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Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity
Commun Nonlinear Sci Numer Simulat 16 (2011) 4189–4196
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Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity R.K. Gupta a,⇑, K. Singh b a b
School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, Punjab, India Department of Mathematics, Jaypee University of Information Technology, Waknaghat, P.O. Dumehar Bani, Kandaghat, Dist. Solan 173215, H.P., India
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 September 2010 Received in revised form 2 March 2011 Accepted 2 March 2011 Available online 12 March 2011 Keywords: Symmetry analysis Exact solutions Cylindrically symmetric space–time
The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In general relativity the investigation of null fields has acquired considerable interest in connection with the study of gravitational radiation. In the invariant theory of gravitational radiation, the Riemann tensor plays the central part. The algebraic and differential properties of this tensor have been discussed with a view to characterizing wave fields in general relativity. The essential idea in the Riemann tensor analysis as applied to radiation theory is that in a gravitational radiation field, the Riemann tensor will lie in some special relationship to the null cone. One such relationship is [1]
Rhijk þ iRhijk xk ¼ 0;
ð1:1Þ
where xk is a null vector, Rhijk is the Riemann tensor and is its dual. Eq. (1.1) imposes on the Riemann tensor a very severe restrictions, which is satisfied only asymptotically in the wave zone of a radiating system and exactly in plane gravitational waves and a few other special cases. From (1.1), it follows that Rhijk
Rij ¼ rxi xj ;
ð1:2Þ i
where r is a scalar. Since x is a null vector, from (1.2) the spur of the Ricci tensor R vanishes identically and through Einstein field equations the energy momentum tensor can be written as
8PT ij ¼ rxi xj :
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (R.K. Gupta). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.03.006
ð1:3Þ
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In the view of (1.3) the four eigen values s(i) determined by the equation
jT ij sg ij j ¼ 0 turn out to be zero and then the equation Tijvj = sgijvj admits only one real eigenvector which is null. The energy momentum tensor given by (1.3) represents some kind of incoherent superposition of wave packets of radiation and any zero rest mass field can be the source of radiation. We propose, in the present paper, to drive certain solutions of geometrical Eq. (1.2), when r – 0, for the cylindrical symmetric space time with two degrees of freedom. 1.1. The metric form and the Field equations We consider the cylindrical symmetric space time [2] 2
2
2
2
ds ¼ expð2w 2uÞðdt dr Þ ðv 2 expð2uÞ þ r 2 expð2uÞÞd/2 expð2uÞdz 2v expð2uÞd/dz;
ð1:4Þ
where u, v and w are functions of r and t only. When v = 0, (1.4) reduces to the well known Einstein Rosen metric with one degree of freedom. The nonzero components of the Ricci tensor obtained from (1.4) are
w1 u1 expð4uÞ 2 v4; þ w44 þ u11 þ u44 þ 2u24 þ 2r 2 r r w1 u1 expð4uÞ 2 ¼ w11 v 1; w44 u11 þ u44 þ 2u24 þ 2r 2 r r w4 expð4uÞ ¼ v1v4; þ 2u1 u4 þ 2 r 2r u1 expð4uÞ 2 2 ¼ expð4u 2wÞ u11 þ u44 v v ; 1 4 2r 2 r expð4u 2wÞ v ¼ v R33 þ v 11 1 v 44 þ 4ðu1 v 1 u4 v 4 Þ ; 2 r ¼ 2v R23 ðv 2 þ r2 expð4uÞÞR33 :
R44 ¼ w11 R11 R14 R33 R23 R22
ð1:5Þ
Here and in what follows, the subscripts 1 and 4 after u, v and w represent the partial differentiation with respect to r and t respectively. If the direction of propagation of the wave is the positive r-direction, we have here x2 = x3 = 0, x1 = x4 and, therefore, from (1.2) and (1.4) we get the field equations
R11 R44 ¼ 0;
ð1:6aÞ
R14 þ R44 ¼ 0;
ð1:6bÞ
R22 ¼ R23 ¼ R33 ¼ 0:
ð1:6cÞ
Making use of expressions for Rij given in (1.5), the relations (1.6a)–(1.6c) give the four differential equations
1 1 u11 þ u1 u44 ¼ r 2 expð4uÞ v 21 v 24 ; r 2 1 v 11 v 1 v 44 ¼ 4ðu4 v 4 u1 v 1 Þ; r expð4uÞ ðv 1 þ v 4 Þ2 ; w1 þ w4 rðu1 þ u4 Þ2 ¼ 4r expð4uÞ 2 v 1 v 24 : w11 w44 þ u21 u24 ¼ 4r2
ð1:7Þ ð1:8Þ ð1:9Þ ð1:10Þ
So, we have four Eqs. (1.7)–(1.10) for the determination of three unknowns u, v, and w and one can easily verify that these all are consistent. Therefore, we drop Eq. (1.10) and solve the remaining equations for u, v and w. It may be pointed out that Eqs. (1.7,1.8) are a set of coupled, second order, nonlinear partial differential equations in u and v, hence we will concentrate on these two equations and Eq. (1.9) is first order linear partial differential equation in w, which we will solve once we get u and v. Rewriting the Eqs. (1.7,1.8)
1 1 urr þ ur utt ¼ r 2 expð4uÞ v 2r v 2t ; r 2 1 v rr v r v tt ¼ 4ðut v t ur v r Þ: r
ð1:11Þ
The system (1.11) also represents another important class of Einstein field equations for vacuum [2]. No systematic investigation of this problem has yet been made (for more details, please refer to [2] pp - 350–351). Examples are the the counterpart of the Kerr solution [3] and of the Tomimatsu–Sato solutions [4]. Due to nonlinearity of exponential order, it is
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difficult to solve system (1.11) and hence study of symmetries and exact solutions of system (1.11) is of great importance and desirable. The heart of the classification schemes for the solutions of these equations are the symmetry methods based on the Lie groups. The most effective methods, for finding symmetry reductions and constructing exact solutions, include the Lie’s approach [5–9], the nonclassical approach [10–12], the generalized conditional symmetry method [13,14], symmetry reduction method [6,15,16] and isovector field method [17,18]. In this paper, we will discuss system (1.11) with the aid of symmetry reduction method. To determine the underlying symmetry group for Eqs. (1.11), we utilize a method which is based on the Fréchet derivative of the differential operators associated with the system (1.11). The technique has earlier been used to obtain the exact solutions of various nonlinear partial differential equations [6,15]. However, in order to help the reader and to relate it to the familiar one-parameter group of transformations, we provide here some basic knowledge about the approach. Let the system (1.11) be defined in terms of the nonlinear operators N1 and N2 as follows:
1 1 N1 ðu; v Þ ¼ urr þ ur utt r 2 expð4uÞ v 2r v 2t ¼ 0; r 2 1 N2 ðu; v Þ ¼ v rr v r v tt 4ðut v t ur v r Þ ¼ 0: r
ð1:11Þ
Next, we define an operator called symmetry operator for the system (1.11)⁄ given by
S ¼ ðS1 ; S2 Þ; where
@u @u Þ Þ; þ BðX; g þ C 1 ðX; g @t @r @v @v Þ Þ Þ; S2 ðv Þ ¼ AðX; g þ BðX; g þ C 2 ðX; g @t @r
Þ S1 ðuÞ ¼ AðX; g
ð1:12Þ
with
¼ ðu; v Þ: X ¼ ðt; rÞ; g Þ ¼ ðN 1 ; N 2 Þ, in the direction of g 1 is given by The Fréchet derivative of Nðg
þ g 1Þ d Nðg ; g 1 ¼ F N; g j ¼ 0: d
ð1:13Þ
For invariance of Eqs. (1.11), we require that the Fréchet derivative (1.13) must vanish on the solution set sof (1.11) in the direction of the symmetry operator S. That is, we must have
; SÞjr ¼ 0: FðN; g
ð1:14Þ
The associated Lie algebra of infinitesimal symmetries of (1.11) is then the set of vector fields of the form
Þ V ¼ AðX; g
@ @ @ @ Þ C 1 ðX; g Þ Þ : þ BðX; g C 2 ðX; g @t @r @u @v
ð1:15Þ
Equivalently, the one-parameter group of point transformations of (1.11) is as follows:
Þ þ oð2 Þ; t ¼ t þ AðX; g Þ þ oð2 Þ; r ¼ r þ BðX; g
ð1:16Þ
2
g ¼ g CðX; g Þ þ oð Þ; where
¼ ðu ; v Þ: C ¼ ðC 1 ; C 2 Þ; g The importance of the Eqs. (1.11) and the need to have some exact solutions are the main motives behind the present study. To have an insight, the explicit analytic solutions of the system (1.11) may enable one to better understand the phenomena which it describes. A detailed systematic analysis that leads to an exact analytic solution for (1.11) has not been performed [2] and is therefore desirable. The paper has been organized as follows. 2 is entirely devoted to showing how the powerful symmetry reduction method can be used to generate various symmetries of the system (1.11). In 3, we present the reduced systems of ordinary differential equations (ODEs) and their exact solutions. The final section contains the discussion and concluding remarks.
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2. The symmetry and optimal system The method for determining the symmetry group of (1.11) mainly consists of finding the coefficients A, B, C1, C2 in the two symmetry operators S1, S2 as defined by (1.12). These coefficients are to be determined from the invariance condition (1.14). Accordingly, we first find the Fréchet derivative
; g 1 Þ ¼ ðF 1 ðN1 ; g ; g 1 Þ; F 2 ðN2 ; g ; g 1 ÞÞ; FðN; g
ð2:1Þ
where
þ g 1 Þ d½N1 ðg j ¼ 0 d þ g 1 Þ d½N2 ðg ; g 1 Þ ¼ F 2 ðN2 ; g j ¼ 0: d ; g 1 Þ ¼ F 1 ðN1 ; g
ð2:2aÞ ð2:2bÞ
1 is substituted by S ¼ ðS1 ; S2 Þ in order to evaluate With the help of Eqs. (2.2), the Fréchet derivatives are obtained and then g them in the direction of the symmetry operator. This leads to the following:
1 1 ; SÞ ¼ ½S1 rr þ ½S1 r þ ½S1 tt þ expð4uÞ v t ½S2 t v r ½S2 r þ 2½S1 v 2t v 2r ; F 1 ðN1 ; g r r2 1 ; SÞ ¼ ½S2 rr ½S2 r þ ½S2 tt þ 4 ur ½S2 r þ v r ½S1 r ut ½S2 t v t ½S1 t : F 2 ðN2 ; g r For invariance of system (1.11), the following conditions must be satisfied:
; SÞjðN ;N Þ¼0 ¼ 0; F 1 ðN1 ; g 1 2
ð2:3aÞ
; SÞjðN ;N Þ¼0 ¼ 0: F 2 ðN2 ; g 1 2
ð2:3bÞ
The Eqs. (2.3), when expanded, result into the polynomial expressions in various partial derivatives of u(r, t) and v(r, t) with respect to the spatial variable. The calculations involved are tedious, however, to keep the interest of the reader we list here a much simplified set of determining equations for the group infinitesimals A, B, C1 and C2 which we get from (2.3a) after equating the coefficients of various derivative terms to zero:
Au ¼ 0;
Av ¼ 0;
Bu ¼ 0;
Bv ¼ 0;
C 1uu ¼ 0; Ar Bt ¼ 0; At Br ¼ 0; Arr þ r 1 Ar Att 2C 1tu ¼ 0; Brr r 1 Br Btt þ 2C 1ru þ Br 2 ¼ 0;
ð2:4Þ
C 1rr þ r1 C 1r C 1tt ¼ 0; 2C 1rv þ 2r1 C 1v r2 expð4uÞC 2r ¼ 0; 2C 1uv 4C 1v r2 expð4uÞC 2u ¼ 0; 2C 1tv r 2 expð4uÞC 2t ¼ 0; 1 C 1vv þ r2 expð4uÞðC 1u 2C 2v 4C 1 2Br 1 Þ ¼ 0: 2 Similarly, the Eq. (2.3b) brings-in the following additional equations. It is being mentioned here that these equations have been obtained keeping in view the consequences on the infinitesimals as affected by the set of Eq. (2.4).
C 2uu þ 4C 2u ¼ 0; C 2tu þ 2C 2t ¼ 0; 2C 2vv þ 8C 1v þ r 2 expð4uÞC 2u ¼ 0; C 2rr r1 C 2r C 2tt ¼ 0; C 2ru r1 C 2u þ 2C 2r ¼ 0;
ð2:5Þ
C 2uv þ 2C 1u ¼ 0; Arr r 1 Ar Att 2C 2tv 4C 1t ¼ 0; Brr þ r 1 Br Btt þ 2C 2rv Br 2 þ 4C 1r ¼ 0: Eqs. (2.4,2.5) enable us to derive the infinitesimals A, B, C1 and C2. Without presenting any calculations, we provide the results obtained as follow:
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A ¼ a2 t þ a5 ; B ¼ a2 r;
ð2:6Þ
C 1 ¼ a1 v þ a3 ; C 2 ¼ a1 r 2 expð4uÞ a1 v 2 ða2 þ 2a3 Þv þ a4 ;
where aj, j = 1, 2, 3 . . . 5 are arbitrary constants. The symmetries under which the Eq. (1.11) is invariant can be spanned by the following five linearly independent infinitesimal generators:
@ @ þ ðv 2 r 2 expð4uÞÞ @u @v @ @ @ ¼t þr þv ða2 Þ; @t @r @v @ @ ¼ ða3 Þ; þ 2v @u @v @ ¼ ða4 Þ; @t @ ¼ ða5 Þ: @v
V 1 ¼ v V2 V3 V4 V5
ða1 Þ;
ð2:7Þ
Using these generators one can obtain a reduction of Eqs. (1.11) to a system of ODEs after getting the similarity variables and the form by solving the characteristic equations
dt dr du dv ¼ ¼ ¼ : A B C 1 C 2
ð2:8Þ
In general, one may obtain the reduced system of ODEs from any linear combination of generators Vj, j = 1, 2, . . ., 5. Since there exist infinite possibilities for such combinations, a systematic procedure to classify these reductions is based on the property that the transformations of the symmetry group will transform solutions of Eqs. (1.11) into other solutions. Therefore, we classify the symmetry algebra of system into conjugacy inequivalent sub algebra under the adjoint action of the symmetry group. We will work out first an optimal system and then embark upon the various reductions associated with generators in the optimal system. We begin by considering a general element V = a1V1 + a2V2 + a3V3 + a4V4 + a5V5 of symmetry algebra and subject it to various adjoint transformations to simplify it as much as possible [19]. The adjoint action is given by the Lie series
AdðexpðV i ÞÞV j ¼ V j ½V i ; V j þ 2 ½V i ; ½V i ; V j ; where [Vi, Vj] = ViVj VjVi is the commutator for the Lie algebra, and is a parameter. The commutator table and the adjoint table for Lie algebra (2.6) are as follows: Commutator table Index
V1
V2
V3
V4
V5
V1 V2 V3 V4 V5
0 V1 2V1 0 V3
V1 0 0 V4 V5
2V1 0 0 0 2V5
0 V4 0 0 0
V3 V5 2V5 0 0
Adjoint table
Index
V1
V2
V3
V4
V5
V1 V2 V3 V4 V5
V1 V1 exp () V1 exp (2) V1 V1 + V3 2V5
V2 + V1 V2 V2 V2 V4 V2 V5
V3 + 2V1 V3 V3 V3 V3 2V5
V4 V4 exp V4 V4 V4
V5 V3 2V1 V5 exp V5 exp (2) V5 V5
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The optimal system consists of the following basic vector fields:
ðiÞ V 1 þ aV 2 þ bV 5 ; ðiiÞ V 2 þ cV 3 ; ðiiiÞ V 3 þ dV 4 ;
ð2:9Þ
ðivÞ V 4 þ eV 5 ; ðvÞ V 5 ; where a, b, c, d and e are arbitrary constants. 3. The reduced systems and solutions In this section, the primary focus is on the reductions associated with the vector fields (2.7) and attempt to find some exact solutions. Unfortunately, we are not able to solve characteristic equations corresponding to case (i), this will be taken as future endeavor. Corresponding to other cases, reductions are as follows: Case (ii) On using the characteristic Eqs. (2.8), the similarity variable and the form of similarity solution is as follows:
r n¼ ; t
uðr; tÞ ¼ FðnÞ c lnðtÞ;
v ðr; tÞ ¼ t1þ2c GðnÞ:
On using these in Eq. (1.11), the reduced system of ODEs is given by
2n2 F 00 2nF 0 þ n4 F 00 þ 4n3 F 0 þ 2n2 c þ expð4uÞ½G02 G2 ð1 þ 4c þ 4c2 Þ þ GG0 ð2n þ 4nc n2 Þ ¼ 0; nG00 þ G0 2nGc 4nGc2 þ n3 G00 4n2 GF 0 ð1 þ 2cÞ þ 4ðn3 nÞG0 F 0 ¼ 0:
ð3:1Þ
0
Let G(n) = exp(2F(n))H(n), F (n) = N(n). Then our system (3.1) becomes 2nN0 H 4nN2 H þ nH00 þ 2NH H0 þ 2nHc þ 4nHc2 þ 2n3 HN0 þ 2n3 HN 0 þ 4n3 H0 N 2 n3 H00 þ 4n2 HN þ 8n2 HNc ¼ 0; 2n2 N 0 2nN þ 2n4 N 0 þ 4n3 N þ 2n2 c þ 4H2 N 2 4NHH0 þ H02 H2 4H2 c2 4ð1 þ 2cÞnNH2 þ 2n2 NH2 þ 2nHH0 ð1 þ 2cÞ n2 HH0 ¼ 0:
Due to complexity of the system of ODEs, we are not capable for determining its solution. Case (iii) For this vector field, the form of the similarity variable and similarity solution is as follows:
n ¼ r; uðr; tÞ ¼ FðnÞ t=d; v ðr; tÞ ¼ exp
2t GðnÞ: d
The reduced ODEs in this case is as follows: 2 00
0
2n F þ 2nF þ expð4FÞ nG00 G0 þ
4nG 2
d
" 4G2 2
d
# G
02
¼ 0; ð3:2Þ
þ 4nG0 F 0 ¼ 0:
Solution of this system (3.2) of ODEs is as follow: 0 Let G(n) = exp(2F(n))H(n), F (n) = N(n). Using these substitutions our system reduce to
2n2 N0 þ 2nN þ
4H2 2
d
4H2 N2 þ 4NHH0 H02 ¼ 0;
2nN0 H þ 4nHN2 nH00 2HN þ H0
4nH 2
d
ð3:3Þ ¼ 0:
Now using MAPLE we examine this system. We arrive at following cases: Case (iii.1) H(n) = 0, that is not a physically interesting case. Case (iii.2) H(n) = ni, where i represents the complex number iota. With this, our system (3.3) reduced to single equation:
2n2 N0 2nN
4n2 2
d
þ 4n2 N2 þ 1 ¼ 0;
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which can be further solve to give solution:
NðnÞ ¼
J 0 2n d 2nJ 1 2n CY 0 2n d 1 2nCY 1 2n d d d d ; 2 n½CY 0 ð2nÞ þ J 0 ð2nÞd
where C is arbitrary constant and Jv(x) and Yv(x) are the modified Bessel functions of the first and second kinds, respectively. They satisfy the modified Bessel equation:
x2 Y 00 þ xY 0 ðx2 þ v 2 ÞY ¼ 0: Using these results we get final solution of system (1.11):
Z
½2rCY 1 2rd J 0 2rd d 2rJ1 2rd CY 0 2rd d dr þ D; r½CY 0 ð2rÞ þ J 0 ð2rÞd
Z 2rCY 1 2rd J 0 2rd d 2rJ1 2rd CY 0 2rd d 2t v ðr; tÞ ¼ ir exp exp dr 2D : r½CY 0 ð2rÞ þ J 0 ð2rÞd d uðr; tÞ ¼ t=d
1 2
ð3:4Þ
Making substitutions of these expressions for u(r, t) and v(r, t) in (1.9) and solving for w(r,t), we get
wðx; tÞ ¼
Z ð4rCY 0 ð2rÞ þ CY 0 ð2rÞd 2CY 0
ð2rÞ d
d þ 4CrY 1 2rd 2J 0 2rd d þ 4rJ0 ð2rÞ 4rJ1 2rd þ J 0 ð2rÞdÞ
4ðrðCY 0 ð2rÞÞ þ J 0 ð2rÞdÞ
dr þ f ðt rÞ;
where D is arbitrary constant and f(t r) is an arbitrary function of (t r). Case (iii.3) H(n) = ni With this, our system (3.3) reduced to single equation:
2n2 N 0 2nN
4n2 2
d
þ 4n2 N2 þ 1 ¼ 0;
which can be further solve to give solution:
NðnÞ ¼
J 0 2n d 2nJ 1 2n CY 0 2n d 1 2nCY 1 2n d d d d ; 2 n½CY 0 ð2nÞ þ J 0 ð2nÞd
where C is arbitrary constant and Jv(x) and Yv(x) are the modified Bessel functions of the first and second kinds, respectively. They satisfy the modified Bessel equation:
x2 Y 00 þ xY 0 ðx2 þ v 2 ÞY ¼ 0: Using these results we get final solution of system (1.11):
J 0 2rd d 2rJ1 2rd CY 0 2rd d dr þ D; r½CY 0 ð2rÞ þ J 0 ð2rÞd
Z 2rCY 1 2rd J 0 2rd d 2rJ 1 2rd CY 0 2rd d 2t v ðr; tÞ ¼ ir exp exp dr 2D : d r½CY 0 ð2rÞ þ J 0 ð2rÞd uðr; tÞ ¼ t=d
1 2
Z
2rCY 1
2r d
Making substitutions of these expressions for u(r, t) and v(r, t) in (1.9) and solving for w(r, t), we get
wðx; tÞ ¼
Z ð4rCY 0 ð2rÞ þ CY 0 ð2rÞd 2CY 0
ð2rÞ d
d þ 4CrY 1 2rd 2J 0 2rd d þ 4rJ0 ð2rÞ 4rJ1 2rd þ J 0 ð2rÞdÞ
4ðrðCY 0 ð2rÞÞ þ J 0 ð2rÞdÞ
dr þ f ðt rÞ;
where D is arbitrary constant and f(t r) is an arbitrary function of (t r). (only change is negative sign in v) Case (iii.4) H(n) is solution of following ODE: 2
2
2
2
2
n2 d ðn2 þ H2 ÞH00 þ 8n2 H3 þ 4H5 n3 H0 d þ nH2 H0 d n2 HH02 d þ 4n4 H þ Cn2 Hd ¼ 0; where C is arbitrary constant, and N(n) is given by
NðnÞ ¼
1
2
2Hðn2 þ H2 Þd
2
2
2
2
½nHd þ H2 H0 d þ sqrtðn2 H2 d þ nH3 H0 d þ 4n4 H2 n3 HH0 d þ n4 HH00 d n2 H2 H02 d þ 8n2 H4
2
þ n2 H3 H00 d þ 4H6 Þ: Case (iv) Similarity variable and similarity solution is
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n ¼ r;
uðr; tÞ ¼ FðnÞ;
v ðr; tÞ ¼ GðnÞ
t : e
On using these in Eq. (1.11), the corresponding system of reduced ODEs is given by
1 1 F 00 þ n1 F 0 n2 expð4FÞ G02 2 ¼ 0; 2 e G00 n1 G0 þ 4F 0 G0 ¼ 0: This can be further reduced with the help of MAPLE:
GðnÞ ¼
Z
gðnÞdn þ C;
where g(n) is solution of following differential equation.
ngg 00 e2 ¼ 2g 3 expð4DÞe2 gg 0 e2 þ ng 02 e2 þ 2g expð4DÞ; and F(n) is
FðnÞ ¼
1 1 lnðG0 ðnÞÞ þ lnðnÞ þ D; 4 4
where C and D are arbitrary constants. Case (v) Corresponding to this vector field, we get only constant solutions of the Eq. (1.11). 4. Discussion and concluding remarks We have deduced exact solutions of Field equations in general relativity corresponding to cylindrical symmetric space– time. The symmetry method based on the Fréchet derivative of the differential operators is utilized to investigate the symmetries and invariant solutions of these Field equations. The infinitesimal generators in optimal system of sub algebras of the full Lie algebra of the coupled system of nonlinear partial differential equations of second order of Field equations are considered for reductions and exact solutions. We completely solved the determining equations for the infinitesimal generators of Lie groups and obtained all linearly independent vector fields of Field equations corresponding to cylindrical symmetric space–time. Then we employed these vector fields to generate similarity (group-invariant) solutions. Thus we found some new exact solutions that might prove to be interesting for further applications. It is worth mentioning here that the authenticity of all the solutions has been checked with the aid of software Maple. Acknowledgments The authors gratefully acknowledge the critical comments and suggestions made by the anonymous referees. References [1] Rao JK. Cylindrical symmetric null fields in general relativity. Proc Natn Inst Sci India A 1968;1(3):367. [2] Stephani H, Kramer D, MacCallum M, Hoenselaers C, Herlt E. Exact solutions of Einstein’s field equations. Secone ed. Cambridge: Cambridge University Press; 2003. pp-350. [3] Piran T, Safier PN, Katz J. Cylinderical gravitational waves with two degrees of freedom: an exact solution. Phys Rev D 1986;34(2):331. [4] Papadopoulos D, Xanthopoulos BC. Tomimatsu–Sato solutions describe cosmic strings interacting with gravitational waves. Phys Rev D 1990;41(8):2512. [5] Bluman GW, Anco SC. Symmetry and integration methods for differential equations. Appl Math Sci. Springer; 2002. [6] Singh K, Gupta RK. Lie symmetries and exact solutions of a new generalized Hirota–Satsuma coupled KdV system with variable coefficients. Int J Eng Sci 2006;44(3-4):241. [7] Xiaoda Ji. Lie symmetry analysis and some new exact solutions of the WuZhang equation. J Math Phys 2004;45(1):448. [8] Zhao Li, Fu Jing-Li, Chen Ben-Yong. Lie symmetries and conserved quantities for a two-dimensional nonlinear diffusion equation of concentration. Chinese Phys B 2010;19(1):010301-1. [9] Ma WX, Chen M. Direct search for exact solutions to the nonlinear Schrödinger equation. Appl Math Comp 2009;215(8):2835. [10] Bluman GW, Cole JD. Similarity methods for differential equations. New York: Springer Verlag; 1974. [11] Gandarias ML, Bruzon MS. Classical and nonclassical symmetries of a generalized Boussinesq equation. J Nonlinear Math Phys 1998;5(1):8. [12] Quin S. Nonclassical symmetry reductions for coupled KdV equation. Int J Nonlinear Sci 2006;2(2):97–103. [13] Bhutani OP, Singh K. Generalized similarity solutions for the type D fluid in five dimensional flat space. J Math Phys 1998;39(6):3203. [14] Bhutani OP, Singh K, Kalra DK. On certain classes of exact solutions of Einstein equations for rotating fields in conventional and nonconventional form. Int J Eng Sci 2003;41(7):769. [15] Singh K, Gupta RK. Exact solutions of a variant Boussinesq system. Int J Eng Sci 2006;44(18-19):1256. [16] Steinberg S. Symmetry methods in differential equations. Technical Report No. 367, The University of New Mexico, 1979. [17] Ali AT. New exact solutions of the Einstein vacuum equations for rotating axially symmetric fields. Phys Scr 2009;79:1. [18] Attallaha SK, El-Sabbagh MF, Ali AT. Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields. Commun Nonlinear Sci Numer Simul 2007;12:1153. [19] Olver PJ. Application of Lie groups to differential equations in graduate texts in mathematics, vol. 107. New York: Springer; 1993.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity R.K. Gupta a,⇑, K. Singh b a b
School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, Punjab, India Department of Mathematics, Jaypee University of Information Technology, Waknaghat, P.O. Dumehar Bani, Kandaghat, Dist. Solan 173215, H.P., India
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 September 2010 Received in revised form 2 March 2011 Accepted 2 March 2011 Available online 12 March 2011 Keywords: Symmetry analysis Exact solutions Cylindrically symmetric space–time
The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In general relativity the investigation of null fields has acquired considerable interest in connection with the study of gravitational radiation. In the invariant theory of gravitational radiation, the Riemann tensor plays the central part. The algebraic and differential properties of this tensor have been discussed with a view to characterizing wave fields in general relativity. The essential idea in the Riemann tensor analysis as applied to radiation theory is that in a gravitational radiation field, the Riemann tensor will lie in some special relationship to the null cone. One such relationship is [1]
Rhijk þ iRhijk xk ¼ 0;
ð1:1Þ
where xk is a null vector, Rhijk is the Riemann tensor and is its dual. Eq. (1.1) imposes on the Riemann tensor a very severe restrictions, which is satisfied only asymptotically in the wave zone of a radiating system and exactly in plane gravitational waves and a few other special cases. From (1.1), it follows that Rhijk
Rij ¼ rxi xj ;
ð1:2Þ i
where r is a scalar. Since x is a null vector, from (1.2) the spur of the Ricci tensor R vanishes identically and through Einstein field equations the energy momentum tensor can be written as
8PT ij ¼ rxi xj :
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (R.K. Gupta). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.03.006
ð1:3Þ
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In the view of (1.3) the four eigen values s(i) determined by the equation
jT ij sg ij j ¼ 0 turn out to be zero and then the equation Tijvj = sgijvj admits only one real eigenvector which is null. The energy momentum tensor given by (1.3) represents some kind of incoherent superposition of wave packets of radiation and any zero rest mass field can be the source of radiation. We propose, in the present paper, to drive certain solutions of geometrical Eq. (1.2), when r – 0, for the cylindrical symmetric space time with two degrees of freedom. 1.1. The metric form and the Field equations We consider the cylindrical symmetric space time [2] 2
2
2
2
ds ¼ expð2w 2uÞðdt dr Þ ðv 2 expð2uÞ þ r 2 expð2uÞÞd/2 expð2uÞdz 2v expð2uÞd/dz;
ð1:4Þ
where u, v and w are functions of r and t only. When v = 0, (1.4) reduces to the well known Einstein Rosen metric with one degree of freedom. The nonzero components of the Ricci tensor obtained from (1.4) are
w1 u1 expð4uÞ 2 v4; þ w44 þ u11 þ u44 þ 2u24 þ 2r 2 r r w1 u1 expð4uÞ 2 ¼ w11 v 1; w44 u11 þ u44 þ 2u24 þ 2r 2 r r w4 expð4uÞ ¼ v1v4; þ 2u1 u4 þ 2 r 2r u1 expð4uÞ 2 2 ¼ expð4u 2wÞ u11 þ u44 v v ; 1 4 2r 2 r expð4u 2wÞ v ¼ v R33 þ v 11 1 v 44 þ 4ðu1 v 1 u4 v 4 Þ ; 2 r ¼ 2v R23 ðv 2 þ r2 expð4uÞÞR33 :
R44 ¼ w11 R11 R14 R33 R23 R22
ð1:5Þ
Here and in what follows, the subscripts 1 and 4 after u, v and w represent the partial differentiation with respect to r and t respectively. If the direction of propagation of the wave is the positive r-direction, we have here x2 = x3 = 0, x1 = x4 and, therefore, from (1.2) and (1.4) we get the field equations
R11 R44 ¼ 0;
ð1:6aÞ
R14 þ R44 ¼ 0;
ð1:6bÞ
R22 ¼ R23 ¼ R33 ¼ 0:
ð1:6cÞ
Making use of expressions for Rij given in (1.5), the relations (1.6a)–(1.6c) give the four differential equations
1 1 u11 þ u1 u44 ¼ r 2 expð4uÞ v 21 v 24 ; r 2 1 v 11 v 1 v 44 ¼ 4ðu4 v 4 u1 v 1 Þ; r expð4uÞ ðv 1 þ v 4 Þ2 ; w1 þ w4 rðu1 þ u4 Þ2 ¼ 4r expð4uÞ 2 v 1 v 24 : w11 w44 þ u21 u24 ¼ 4r2
ð1:7Þ ð1:8Þ ð1:9Þ ð1:10Þ
So, we have four Eqs. (1.7)–(1.10) for the determination of three unknowns u, v, and w and one can easily verify that these all are consistent. Therefore, we drop Eq. (1.10) and solve the remaining equations for u, v and w. It may be pointed out that Eqs. (1.7,1.8) are a set of coupled, second order, nonlinear partial differential equations in u and v, hence we will concentrate on these two equations and Eq. (1.9) is first order linear partial differential equation in w, which we will solve once we get u and v. Rewriting the Eqs. (1.7,1.8)
1 1 urr þ ur utt ¼ r 2 expð4uÞ v 2r v 2t ; r 2 1 v rr v r v tt ¼ 4ðut v t ur v r Þ: r
ð1:11Þ
The system (1.11) also represents another important class of Einstein field equations for vacuum [2]. No systematic investigation of this problem has yet been made (for more details, please refer to [2] pp - 350–351). Examples are the the counterpart of the Kerr solution [3] and of the Tomimatsu–Sato solutions [4]. Due to nonlinearity of exponential order, it is
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difficult to solve system (1.11) and hence study of symmetries and exact solutions of system (1.11) is of great importance and desirable. The heart of the classification schemes for the solutions of these equations are the symmetry methods based on the Lie groups. The most effective methods, for finding symmetry reductions and constructing exact solutions, include the Lie’s approach [5–9], the nonclassical approach [10–12], the generalized conditional symmetry method [13,14], symmetry reduction method [6,15,16] and isovector field method [17,18]. In this paper, we will discuss system (1.11) with the aid of symmetry reduction method. To determine the underlying symmetry group for Eqs. (1.11), we utilize a method which is based on the Fréchet derivative of the differential operators associated with the system (1.11). The technique has earlier been used to obtain the exact solutions of various nonlinear partial differential equations [6,15]. However, in order to help the reader and to relate it to the familiar one-parameter group of transformations, we provide here some basic knowledge about the approach. Let the system (1.11) be defined in terms of the nonlinear operators N1 and N2 as follows:
1 1 N1 ðu; v Þ ¼ urr þ ur utt r 2 expð4uÞ v 2r v 2t ¼ 0; r 2 1 N2 ðu; v Þ ¼ v rr v r v tt 4ðut v t ur v r Þ ¼ 0: r
ð1:11Þ
Next, we define an operator called symmetry operator for the system (1.11)⁄ given by
S ¼ ðS1 ; S2 Þ; where
@u @u Þ Þ; þ BðX; g þ C 1 ðX; g @t @r @v @v Þ Þ Þ; S2 ðv Þ ¼ AðX; g þ BðX; g þ C 2 ðX; g @t @r
Þ S1 ðuÞ ¼ AðX; g
ð1:12Þ
with
¼ ðu; v Þ: X ¼ ðt; rÞ; g Þ ¼ ðN 1 ; N 2 Þ, in the direction of g 1 is given by The Fréchet derivative of Nðg
þ g 1Þ d Nðg ; g 1 ¼ F N; g j ¼ 0: d
ð1:13Þ
For invariance of Eqs. (1.11), we require that the Fréchet derivative (1.13) must vanish on the solution set sof (1.11) in the direction of the symmetry operator S. That is, we must have
; SÞjr ¼ 0: FðN; g
ð1:14Þ
The associated Lie algebra of infinitesimal symmetries of (1.11) is then the set of vector fields of the form
Þ V ¼ AðX; g
@ @ @ @ Þ C 1 ðX; g Þ Þ : þ BðX; g C 2 ðX; g @t @r @u @v
ð1:15Þ
Equivalently, the one-parameter group of point transformations of (1.11) is as follows:
Þ þ oð2 Þ; t ¼ t þ AðX; g Þ þ oð2 Þ; r ¼ r þ BðX; g
ð1:16Þ
2
g ¼ g CðX; g Þ þ oð Þ; where
¼ ðu ; v Þ: C ¼ ðC 1 ; C 2 Þ; g The importance of the Eqs. (1.11) and the need to have some exact solutions are the main motives behind the present study. To have an insight, the explicit analytic solutions of the system (1.11) may enable one to better understand the phenomena which it describes. A detailed systematic analysis that leads to an exact analytic solution for (1.11) has not been performed [2] and is therefore desirable. The paper has been organized as follows. 2 is entirely devoted to showing how the powerful symmetry reduction method can be used to generate various symmetries of the system (1.11). In 3, we present the reduced systems of ordinary differential equations (ODEs) and their exact solutions. The final section contains the discussion and concluding remarks.
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2. The symmetry and optimal system The method for determining the symmetry group of (1.11) mainly consists of finding the coefficients A, B, C1, C2 in the two symmetry operators S1, S2 as defined by (1.12). These coefficients are to be determined from the invariance condition (1.14). Accordingly, we first find the Fréchet derivative
; g 1 Þ ¼ ðF 1 ðN1 ; g ; g 1 Þ; F 2 ðN2 ; g ; g 1 ÞÞ; FðN; g
ð2:1Þ
where
þ g 1 Þ d½N1 ðg j ¼ 0 d þ g 1 Þ d½N2 ðg ; g 1 Þ ¼ F 2 ðN2 ; g j ¼ 0: d ; g 1 Þ ¼ F 1 ðN1 ; g
ð2:2aÞ ð2:2bÞ
1 is substituted by S ¼ ðS1 ; S2 Þ in order to evaluate With the help of Eqs. (2.2), the Fréchet derivatives are obtained and then g them in the direction of the symmetry operator. This leads to the following:
1 1 ; SÞ ¼ ½S1 rr þ ½S1 r þ ½S1 tt þ expð4uÞ v t ½S2 t v r ½S2 r þ 2½S1 v 2t v 2r ; F 1 ðN1 ; g r r2 1 ; SÞ ¼ ½S2 rr ½S2 r þ ½S2 tt þ 4 ur ½S2 r þ v r ½S1 r ut ½S2 t v t ½S1 t : F 2 ðN2 ; g r For invariance of system (1.11), the following conditions must be satisfied:
; SÞjðN ;N Þ¼0 ¼ 0; F 1 ðN1 ; g 1 2
ð2:3aÞ
; SÞjðN ;N Þ¼0 ¼ 0: F 2 ðN2 ; g 1 2
ð2:3bÞ
The Eqs. (2.3), when expanded, result into the polynomial expressions in various partial derivatives of u(r, t) and v(r, t) with respect to the spatial variable. The calculations involved are tedious, however, to keep the interest of the reader we list here a much simplified set of determining equations for the group infinitesimals A, B, C1 and C2 which we get from (2.3a) after equating the coefficients of various derivative terms to zero:
Au ¼ 0;
Av ¼ 0;
Bu ¼ 0;
Bv ¼ 0;
C 1uu ¼ 0; Ar Bt ¼ 0; At Br ¼ 0; Arr þ r 1 Ar Att 2C 1tu ¼ 0; Brr r 1 Br Btt þ 2C 1ru þ Br 2 ¼ 0;
ð2:4Þ
C 1rr þ r1 C 1r C 1tt ¼ 0; 2C 1rv þ 2r1 C 1v r2 expð4uÞC 2r ¼ 0; 2C 1uv 4C 1v r2 expð4uÞC 2u ¼ 0; 2C 1tv r 2 expð4uÞC 2t ¼ 0; 1 C 1vv þ r2 expð4uÞðC 1u 2C 2v 4C 1 2Br 1 Þ ¼ 0: 2 Similarly, the Eq. (2.3b) brings-in the following additional equations. It is being mentioned here that these equations have been obtained keeping in view the consequences on the infinitesimals as affected by the set of Eq. (2.4).
C 2uu þ 4C 2u ¼ 0; C 2tu þ 2C 2t ¼ 0; 2C 2vv þ 8C 1v þ r 2 expð4uÞC 2u ¼ 0; C 2rr r1 C 2r C 2tt ¼ 0; C 2ru r1 C 2u þ 2C 2r ¼ 0;
ð2:5Þ
C 2uv þ 2C 1u ¼ 0; Arr r 1 Ar Att 2C 2tv 4C 1t ¼ 0; Brr þ r 1 Br Btt þ 2C 2rv Br 2 þ 4C 1r ¼ 0: Eqs. (2.4,2.5) enable us to derive the infinitesimals A, B, C1 and C2. Without presenting any calculations, we provide the results obtained as follow:
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A ¼ a2 t þ a5 ; B ¼ a2 r;
ð2:6Þ
C 1 ¼ a1 v þ a3 ; C 2 ¼ a1 r 2 expð4uÞ a1 v 2 ða2 þ 2a3 Þv þ a4 ;
where aj, j = 1, 2, 3 . . . 5 are arbitrary constants. The symmetries under which the Eq. (1.11) is invariant can be spanned by the following five linearly independent infinitesimal generators:
@ @ þ ðv 2 r 2 expð4uÞÞ @u @v @ @ @ ¼t þr þv ða2 Þ; @t @r @v @ @ ¼ ða3 Þ; þ 2v @u @v @ ¼ ða4 Þ; @t @ ¼ ða5 Þ: @v
V 1 ¼ v V2 V3 V4 V5
ða1 Þ;
ð2:7Þ
Using these generators one can obtain a reduction of Eqs. (1.11) to a system of ODEs after getting the similarity variables and the form by solving the characteristic equations
dt dr du dv ¼ ¼ ¼ : A B C 1 C 2
ð2:8Þ
In general, one may obtain the reduced system of ODEs from any linear combination of generators Vj, j = 1, 2, . . ., 5. Since there exist infinite possibilities for such combinations, a systematic procedure to classify these reductions is based on the property that the transformations of the symmetry group will transform solutions of Eqs. (1.11) into other solutions. Therefore, we classify the symmetry algebra of system into conjugacy inequivalent sub algebra under the adjoint action of the symmetry group. We will work out first an optimal system and then embark upon the various reductions associated with generators in the optimal system. We begin by considering a general element V = a1V1 + a2V2 + a3V3 + a4V4 + a5V5 of symmetry algebra and subject it to various adjoint transformations to simplify it as much as possible [19]. The adjoint action is given by the Lie series
AdðexpðV i ÞÞV j ¼ V j ½V i ; V j þ 2 ½V i ; ½V i ; V j ; where [Vi, Vj] = ViVj VjVi is the commutator for the Lie algebra, and is a parameter. The commutator table and the adjoint table for Lie algebra (2.6) are as follows: Commutator table Index
V1
V2
V3
V4
V5
V1 V2 V3 V4 V5
0 V1 2V1 0 V3
V1 0 0 V4 V5
2V1 0 0 0 2V5
0 V4 0 0 0
V3 V5 2V5 0 0
Adjoint table
Index
V1
V2
V3
V4
V5
V1 V2 V3 V4 V5
V1 V1 exp () V1 exp (2) V1 V1 + V3 2V5
V2 + V1 V2 V2 V2 V4 V2 V5
V3 + 2V1 V3 V3 V3 V3 2V5
V4 V4 exp V4 V4 V4
V5 V3 2V1 V5 exp V5 exp (2) V5 V5
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The optimal system consists of the following basic vector fields:
ðiÞ V 1 þ aV 2 þ bV 5 ; ðiiÞ V 2 þ cV 3 ; ðiiiÞ V 3 þ dV 4 ;
ð2:9Þ
ðivÞ V 4 þ eV 5 ; ðvÞ V 5 ; where a, b, c, d and e are arbitrary constants. 3. The reduced systems and solutions In this section, the primary focus is on the reductions associated with the vector fields (2.7) and attempt to find some exact solutions. Unfortunately, we are not able to solve characteristic equations corresponding to case (i), this will be taken as future endeavor. Corresponding to other cases, reductions are as follows: Case (ii) On using the characteristic Eqs. (2.8), the similarity variable and the form of similarity solution is as follows:
r n¼ ; t
uðr; tÞ ¼ FðnÞ c lnðtÞ;
v ðr; tÞ ¼ t1þ2c GðnÞ:
On using these in Eq. (1.11), the reduced system of ODEs is given by
2n2 F 00 2nF 0 þ n4 F 00 þ 4n3 F 0 þ 2n2 c þ expð4uÞ½G02 G2 ð1 þ 4c þ 4c2 Þ þ GG0 ð2n þ 4nc n2 Þ ¼ 0; nG00 þ G0 2nGc 4nGc2 þ n3 G00 4n2 GF 0 ð1 þ 2cÞ þ 4ðn3 nÞG0 F 0 ¼ 0:
ð3:1Þ
0
Let G(n) = exp(2F(n))H(n), F (n) = N(n). Then our system (3.1) becomes 2nN0 H 4nN2 H þ nH00 þ 2NH H0 þ 2nHc þ 4nHc2 þ 2n3 HN0 þ 2n3 HN 0 þ 4n3 H0 N 2 n3 H00 þ 4n2 HN þ 8n2 HNc ¼ 0; 2n2 N 0 2nN þ 2n4 N 0 þ 4n3 N þ 2n2 c þ 4H2 N 2 4NHH0 þ H02 H2 4H2 c2 4ð1 þ 2cÞnNH2 þ 2n2 NH2 þ 2nHH0 ð1 þ 2cÞ n2 HH0 ¼ 0:
Due to complexity of the system of ODEs, we are not capable for determining its solution. Case (iii) For this vector field, the form of the similarity variable and similarity solution is as follows:
n ¼ r; uðr; tÞ ¼ FðnÞ t=d; v ðr; tÞ ¼ exp
2t GðnÞ: d
The reduced ODEs in this case is as follows: 2 00
0
2n F þ 2nF þ expð4FÞ nG00 G0 þ
4nG 2
d
" 4G2 2
d
# G
02
¼ 0; ð3:2Þ
þ 4nG0 F 0 ¼ 0:
Solution of this system (3.2) of ODEs is as follow: 0 Let G(n) = exp(2F(n))H(n), F (n) = N(n). Using these substitutions our system reduce to
2n2 N0 þ 2nN þ
4H2 2
d
4H2 N2 þ 4NHH0 H02 ¼ 0;
2nN0 H þ 4nHN2 nH00 2HN þ H0
4nH 2
d
ð3:3Þ ¼ 0:
Now using MAPLE we examine this system. We arrive at following cases: Case (iii.1) H(n) = 0, that is not a physically interesting case. Case (iii.2) H(n) = ni, where i represents the complex number iota. With this, our system (3.3) reduced to single equation:
2n2 N0 2nN
4n2 2
d
þ 4n2 N2 þ 1 ¼ 0;
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which can be further solve to give solution:
NðnÞ ¼
J 0 2n d 2nJ 1 2n CY 0 2n d 1 2nCY 1 2n d d d d ; 2 n½CY 0 ð2nÞ þ J 0 ð2nÞd
where C is arbitrary constant and Jv(x) and Yv(x) are the modified Bessel functions of the first and second kinds, respectively. They satisfy the modified Bessel equation:
x2 Y 00 þ xY 0 ðx2 þ v 2 ÞY ¼ 0: Using these results we get final solution of system (1.11):
Z
½2rCY 1 2rd J 0 2rd d 2rJ1 2rd CY 0 2rd d dr þ D; r½CY 0 ð2rÞ þ J 0 ð2rÞd
Z 2rCY 1 2rd J 0 2rd d 2rJ1 2rd CY 0 2rd d 2t v ðr; tÞ ¼ ir exp exp dr 2D : r½CY 0 ð2rÞ þ J 0 ð2rÞd d uðr; tÞ ¼ t=d
1 2
ð3:4Þ
Making substitutions of these expressions for u(r, t) and v(r, t) in (1.9) and solving for w(r,t), we get
wðx; tÞ ¼
Z ð4rCY 0 ð2rÞ þ CY 0 ð2rÞd 2CY 0
ð2rÞ d
d þ 4CrY 1 2rd 2J 0 2rd d þ 4rJ0 ð2rÞ 4rJ1 2rd þ J 0 ð2rÞdÞ
4ðrðCY 0 ð2rÞÞ þ J 0 ð2rÞdÞ
dr þ f ðt rÞ;
where D is arbitrary constant and f(t r) is an arbitrary function of (t r). Case (iii.3) H(n) = ni With this, our system (3.3) reduced to single equation:
2n2 N 0 2nN
4n2 2
d
þ 4n2 N2 þ 1 ¼ 0;
which can be further solve to give solution:
NðnÞ ¼
J 0 2n d 2nJ 1 2n CY 0 2n d 1 2nCY 1 2n d d d d ; 2 n½CY 0 ð2nÞ þ J 0 ð2nÞd
where C is arbitrary constant and Jv(x) and Yv(x) are the modified Bessel functions of the first and second kinds, respectively. They satisfy the modified Bessel equation:
x2 Y 00 þ xY 0 ðx2 þ v 2 ÞY ¼ 0: Using these results we get final solution of system (1.11):
J 0 2rd d 2rJ1 2rd CY 0 2rd d dr þ D; r½CY 0 ð2rÞ þ J 0 ð2rÞd
Z 2rCY 1 2rd J 0 2rd d 2rJ 1 2rd CY 0 2rd d 2t v ðr; tÞ ¼ ir exp exp dr 2D : d r½CY 0 ð2rÞ þ J 0 ð2rÞd uðr; tÞ ¼ t=d
1 2
Z
2rCY 1
2r d
Making substitutions of these expressions for u(r, t) and v(r, t) in (1.9) and solving for w(r, t), we get
wðx; tÞ ¼
Z ð4rCY 0 ð2rÞ þ CY 0 ð2rÞd 2CY 0
ð2rÞ d
d þ 4CrY 1 2rd 2J 0 2rd d þ 4rJ0 ð2rÞ 4rJ1 2rd þ J 0 ð2rÞdÞ
4ðrðCY 0 ð2rÞÞ þ J 0 ð2rÞdÞ
dr þ f ðt rÞ;
where D is arbitrary constant and f(t r) is an arbitrary function of (t r). (only change is negative sign in v) Case (iii.4) H(n) is solution of following ODE: 2
2
2
2
2
n2 d ðn2 þ H2 ÞH00 þ 8n2 H3 þ 4H5 n3 H0 d þ nH2 H0 d n2 HH02 d þ 4n4 H þ Cn2 Hd ¼ 0; where C is arbitrary constant, and N(n) is given by
NðnÞ ¼
1
2
2Hðn2 þ H2 Þd
2
2
2
2
½nHd þ H2 H0 d þ sqrtðn2 H2 d þ nH3 H0 d þ 4n4 H2 n3 HH0 d þ n4 HH00 d n2 H2 H02 d þ 8n2 H4
2
þ n2 H3 H00 d þ 4H6 Þ: Case (iv) Similarity variable and similarity solution is
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n ¼ r;
uðr; tÞ ¼ FðnÞ;
v ðr; tÞ ¼ GðnÞ
t : e
On using these in Eq. (1.11), the corresponding system of reduced ODEs is given by
1 1 F 00 þ n1 F 0 n2 expð4FÞ G02 2 ¼ 0; 2 e G00 n1 G0 þ 4F 0 G0 ¼ 0: This can be further reduced with the help of MAPLE:
GðnÞ ¼
Z
gðnÞdn þ C;
where g(n) is solution of following differential equation.
ngg 00 e2 ¼ 2g 3 expð4DÞe2 gg 0 e2 þ ng 02 e2 þ 2g expð4DÞ; and F(n) is
FðnÞ ¼
1 1 lnðG0 ðnÞÞ þ lnðnÞ þ D; 4 4
where C and D are arbitrary constants. Case (v) Corresponding to this vector field, we get only constant solutions of the Eq. (1.11). 4. Discussion and concluding remarks We have deduced exact solutions of Field equations in general relativity corresponding to cylindrical symmetric space– time. The symmetry method based on the Fréchet derivative of the differential operators is utilized to investigate the symmetries and invariant solutions of these Field equations. The infinitesimal generators in optimal system of sub algebras of the full Lie algebra of the coupled system of nonlinear partial differential equations of second order of Field equations are considered for reductions and exact solutions. We completely solved the determining equations for the infinitesimal generators of Lie groups and obtained all linearly independent vector fields of Field equations corresponding to cylindrical symmetric space–time. Then we employed these vector fields to generate similarity (group-invariant) solutions. Thus we found some new exact solutions that might prove to be interesting for further applications. It is worth mentioning here that the authenticity of all the solutions has been checked with the aid of software Maple. Acknowledgments The authors gratefully acknowledge the critical comments and suggestions made by the anonymous referees. References [1] Rao JK. Cylindrical symmetric null fields in general relativity. Proc Natn Inst Sci India A 1968;1(3):367. [2] Stephani H, Kramer D, MacCallum M, Hoenselaers C, Herlt E. Exact solutions of Einstein’s field equations. Secone ed. Cambridge: Cambridge University Press; 2003. pp-350. [3] Piran T, Safier PN, Katz J. Cylinderical gravitational waves with two degrees of freedom: an exact solution. Phys Rev D 1986;34(2):331. [4] Papadopoulos D, Xanthopoulos BC. Tomimatsu–Sato solutions describe cosmic strings interacting with gravitational waves. Phys Rev D 1990;41(8):2512. [5] Bluman GW, Anco SC. Symmetry and integration methods for differential equations. Appl Math Sci. Springer; 2002. [6] Singh K, Gupta RK. Lie symmetries and exact solutions of a new generalized Hirota–Satsuma coupled KdV system with variable coefficients. Int J Eng Sci 2006;44(3-4):241. [7] Xiaoda Ji. Lie symmetry analysis and some new exact solutions of the WuZhang equation. J Math Phys 2004;45(1):448. [8] Zhao Li, Fu Jing-Li, Chen Ben-Yong. Lie symmetries and conserved quantities for a two-dimensional nonlinear diffusion equation of concentration. Chinese Phys B 2010;19(1):010301-1. [9] Ma WX, Chen M. Direct search for exact solutions to the nonlinear Schrödinger equation. Appl Math Comp 2009;215(8):2835. [10] Bluman GW, Cole JD. Similarity methods for differential equations. New York: Springer Verlag; 1974. [11] Gandarias ML, Bruzon MS. Classical and nonclassical symmetries of a generalized Boussinesq equation. J Nonlinear Math Phys 1998;5(1):8. [12] Quin S. Nonclassical symmetry reductions for coupled KdV equation. Int J Nonlinear Sci 2006;2(2):97–103. [13] Bhutani OP, Singh K. Generalized similarity solutions for the type D fluid in five dimensional flat space. J Math Phys 1998;39(6):3203. [14] Bhutani OP, Singh K, Kalra DK. On certain classes of exact solutions of Einstein equations for rotating fields in conventional and nonconventional form. Int J Eng Sci 2003;41(7):769. [15] Singh K, Gupta RK. Exact solutions of a variant Boussinesq system. Int J Eng Sci 2006;44(18-19):1256. [16] Steinberg S. Symmetry methods in differential equations. Technical Report No. 367, The University of New Mexico, 1979. [17] Ali AT. New exact solutions of the Einstein vacuum equations for rotating axially symmetric fields. Phys Scr 2009;79:1. [18] Attallaha SK, El-Sabbagh MF, Ali AT. Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields. Commun Nonlinear Sci Numer Simul 2007;12:1153. [19] Olver PJ. Application of Lie groups to differential equations in graduate texts in mathematics, vol. 107. New York: Springer; 1993.