal vector field if it satisfies the

equation (2). One can write (2) explicitly using (5), and we have

V ′(r)X 1

+

2

X

0 ,0

=φ

(7)

X ,10 − eV (r ) X ,01 = 0 (8)

eU (r ) X ,20 − eV (r ) X ,02 = 0 (9)

eW (r ) X ,30 − eV (r ) X ,03 = 0

(10)

2

X

1 ,1

=φ

(11)

eU (r ) X ,21 + X ,12 = 0

(12)

eW (r ) X ,31 + X ,13 = 0

(13)

U ′(r)X 1

+

2

X

2 ,2

=φ

(14)

eW (r ) X ,32 + eU (r ) X ,23 = 0

(15)

W ′(r)X 1

+

2

X

3 ,3

=

φ,

(16)

where φ = φ(t, r,θ , z). Equations (11), (8), (12) and (13) give

26

Ghulam Shabbir, Shaukat Iqbal

∫ ∫ ∫ X 0 =

e−V (r ) ⎜⎛ 1

φ dr ⎟⎞dr + A1(t,θ , z) e−V (r)dr + A2 (t,θ , z)

t

t

⎫ ⎪

⎝2

⎠

⎪

∫ X 1 = 1 φ dr +A1(t,θ , z)

⎪ ⎪

2

⎪⎬, (17)

X 2 = − e−U (r)⎜⎛ 1 φ dr ⎟⎞dr − A1 (t,θ , z) e−U (r)dr + A3 (t,θ , z)⎪

∫ ∫ ∫ ⎝ 2 θ ⎠

θ

⎪

⎪

X 3 = − e−W (r)⎜⎛ 1 φ dr ⎟⎞dr − A1 (t,θ , z) e−W (r)dr + A4 (t,θ , z)⎪

∫ ∫ ∫ ⎝ 2 z ⎠

z

⎪⎭

where A1(t,θ , z), A2 (t,θ , z), A3 (t,θ , z) and A4 (t,θ , z) are functions of integration. In order to determine A1(t,θ , z), A2 (t,θ , z), A3 (t,θ , z) and A4 (t,θ , z) we need to

integrate the remaining six equations. To avoid details, here we will present only results, when the above space-time (5) admits proper conformal vector fields. It follows after some tedious and lengthy calculations that there exist two cases when the above space-time (5) admits proper conformal Killing vector fields which are:

Case (1)

In this case the space-time (5) becomes ds 2 = dr 2 + M 2 (r)(−e −2d7 N (r) dt 2 + e −2d11 N (r) dθ 2 + e −2d14 N (r) dz 2 ),

(18)

where M (r) = 12 ∫φ(r)dr + d9 and N (r) = ∫ M1(r) dr. The conformal

vector fields in this case are

X 0 = d7t + d8 , X 1 = M (r), X 2 = d11θ + d12 , X 3 = d14 z + d15 ,

(19)

where d7 , d8 , d9 , d11 , d12 , d14 , d15 ∈ IR(d7 ≠ 0, d11 ≠ 0, d14 ≠ 0, d7 ≠ d11 , d7 ≠ d14 ,

d ≠ d ) and the conformal factor is φ(r) = 2 dM . One can write the above

11

14

dr

equation (19), after subtracting Killing vector fields (which are given in equation

(6)) as

X = (d7t, M (r), d11θ , d14 z).

(20)

Here, the above space-time (18) admits four independent conformal vector

fields (see equation (19)) in which one is proper conformal which is given in

A note on proper conformal vector fields in cylindrically symmetric static space-times 27

equation (20) and three are independent Killing vector fields which are given in equation (6).

Case (2)

This case is the sub case of case (1). In this case we consider

d11 = d14 and d7 ≠ d11 and the space-time (18) becomes

ds 2 = dr 2 + M 2 (r)(−e −2d7 N (r) dt 2 + e −2d11 N (r) (dθ 2 + dz 2 )).

(21)

The above space-time (21) admits four independent Killing vector fields

which are

∂ , ∂ , ∂ , z ∂ −θ ∂ .

(22)

∂t ∂θ ∂z ∂θ ∂z

The conformal vector fields in this case are

X 0 = d7t + d8 , X 1 = M (r), X 2 = d11θ + d16 z + d12 , X 3 = d11z − d16θ + d15 , (23)

where d7 , d8 , d11, d12 , d15 , d16 ∈ IR(d7 ≠ 0, d11 ≠ 0, d7 ≠ d11 ) and the conformal

factor is φ(r) = 2 dM . One can write the above equation (23), after subtracting dr

Killing vector fields (see equation (22)) as

X = (d7t, M (r), d11θ , d11z).

(24)

The above space-time (21) admits five independent conformal vector fields in

which four independent Killing vector fields which are given in equation (22) and

one proper conformal vector field which is given in equation (24). The cases when d7 = d11, d7 ≠ d14 and d7 = d14 , d11 ≠ d14 are exactly the same.

R E F E R E N C E S

[1] G. S. Hall, conformal vector fields and conformal type collineations in space-times, General Relativity and Gravitation, 32 (2000) 933-941.

[2] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselears and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press, 2003.

[3] G. S. Hall, symmetries and curvature structure in general relativity, World Scientific, 2004. [4] A. D. Catalano, Closed conformal vector fields on pseudo Riemannian manifolds, International

journal of Mathematics and Mathematical Sciences, 2006 (2006) 1-8. [5] M. Steller, conformal vector fields on space-times, Ann. Global Anal. Geometry, 29 (2006)

293-317.

28

Ghulam Shabbir, Shaukat Iqbal

[6] B. Biadabad, conformal vector fields on tangent bundle of Finsler manifolds, Balkan J. of Geom. and its application, 11 (2006) 28-35.

[7] D. S. Kim, Y. H. Kim and H. S. Park, Einstein spaces and conformal vector fields, Journal of the Korean Mathematics, 43 (2006) 133-145.

[8] F. M. Sim and C. Tezer, conformal vector fields with respect to their Sasaki metric tensor, J. Geometry, 84 (2005) 133-151.