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International Journal of TheoreticalPhysics ISSN 0020-7748 Int J Theor PhysDOI 10.1007/s10773-014-2338-8

Symmetries of Type N Pure RadiationFields

Zafar Ahsan & Musavvir Ali

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Int J Theor PhysDOI 10.1007/s10773-014-2338-8

Symmetries of Type N Pure Radiation Fields

Zafar Ahsan ·Musavvir Ali

Received: 29 March 2014 / Accepted: 9 September 2014© Springer Science+Business Media New York 2014

Abstract The geometrical symmetries corresponding to the continuous groups ofcollineations and motions generated by a null vector l are considered. These symmetrieshave been translated into the language of Newman-Penrose formalism for pure radiation(PR) type N fields. It is seen that for such fields, conformal, special conformal and homo-thetic motions degenerate to motion. The concept of free curvature, matter curvature andmatter affine collineations have been discussed and the conditions under which PR type Nfields admit such collineations have been obtained. Moreover, it is shown that the projectivecollineation degenerate to matter affine, special projective, conformal, special conformal,null geodesic and special null geodesic collineations. It is also seen that type N pureradiation fields admit Maxwell collineation along the propagation vector l.

Keywords Type N PR fields · NP-formalism · Collineations

1 Introduction

Since it was initiated some forty years back, Petrov type N solutions of Einstein vacuumequations are among the most interesting but rather difficult and little explored of all emptyspacetime metrics ([9, 15]). From the physical point of view, they represent spacetime filledup entirely with gravitational radiation while mathematically they form a class of solutionsof Einstein equations which should be possible to be determined explicitly. The behavior ofthe gravitational radiation from a bounded source is an important physical problem. Evenreasonably far from the source, however, twisting typeN solutions of the vacuum field equa-tions are required for an exact description of that radiation. Such solutions would providesmall laboratories in which to understand better the complete nature of singularities of type

Z. Ahsan (�) · M. AliDepartment of Mathematics, Aligarh Muslim University, Aligarh, 202002, Indiae-mail: zafar.ahsan@rediffmail.com

M. Alie-mail: musavvirali.maths@amu.ac.in

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N solutions and could also be used to check numerical solutions that include gravitationalradiation.

Different types of symmetries of Petrov type N gravitational fields has been the sub-ject of interest since last few decades (cf., [2]) but a complete analysis of collineationsis not found in the literature (as far as we know). Moreover, in general theory of rel-ativity the curvature tensor describing the gravitational field consists of two parts viz.,the matter part and the free gravitational part. The interaction between these two parts isdescribed through Bianchi identities. For a given distribution of matter, the construction ofgravitational potential satisfying Einstein’s field equations is the principal aim of all inves-tigations in gravitational physics and this has often been achieved by imposing symmetrieson the geometry compatible with the dynamics of the chosen distribution of matter. Thegeometrical symmetries of the spacetime are expressible through the equation

£ξA− 2�A = 0 (1)

where A represents a geometrical/physical quantity, £ξ denotes the Lie derivative of A withrespect to the vector field ξ (this vector may be time-like, space-like or null) and � is ascalar.

One of the most simple and widely used example is the metric inheritance symmetry forwhich A = gij in (1). In this case, the vector field ξa is called the conformal Killing vectorwhich includes a homothetic vector and a Killing vector according as �;a = 0 and � = 0,respectively.

In a series of papers ([4, 5, 10, 11]) Katzin, Levine, Davis and collaborators have iden-tified many symmetries for the gravitational field with their interrelationships and haveobtained the corresponding weak conservation laws as the integrals of the geodesic equa-tion. Different types of matter distribution compatible with geometrical symmetries havebeen the subject of interest of several investigators for quite sometime and in this con-nection, Oliver and Davis [14], for the perfect fluid spacetimes, have studied the time-likesymmetries with special reference to conformal motion and family of contracted Riccicollineation. The perfect fluid spacetimes including electromagnetic field which admit sym-metry mapping belonging to the family of contracted Ricci collineation, have been studiedby Norris et al. [13]. The role of geometrical symmetries in the study of perfect fluid space-times, with an emphasis on conformal collineation has been explored by Duggal [7] andDuggal and Sharma [8] (see also [1]). The geometrical symmetry £ξRij = 2�Rij , known asRicci inheritance, has been studied by Ahsan [3] who obtained the necessary and sufficientconditions for perfect fluid spacetimes to admit such symmetries in terms of the kinematicalquantities.

In this paper, we consider the free gravitational field to be the transverse gravitationalwave zone which can be identified as Petrov type N fields and we focus our attention on theinteraction of pure electromagnetic radiation field and pure gravitational radiation field. Forthe sake of brevity, we call such interacting radiation fields as pure radiation or PR fields.For such fields, using NP-formalism [12] a systematic and detailed study of different typesof collineations has been made here.

2 NP-Formalism and Type N Pure Radiation Fields

Consider a four dimensional Lorentzian manifold M which admits a Lorentzian metric gij .Let Za

μ = {la, na, ma, ma} be the complex null tetrad (μ = 1, 2,3, 4) where la, na arereal null vectors and ma, ma are the complex null vectors. All the inner products between

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the tetrad vectors vanish except lana = 1 = −mama . In NP-formalism the expressions formetric tensor gij , can be expressed as

gij = 2l(inj) − 2m(imj) (2)

Using the definition of the spin-coefficients, the covariant derivatives of null vectors aregiven by ([6])

∇j li = li;j = (γ + γ )lj li − τ ljmi − τ lj mi + (ε + ε)nj li

−2κnjmi − κnj mi − (α + β)mj li − (α + β)mj li

+σmjmi + σmj mi + ρmjmi + ρmjmi (3a)

∇j ni = ni;j = −(γ + γ )ljni − νljmi − νlj mi − (ε + ε)nj ni

+πnjmi + πnj mi + (α + β)mjni + (α + β)mjni

−λmjmi − λmj mi − μmjmi − ρmj mi (3b)

∇jmi = mi;j = νlj li − τ lj ni + (γ − γ )ljmi + πnj li − κnjni

+(ε − ε)njmi − μmj li + ρmjmi + (β − α)mj mi

+(α − β)mjmi − λmj li + σmjni (3c)

Moreover, from the definition of the differential operators

D ≡ li∇i , � ≡ ni∇i , δ ≡ mi∇i , δ ≡ mi∇i (4)

it is easy to write

∇i = li�+ niD − miδ −miδ (5)

In terms of the complex null tetrad Zaμ, the Maxwell scalars characterizing the null

electromagnetic fields with la as the propagation vector are [6]

�0 = �1 = 0, � = �2 �= 0 (6)

and the electromagnetic field tensor is

Fij = �l[imj ] + �l[i mj ] (7)

where � = 2Fijminj , a square bracket denote the skew symmetrization and a parenthesis

denotes the symmetrization.The Weyl scalar characterizing a pure (null) gravitational field with propagation la is

�4 = � �= 0, �i = 0, i = 0, 1, 2, 3 (8)

Since we are considering null electromagnetic field, lμ can be scaled, i.e., ε = 0 andfrom equation (8), the Goldberg-Sachs Theorem demands that the propagation vector lμ isgeodetic and shear-free. Thus, we have

κ = σ = ε = 0 (9)

and the source-free Maxwell equations are

Dφ = ρφ

δφ = (τ − 2β)φ(10)

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For pure radiation type N fields, NP field equations, Bianchi identities, coupled Bianchiidentities and commutator relations, using (6)–(9), are given by ([12])

Dρ = ρ2 (11a)

Dτ = (τ + π)ρ (11b)

Dα = ρ(α + π) (11c)

Dβ = ρβ (11d)

0 = (τ + π)α + (τ + π)β + τπ (11e)

Dλ− δπ = ρλ+ π2 + (α − β)π (11f)

Dμ− δπ = ρμ+ ππ − π(α − β) (11g)

Dν −�π = (τ + π)μ+ (τ + π)λ (11h)

�λ− δν = −(μ+ μ)λ+ (3α + β + π − τ )ν −� (11i)

δρ = ρ(α + β)+ (ρ − ρ)τ (11j)

δα − δβ = μρ + αα + ββ − 2αβ (11k)

δλ− δμ = −(ρ − ρ)ν + (μ− μ)π + μ(α + β)+ λ(α − 3β) (11l)

δν −�μ = μ2 + λλ− νπ + (τ − 3β − α)ν +�� (11m)

�β = −μτ − βμ− αλ (11n)

δτ = λρ + (τ + β − α)τ (11o)

�ρ − δτ = −(ρμ)+ (β − α − τ )τ (11p)

�α = ρν − (τ + β)λ− μα (11q)

D� = ρ� (12a)

δ� = (τ − 4β)� (12b)

D�� = ρ�� (12c)

δ� − δ(��) = (τ − 4β)� + (2α − τ + 2β)�� (12d)

(�D −D�)η = [−(τ + π )δ − (τ + π)δ]η (13a)

(δD −Dδ)η = [(α + β − π )D − ρδ]η (13b)

(δ�−�δ)η = [−νD + (τ − α − β)�+ λδ + μδ]η (13c)

(δδ − δδ)η = [(μ− μ)D + (ρ − ρ)�− (α − β)δ − (β − α)δ]η (13d)

3 Collineations for Pure Radiation Fields

In this section, we shall translate different types of collineations in the language of NP-formalism and obtain a number of results about such collineations. We have

(i) MotionA symmetry of a spacetime is called as motion if the Lie derivative of metric tensor gij

along the vector ξa vanishes, i.e.,

£ξ gij = ξi;j + ξj ;i = 0 (14)

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Since for null fields it is possible to choose scaling li → ϕli so that γ = 0 and ε = 0. Thustaking ξ i = li in (14) and using (3a), we get

Theorem 1 Type N pure radiation fields admit motion if and only if τ + α + β = 0 and la

is expansion-free.

Remark 1 It may be noted that

(ii) Conformal motions £ξ gij = 2�gij ,(iii) Special conformal motion £ξ gij = 2�gij , �;jk = 0(iv) Homothetic motion £ξ gij = 2�gij � = constant

all degenerate to motion.

The Ricci tensor for the source-free null electromagnetic fields with propagation vectorla is

Rij = χ

2��li lj (15)

It is known that ([15, 16]) the Weyl tensor can be decomposed into null tetrad componentsas

Cabcd = 12 (Cabcd + iC∗

abcd)

= �0UabUcd +�1(UabWcd +WabUcd)

+�2(UabVcd + VabUcd +WabWcd)

+�3(VabWcd +WabVcd)+ �4VabVcd

(16)

where

Uab = namb −manb,

Vab = malb − mbla,

Wab = lanb − lbna +mamb − mamb (17)

and

UabVab = 2, WabW

ab = −4 (18)

so that the Weyl tensor characterizing transverse gravitational field of type N , using (8) and(17), is given by

Cabcd = 2Re�VabVcd (19)

From (7), (15) and (19), we have

Fij li = 0, Fijn

i = �

2mj + �

2mj , Fijm

i = − �

2lj , Fij m

i = −�

2lj

Rij li = 0, Rij n

i = χ

2��lj , Rijm

i = 0, Rij mi = 0

Cijklli = 0, Cijkln

i = −2Re�mjVkl,

Cijklmi = −2Re�ljVkl, Cijklm

i = 0 (20)

These equations show that li is the common propagation vector for electromagnetic field,Ricci tensor and Weyl tensor.

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(v) Ricci collineation

A symmetry of a spacetime is called as Ricci collineation if The Lie derivative of Riccitensor Rij along vector ξ vanishes, i.e., £ξRij = 0

The Lie derivative of Ricci tensor along the propagation la , from the definition, is givenby

£lRij = DRij + Riklk ;j + Rkj l

k ;i (21)

where D ≡ li∇i . From (3a), (11a–11q) and the properties of tetrad {la , na, ma, ma},equation (21) reduces to

£lRij = (ρ + ρ)Rij (22)

Thus, we have

Theorem 2 Type N pure radiation fields admit Ricci collineation if and only if thepropagation vector is expansion-free.

(vi) Free and matter curvature collineations

The Riemann curvature tensor Rabcd is defined as

Rabcd = Cabcd + 1

2(gacRbd − gbcRad + gbdRac − gadRbc)

+R

6(gadgbc − gacgbd) (23)

while from (2) and (15), for pure radiation fields, we have

R = gijRij = 0 (24)

Therefore, for type N PR fields, the curvature tensor is given by

Rabcd = Cabcd + 12 (gacRbd − gbcRad + gbdRac − gadRbc)

= Rabcd(F)

+ Rabcd(M)

(25)

whereRabcd(F)

= Cabcd (26)

denotes the free-gravitational part and

Rabcd(M)

= 1

2(gacRbd − gbcRad + gbdRac − gadRbc) (27)

denotes the matter part of the gravitational field.Thus, for type N pure radiation fields, from (2), (15) and (19), we have

Rabcd(F)

= Cabcd = 2Re�VabVcd (28)

andRabcd(M)

= χ

4��(VabVcd + VabVcd) (29)

Moreover, from (18) and the properties of tetrad, (28) and (29) lead to

Rabcd(F)

la = 0 and Rabcd(M)

la = 0 (30)

which shows that la is the propagation vector for the free and matter part of the gravitationalfield.

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The Lie derivative of Cabcd , by the definition, is given by

£lCebcd = lhCe

bcd;h − Chbcd l

e;h + Cehcdl

h;b + Cebhdl

h;c + Cebchl

h;d (31)

Now from NP Bianchi identities, the definition of li ;j and the properties of PR fields, wehave

lhCebcd;h = DCe

bcd = D{2Re�V ebVcd }

= 2Re�ρV ebVcd (32)

Chbcd l

e;h = 2Re�[(α + β)lble − ρlbm

e]Vcd (33)

Cehcd l

h;b = 2Re�[−τ lble − ρmbl

e]Vcd (34)

etc. With these equations, (31) now reduces to

£lCebcd = 2Re�[(ρ − ρ)V e

b − (α + β + τ )lble + ρmbl

e]Vcd

= £lRebcd

(F)

(35)

Thus, we have

Theorem 3 Type N pure radiation fields admit free curvature collination(i.e., £lRe

bcd(F)

= 0) along the propagation vector la if and only if τ + α + β = 0 and ρ = 0.

Since Rebcd = geaRabcd , taking the Lie derivative of (29) along la and using

Theorems 1 and 2, we have

Theorem 4 Type N pure radiation fields admit matter curvature collination(i.e., £lRe

bcd(M)

= 0) along the propagation vector la if and only if τ + α + β = 0 and la is

non-diverging.

(vii) Curvature collineation

Since Rebcd = geaRabcd , from (28) and (29), (26) can be expressed as

Rebcd = 2Re�(V e

bVcd)+ χ

4��(V e

bVcd + V eb Vcd) (36)

which on using the properties of tetrad vectors and (19), leads to

Rebcd l

c = 0 (37)

which shows that la is the propagation vector for the Riemann curvature tensor. Thus, from(26), Theorems 3 and 4, we have

Theorem 5 Type N pure radiation fields admit curvature collination along la (i.e.,£lRe

bcd = 0) if and only if τ + α + β = 0 and ρ = 0.

Remark 2 It may be noted that the necessary and sufficient conditions for type N pureradiation fields to admit free curvature collineation and curvature collineation are same,

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although the structure of free gravitational part [cf., (28)] of the gravitational field and theRiemann curvature tensor [cf., (36)] are different.

(viii) Affine collineation

The affine collineation (AC) is known to be defined as

£ξ�abc = ξa;cb + Ra

cmbξm = 0 (38)

Choosing ξa = la , (38) for pure radiation type N fields, from (37) reduces to

£l�abc = la;cb = 0 (39)

For type N pure radiation fields, using (3a) and (5), the second order covariant derivative ofla is found to be as

la;cb = la[{τ(α + β)+ τ (α + β)}lbnc−{�(α + β)− 2γ (α + β)}lbmc

−{τ π + π(α + β)}nblc + {ρπ −D(α + β)nbmc}−{τ λ+ λ(α + β)}mblc

−{δ(α + β)+ (α + β)(β − α)+ (α + β)(α + β)+ ρλ}mbmc

−{μτ − μ(α + β)}mblc + ρ(α + β)mbnc

−{δ(α + β)+ (α + β)(α − β)+ (α + β)2 + μρ}mbmc]+na[2τ τ lblc − ρτ lbmc + ρτmblc − ρ2mbmc]+ma[−(�τ + 2γ τ )lblc + {�ρ + τ2 + (α + β)τ}lbmc

+(α + β)τ lbmc −Dτmblc − {δτ + τ (α + β)+ τ (β − α)}mblc

+{τ ρ + ρ(α + β)}mbmc

−{δτ + τ (α + β)+ τ (α − β)}mblc + ρ(α − β)mbmc + τ ρmbmc]+ma[−ρτlbnc + τ τ lbmc + ρπnblc +Dρnbmc

+{δρ + ρ(β − α)+ ρ(β − α)}mbmc + ρλmblc

+ρμmblc − ρ2mbnc + {ρ(α + β)+ ρδ + ρ(α − β)}mbmc]+ c. c.

(40)

Thus, from (39) and (40), we have

Theorem 6 The necessary and sufficient conditions for type N pure radiation fields toadmit affine collineation are that

α + β = 0, ρ = 0, τ = 0 (41)

(ix) Projective collineation

It is known that a projective collineation (PC) along la is defined as

£l�abc = δabA;c + δacA;b (42)

where A is an arbitrary function. Now

A;c = A;igic = (li�+ niD − miδ −miδ)Agic

= (li�+ niD − miδ −miδ)A(linc + ni lc −mimc − mimc)

= �Alc +DAnc − δAmc − δAmc

(43)

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which shows that A;c = 0 if and only if

DA = �A = δA = δA = 0 (44)

That is A is constant.Moreover, from (38) and (42), we have

ξa;cb + Racmbξ

m = δabA;c + δacA;b = 0 (45)

Choosing ξa = la , for type N pure radiation fields we have Racmbl

m = 0 and (45) yields

la;bc = δabA;c + δacA;b = 0 (46)

The tensor (46) are equivalent to

α + β = 0, τ = 0, ρ = 0

DA = �A = δA = δA = 0 (47)

Thus, we have

Theorem 7 For type N pure radiation fields, projective collineation degenerate to affinecollineation.

Remark 3 In a similar way we can show that for type N pure radiation fields, the followingcollineations degenerate to affine collineation.

(x) Special projective collineation (SPC)

£l�abc = δabA;c + δac A;b, A;bc = 0 (48)

where A is arbitrary function.

(xi) Conformal collineation (Conf C)

£l�abc = δabB;c + δac B;b − gbcg

adB;d (49)

where B is arbitrary function.

(xii) Special conformal collineation (S Conf C)

£l�abc = δabB;c + δac B;b − gbcg

adB;d , B;bc = 0 (50)

(xiii) Null geodesic collineation (NC)

£l�abc = gbcg

adE;d (51)

where E is arbitrary function.

(xiv) Special null geodesic collineation (SNC)

£l�abc = gbcg

adE;d , E;bc = 0 (52)

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The double covariant derivative of A, using (5) and (A-1) to (A-12) is given by

A;b;c ≡ A;bc = [−(μ+ μ)nblc + (ρ − ρ)nbnc + {−τ + π + (α − β)}nbmc]DA

+[−(μ+ μ)lblc + (ρ + ρ)lbnc + {−τ + π − (α − β)}lbmc]�A

+[(μ+ μ)mblc + (ρ − ρ)mbnc + {τ − π + (α − β)}mbmc

+{τ − π + (α − β)}mbmc]δA

+ c. c.

(53)

Finally, we have

(xv) Maxwell Collineation (MC)

A spacetime is said to admit Maxwell collineation if there exists a vector field ξa suchthat

£ξFij = Fij ;kξ k + Fikξk;j + Fkj ξ

k;i = 0 (54)

Choosing ξa = la and using (7), (3a) and (A-1)–(A-12), (54) leads to the following

Theorem 8 Type N pure radiation fields satisfying source-free Maxwell equations alwaysadmit Maxwell collineation along the propagation vector field la .

Acknowledgments The authors are grateful to the learned referee for his valuable suggestions.

Appendix

From the definition of the covariant differentiation operators D, �, δ and δ along the direc-tion of the vectors of a complex null tetrad, it is possible to write (3a)–(3c) in the followingcovariant forms

Dla = (ε + ε)la − κma − κma (A-1)

�la = (γ + γ )la − τma − τma (A-2)

δla = (α + β)la − ρma − σma (A-3)

δla = (α + β)la − σma − ρma (A-4)

Dna = −(ε + ε)na + πma + π ma (A-5)

�na = −(γ + γ )na + νma + νma (A-6)

δna = −(α + β)na + μma + λma (A-7)

δna = −(α + β)na + λma + μma (A-8)

Dma = π la − κna + (ε − ε)ma (A-9)

�ma = νla − τna + (γ − γ )ma (A-10)

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δma = λla − σna + (β − α)ma (A-11)

δma = μla − ρna + (α − β)ma (A-12)

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