On conformally related pp-waves

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P R A M A N A @ Indian Academy of Sciences Vol. 56, No+ 5

---- joLIrnaI of May 2001

physics pp. 591-596

On confbrmaliy re]lated pp-waves

V A R S H A D A F T A R D A R - G E J J t

Department of Mathematics, University of Pune, Ganeshkhind, Pune 411 007, India Email: vsgejji @math;unil~une:ernetJn

MS received 28 June 1999; revised 6 December 2000

Abstract. Bfinkmann [1] has shown that conformally related distinct Ricci flat solutions am

PtJ-

waves. Bfinkmann's result has been generalized to include the conformally invariant source tecrns, it has been shown that [4] if 9~: and .~it: (--

"w-29~k,.w:

a scalar function), are distinct metrics having the same Einstein tensor, Gik = Gi~, then both represent (generalized) p/)-waves and w' 0 is a null covafiandy constant vector of flit:. Thus/)p-waves are the only candidates which yield conformatly related nontrivial solutions of Gi~: -- 5f}t: -- Gi~.., with

Tit:

being conformally invariant source.

In this paper the functional form of the conformal factor for the contbrmally related

Pt)-

waves/generalized pp-waves has been obtained. It h0s been shown that the most general/)p wave, conformally related to ds 2 = . - 2 d ~ [ d v - m d y + Hda~] + P-2[d~]2 + dze], turns out to be

(aTe, + b)-~ds ~,

where a~b are constants. Only in the special case when m = 0, H = !, and P = P(?;,

z),

the conformal factor is (a~ + b) -~ or (a(v, + v) + b) -~.

Keywords. Conformal transformations; conformal Killing vectors;/)/)-waves.

PACS Nos 04.20.Jb; 04.30.-w

1. Introduction

The p p - w a v e space-times are p l a n e - f r o n t e d gravitational waves with parallel rays. T h e s e space-times represent the characteristic property of having parallel rays o f plane electro- magnetic waves o f M i n k o w a s k i space. These waves serve as the idealized models of gravitational fields far from isolated radiating bodies. T h e i r u n i q u e role in conformaI trans- itbrmations was first p o i n t e d out by B r i n k m a a n [1, 2]. He proved that c o n f o r m a l l y related Einstein spaces are either Ricci flat p p - w a v e s or c o n f o r m a l l y fiat spaces with one as the de Sitter space and the other being flat. T h u s one can have conformaI[y related Ricci fiat solutions o n l y in case ofp.p-waves. This motivates the study of c o n f o r m a l tra~asformations of pp~waves as they are the o n l y candidates to give non-trivial c o a f o r m a U y related solutions,

A relation b e t w e e n conformal[y related Einstein1 spaces and con formal Killing vector has been established

[3I. We [311

have shown daat any two o f the f o l l o w i n g statements imply the third. (1)

9.i.i

is Einstein. (2) j i j ( = w

29ij,'~v:

a scaIar function) is Einstein.

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w i

is a c o n f o r m a l Killing vector of

g,l.i.

U s i n g this we have given an aIter~mtive p r o o f of B r i n k m a u n ' s theorem. H e n c e the problem of finding co[lforma/ly related p p - w a v e s is reduced to that o f finding covariantly cot~stant mill vectors of p / > w a v e metric [3,4].

5 9 I

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Vat sha Dqfi~rdar-Gejfi

B r i n k m a n n ' s work has further been generalized [41 to include c o n f o r m a l l y invariant source terms such as massless K l e i n Z G o r d o n , null radiatioa fields. It has been proved that [4] if .(]ik and ~ f ~ ( =

'w-~g.i,,)

are distinct metrics witt~

Gi~:

= G~:, then both represent (generalized) 1)p-waves and 't6' .i is a null covariantly constant vector.

in the present p a p e r we obtain the functional l'orm of the c o n f o r m a l factor for the con- f o r m a l l y related

pp-wave/geoeraiized/)p-wave.

It has been shown that the g e n e r a l i z e d

pp-

wave, c o n f o r m a l l y related to d s 2 =

-2&u[dv-'lr~dy+f.fd't,,]-bi~-'e[d~.j "

+ d z 2 ] , turns out to be

(a'~, + b)-~d.s~:

where (L, b are constants. Only in the specia{ case when ~Tz = 0, H = 1, and l " = P ( y , z), the conl'ormal factor is either (az~ + b) - ~ or (a(v, + v) +

b) ~2.

The prese~t p a p e r has been o r g a n i z e d as follows. Section 2 deals with basic definitions and properties of c o n f o r m a l Killing vectors, confornm! h-ansformations and pp~waves. In 9 {}3..we..fi.ncl .coyuia.r~Ttly co~;ts~7{u?t nu!l. yect.ors 9f.1)prw~}y.e lnr w.!)ich. !n. tu.~7!~ .yiebq!S..!he

functional f o r m of the c o n f o r m a l t'actor of e o n I b r m a l t y related/.~p-waves.

2. Preliminaries

2.1 Cot,jbrmal Killing vectors

We present below s o m e basic definitions regarding conformaI K i l l i n g vectors. A confornml Killing vector ( C K V ) { 7/ satisfies

w h e r e c) is called as c o n f o r m a l factor. If q5 = 0, it is called K i l l i n g vector (KV), (/) is a constant, then it is called as h o m o t h e t i c KV. A C K V is called special if ~Si;j = 0 and ,:b,~ 7 i- 0. A C K V is called g r a d i e n t C K V ( G C K V ) if it is a gradient of s o m e scaiar. A vector field is called as covariai~tly constat~t vector (CCV) if 14:;j = 0. N o t e that every C C V is GCKV.

2,2

I.~p- Waves

E x a c t solutions to E i n s t e i n ' s e q u a t i o n s having properties similar to those of plane waves in the {inearized theory o f gravitation, can be obtained by d e m a n d i n g that they are solutions of

[5]

f~ij = 0, ~ y k ~ = 0, & U = 0. (2)

The stronger conditions [5]:

Ri3 = 0,/,;,i;j = 0,

~:~k i

= 0, (3)

h n p / y that the rays associated with the nul[ vector

t~'i

are paral!el. H e n c e the waves are calIed as plane :fionted waves with parallel rays, (pp-waves). Thus a s p a c e - t i m e represents ),al)-wave if it admits a null CCV.

It can be shown that [5, 6] if a s p a c e - t i m e with zero Ricci s c a l a r (Ricci flat space is special case) admits a null CCV, then there exists a s p a c e - t i m e in which the metric takes the l"ollowing l'o,m:

x

592

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May

2~,RH

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Co;!lbrmal/y related pp- wa yes

Cl ,_5, 2 = - 2 7 1 ('u,, ~, z)d'tt" - 2d'ttdv + d//e ? -k dz e , (4)

with '~z and v as retarded/advanced time coordinates (and

-Er,:~:,:,..

+ .H ~.ns~ = 0 [k~r Ricei-f]at solt~r The space-like 2-sur[ace it, v = constant, are called wave surfaces. Since they are flat, the waves are called plane fronted. Ehlers and K u n d t [2] have studied s y m m e t r i e s off)p-waves. T h e y have f o u n d all K i l l i n g s y m m e t r i e s admitted by -/)/)-waves in v a c u u m .

In the case w h e n Ricci scalar is non-zero, tlae space-time achnitting null C C V cakes the fO r [Ii;

d,~ ~ = - 2 d @ l v - mcl~; + 5Zdrc] + P - ~ [ @ ~ + dr"], (5) where '(sT,.~ S:[ al3. d .P ac e f u n c t i o n s of ~, :9, z. f7].- .. . .

2.3

Conjbr, zo! lra~l.sformaUon

We recollect s o m e restllts regarding conforrriaI tranformations, which will be used furtEer.

Consider the c o n f o r m a l transformation of the metric tensor

) ~ = w ~9~,

(6)

which leads to the f o l l o w i n g relation [4]"

~ ' G i k = '~G.ih +

[

20'~v 9ik - 2'~v,i;k,

L

^'l_U ^J ⁽⁷⁾

where

Gik

is Einstein tensor o f the metric

9ik,

a n d Gik is the Einstein tensor o f the metric ,qik, i z i = t o , i , ~ i L , ~ " w , i ; k , ~'UJ ~-

.qi:ttau~.*

Here a s e m i c o l o n refers co a covariat~t derivative for the metric

ffik.

It can be s h o w n that [4] if

9.ik

and .0i~. are distinct solutions having (~Lis,: = G~:M then both admit null CCV, representing generalized/)/)-waves, and w,.i is a nu!] C C V o f 9i~...

Conversely if ~v,i is a null C C V of 9i/r then

~ik

is a/)/)-wave solution, in other words the p r o b l e m o f finding c o n f o r m a l i y related

pp-wave

space-time .{s reduced to findir~g g : a d i e n t null CCVs o f p p - w a v e metrics.

3. C o n ~ b r m a l t r a n s l b r m a t i o x ~ of p p - w a v e s

3.1 Null

CCVs

of pp-wave tnelrics

As discussed in the p r e c e d i n g section, the p r o b l e m of finding eonl:'omxiliy related p p - w a v e space-times reduces to that o f finding gradient ~xtll C C V s of p p - w a v e metrics. N o w to begin with consider the case o f zero Ricci scalar (which includes v a c u u m plo-waves) in which the lap-wave metric has the form [6]: ds ~ = - 2/-S('/z., ~}, z)d'l~ ~ - 2d'~dv + d g e + d z '~.

In order to obtain the conl'ormally related Ep-wave solutions correspoLxting to the /aT)- wave da :~ = - 2 7 I @ , , g , z)d'~ ~ - 2d'~dv + dg :~ + d z 2, we 113] have to soIvr equations

w,h.. i

= 0, viz.

" w . r . - H . ~ w ~, - H,~w,~ - Zt ~z~J : = O,

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Vol. 56, No. 5, ~,4Ja'~' 200] _~ ~)-~

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Va,:rho D a f i a s z l a r - G e j j i

'w ,, .:, - H vw,.o = O, (9)

w,~u~. - H . : ' w , ~ , = O, ( 1 0 )

w,vc,, ~ 'u~,v,v = 'w,v,e = 'tv,~,,~ = w.,,e.a = w,.v,z ~ w,u,, J = O. (11) The set of eqs in (1 t) i m p l i e s

where 7~: is an arbitrary constant and &s, dJ',, <75a are arbitrary functions o17 ~,. Further, eqs (9) and (10) lead to

. . . ,, ; -

a:u = o, . . . (i3)

r

~ k H , + = O. ( 1 4 )

Differentiation of eqs (13) and (14) yields either t i . , j , , a = II,~,z = 0, or k is zero.

Thus for k : 0, w = tr + k:2V + 6a(a/), where ~:1 and ,~2 are arbitrary constants, with 'w{ = ( 6 a , , . , 0 , k u , / q ) . As wi is null; w.ii*J = k~ + k:~ : 0. H e n c e k:! : /~:2 = 0 and w = 63 (*~,), which by eq. (8) impIies 63 ... - 0. In which case 6a -- a'~t -I- b, where a and b are constants, i.e. ~xs = aqx + b.

For k # 0, differentiations of ( 1 3 ) a n d ( 1 4 ) g i v e _~[,.~j,~a = H Z,~ = Hm,.: = 0. The R i e m a n n tensor for t h e / ) / ) - w a v e metric is [6]

,-., <' . < ,

~.~,<..<z t D b}u.[<_.%:]=-<,>'.,s- [~ t,] [<.''d] .... -t-(6[oov~,]~7[cOVd} + ~5~ah+~]3"o ~:v " ~ q H e n c e if t-I,>~s : I - I , , , = H m , ~ = 0, then the s p a c e - t i m e is fiat, this case however, is J:uled out thereby leaving the unique possibility w = c~L + b. Thus the c o n f o r m a l factor in this case is a't, + b.

S o m e r e m a r k s are in order.

(i) U n d e r the following c o o r d i n a t e transformations:

- -(c~,~, + b) -~, ~ = ~,/~ + (:(' + ?)l[2(<~u + b)},

the metric [ 1 / ( a u + b)~]ds 2, will take the form:

d,~ u = -2d,~d,~ - f i ( ~ , ~7, Z,) d,C § d~ ~ -t- d ~ 2 ,

(ii) The a m p l i t u d e A and p o l a r i z a t i o n 0 o f a p p - w a v e is d e f i n e d by [611 A C ~ = (.Hm,, v - /-/ ... ) / 2 + 'i.Hm, ~, H e n c e A = (cvu, -I- b):~A, and (~ = 0. So c o n f o r m a I l y r e l a t e d p p - w a v e s have the same p o l a r i z a t i o n angle.

(iii) Most g e n e r a l l y t h e / ) / > w a v e metric has the form

d.s 2 = -2d'u,[d't2 - 'rrtd v + 7:Id't~] + 1~-2[d'.92 + dz2l, (1.5) where 'rr~, H , and 12 are functions of % 7/, z. Clearly (1, 0, 0, 0) is a null C C V of the metric (t5). Tiffs metric admits one more C C V only in the special case when rra = 0, H = 1, and 1~ = P(:t/, z) (see a p p e n d i x of ref. [7]).

T h u s evetl..fbr t h e m o s t get, crag c a s e the c o , f o , ~ a l f ~ c t o r is ,~ = a~, + b.

594 P r a m a n a - J. Phys., Vol. 56, No. 5, May 2001

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Cvs~j'ormally related I ) p - w a v e s

3.2 The s p e c i a l case: rn, =: 0, H = 1, a n d P = P(gI, z ) Consider ttle case ~a, = 0 , H = 1, and P = P(~j, z).

d s 2 = - 2 d ' ~ d v - 2 d ~ / + P-2[d~/2 + dz2],

where P = .P(g, z) and w satisfies the following set of equations:

71J .u,,t ~ ~ l O , . u . . v ~ 'tU .tt,7: ~ % U , , 8 , z ~ ' ~ I J , v , v =~ 'tU .v .y ~ - ~l) v , z ~ - O :

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9 w , j , ~ + (P,~/t.~)w,,~ + ( P , . ~ / P ) w ~ = 0, (19)

- + ( P = o .

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Equations ia (17) i m p l y ,w = a ~ + cv + ~(71, z), where a and c are constants a n d rk is aa arbitrary f u n c t i o n of its arguments. A d d i n g eqs (18) a n d (20) we get w > v + w .... = 0.

Differentiating eq. (19) with respect to z and eq. (20) with respect to ~j and subtracting one from the other, we get

(21) in view of w,v,s j ^{+ w}.... = 0. As the Ricci scalar for the me~ric (10) (viz, (2/P'))[(P..~,= + i o m , v ) / p _ ( p j + .p~2)/.p:~]), is n o n zero (the ~ = 0 case has been dealt with earlier), eq.

(21) yieIds w ~ a = 0. H e n c e in view o f e q . (l 9) we get w z = 0. So w = ~ u § cv + b, where c.~, b, c are constants. As w,.i is null, we get ( a c - c 2) = 0 and c = 0 or cL = c, y i e l d i n g either w -~ a u + b or w = a(% + v) -I- b.

3.3 Ilhxsrrafi;,e e x a m p l e

The pp~wave ds~ = -[,V ~' ' i' -r- z"]d% ~ - 2 d u d v + d;q ~ + " " dz ~ satisfies E i n s t e i n - K l e i n - G o r d o n equations with massless field & = ,/2%. L i k e w i s e ds~ = (a% + b ) - 2 d s ~ aIso satisfies E i n s t e { n - K l e i n - G o r d o n equations with massless scalar field c) = V@u.. Note that ds~ and ds,5 are different p p - w a v e s , as " " in ds~, I - / = (9 ~ + 9 ^-

52)/a2~ "".

4. Conclusim~

C o n f o r m a l l y related solutions having the same source term are possible in case of pl>waves/genera[ized p p - w a v e s only. In order to obtain c o n f o r m a l l y related iv/J-waves/generalized p p - w a v e s essentially o n e has to determine nulI C C V o f 1)1)- waves~generalized/)p-waves. We herewith identify the functional form of tile c o n f o r m a [ factor.

a 1

It turns out that co~fform~ lly related 7,i)-wa\ e c o n e s p o n d i n g to

P r a m a u a - J. Phys., Vo!. 56, No. 5, May 2001 595

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I/arsha DctftarclLzr-Gej/i

d 9 = - ' ) d , , [ d , . - ,n,~d:~ + H d,~,,] + e - ~ [ d v ~ + d.~i!

h,'ts the l:orm (c~'u, + :~) 2ci.s~. Only w h e n 'of = 0, H = 1, and ? = ]~(~/, z), the conl:ormal factor turns out to be either (a:u, 4- b) - ~ or (a(~z + '~:) F b) :;, as a special case,

R e f e r e n c e s

[1] El W Brinkmann, l~/Iclth. Am~. 94, 119 (t925)

[2] J Ehlevs a~ad W l~[undt, in G~avita:iosl: As~ i/~rocl~ac~io~z to C~):v~t Research edited by L Wi~ten (Wiley, New York, 1962)

[3] V Oaftardar m~d N Dadhich, Ge~erol Relatil'. G~'c~:it. 26, 859 (1994) and the rel~rences therein [4] V Daftarda~-Gejji, Gel~esal Rela:iv. GraviL 30; 695 (199S)

G S Hall and A D bLendall, J. Matt~. Phy.v, 2B, 1837 (1987) [5] t-I Stephani, Genera! Relativio, (Cambridge University Press, 1990) [6] R Maartens and S D Mahara i, Clas.~'. Q~lclJ~fu~l Grctvi:. 8, 503 (1991) [7] A A Coley ~.lld B O J Tupper, J. Math. P]l.ys§ 31,649 (1990)

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