Volume 141, number5,6 PHYSICSLETFERSA 6 November1989

GRAVITATIONAL INTERACTION OF MASSLESS PARTICLES

ValeriaFERRARIInternationalCenterfor RelativisticAstrophysics— ICRA, Dipartimentodi Fisica “G. Marconi’~Universitàdi Roma,Rome,Italy

and

JesusIBAREZ’DepartamentodeFisica, UniversidaddeLasIslasBaleares,Spain

Received21 June1989; revisedmanuscriptreceived31 July 1989;acceptedfor publication6 September1989CommunicatedbyJ.P.Vigier

An exactsolutionof non-vacuumEinsteinequationsdescribingthecollisionof two cloudsofnull particlesandtheirconsequentgravitationalinteractionis presented.As aresult, a sourceis producedwhichsatisfiestheKlein—Gordonequationfor masslessscalarfields,anda physicalsingularityfinally appears.

Thegravitationalinteractionof colliding shellsof coupledto the dust,andthat a sourcecanbe pro-null dust has been recently investigated by ducedwhich behaves,at a macroscopicallevel, asaChandrasekharand Xanthopoulos [1,2], and by fluid with an equationof statedifferent from p =

Drayand‘t Hooft [3]. In refs. [1] and[2], thenull and with an anisotropicdistribution of pressure,dustflies in flat spacetimefollowing the leadingedges beingthe componentsorthogonalto the directionofof impulsive gravitationalwaves,andthe collision propagationdifferent from the parallelcomponent.producesaregionof interactionwhereeithera mix- The energy—momentumtensorsatisfiesthe strongture of null dust moving in oppositedirections is energyconditions.Unlike the solutionsquotedbe-present,or a perfectfluid in which theenergydensity fore, the metric is continuouswith its first deriva-~t is equalto thepressurep. Thepresenceof different tives,thereforeno impulsivewavesarepresenton thefluidsin theregionof interaction,is due,asclarified null hypersurfacesseparatingdifferent regions.by Taub [4,51,to different assumptionson the en- A physical singularity developsin the region ofergy—momentumtensor,which specifythe natureof interaction.the interaction.In ref. [3] the collision of two im- One might ask how is it possiblethat the gravi-pulsiveplanarshellsof null dusthasbeenconsidered, tationalinteractionofnull dustcanproduceasolargewith a result similar to thatdescribedin ref. [1]. varietyof sources,andwhetherthesesolutionsreally

Oneof the most remarkableconsequencesof the representtheinteractionof somefields,andweshallinteractionof null dust, is the possibility, shownin try to answerthis question.ref. [21, of a gravitationallyinducedtransformation The null dust is definedby the following energy—of masslessparticles,describingnull trajectories,into momentumtensor:a perfectfluid, althoughwith a peculiarequationof T~V— ~)k~k~ (1)state.

We shall show in this paperthata similar transi- whereE is a positivefunctionof a null coordinateution canoccuralsowhenno gravitationalwavesare representingthe energyof the dust,andk’~is a null

vector. This definition presentsan intrinsic ambi-GuestoftheDepartmentofPhysicsoftheUniversityofRome. guity, since the energy—momentumtensor (1) can

0375-960l/89/$03.50© ElsevierSciencePublishersB.V. 233(North-HollandPhysicsPublishingDivision)

Volume141, number5,6 PHYSICSLETTERSA 6 November1989

representany kind of masslessfield propagatingas equationsin five dimensions,we can derive a so-a planewave along a definite direction. In this re- lution of the non-vacuumEinsteinequationsin fourspect,it is not surprisingthat many solutionsexist dimensions, provided g44 satisfies the constraintin which two wavesof null dustcollide and,due to (4b), andthe energy—momentumtensorassociatedthe gravitational interaction, produce different to the function0 satisfiesthe energyconditions.sources,asshownin the referencesquotedabove. From eqs.(3), the equationfor the function q is

However,whenthespecific equationssatisfiedbythenull particlesareidentified, the problemis corn- q” ~q— — ~=0, (6)pletelydefined,the solution is unique,andthe am-biguity disappears.Thus, in our case, we have where the dot and the prime indicate,respectively,checkedthat theKlein—Gordonequationfora mass- differentiationwith respectto t andz; the functionless scalarfield is satisfiedboth in the regionof in- f canbe derivedby quadraturefrom the systemteractionandin the precollisionregion, andthat the

f=4t(q’2+q2)—2/3t f=3tqq’ . (7)

scalarfield satisfiesthe Junctionconditionsacrossthe null boundariesseparatingdifferent regiovs. We selectthe solutionThereforewe are able to identify the natureof the

q=c1log~1+c2logj~2+klogt (8)

null dustandwecanconcludethat oursolutionrep-resentsthe gravitationalinteractionof masslesssea- wherelar waves.

• . •

It is interestingto note that the sourcein the re-gionof interactionmimics a non-perfectfluid with ~i2= 1 —z+~,/(1 —z)

2--t2an anisotropicdistribution of pressure,asis shown k— + 9by the analysis of the correspondingenergy—mo- — — ~ c

2)~mentumtensor in a suitably chosenorthonormal and c1 andc2 are constants.We shall considerthisbasis, solution in the region

Thesolution hasbeenobtainedby a dimensional• . . • , • • 0z~t~l t—l.~z~l—t. (10)

reductionof asolutionof Einsteinsequationsin five-dimensionalvacuum. Consequently,f will be

Let us considerthe following form of the metric:

th2=e~[—(dt)2+(dz)2] f=3((kcl+c~)lOg/1l+(kC

2+c4)log~L2

+t2/s[e~(dxi)2+e~(dx2)2+e2~(dx4)21 . 2c’1c2

‘2’ +1 Cui+u2)‘• ~

Themetricfunctionsf~q, areassumedto dependonand z only. It is known [6,71 that the Einstein + (4k

2— 4) log t~. (11)equationswritten in five dimensionsfor thevacuum, I

RAB—O (A, B=0, 4), (3) To investigatethe physical meaningof the solu-tion, we shall expressthecomponentsof theenergy—

are equivalentto the following systemof equations momentumtensorin four dimensions:

R~~=O~q~,~p, (4a) TppØ~Ø;p;p, (12)with respectto an orthonormalbasis.The resulting

Ø’~’.,,~=0(~,v=0, 3) (4b) canonicalform of TM~is definedby the following

where eigenvalues:

ØJ~(I/3e_~ (5)

Therefore,froma givenvacuumsolutionof Einstein’s

234

Volume 141, number5,6 PHYSICSLETTERSA 6 November1989

e-1 12 I= { — O;0;o+ 033

t 0-1E(0;o;o+0;3;3)2_40,2o;31h/’2}, (13a) iie—f

23{0;0;O+0;3;3 6

+[(0;o;o+0;3;3)240~o;31h/2}, (13b)

(13c) :To ensuretherealityof theeigenvalues,andto makethem finite on the null boundaries

0z=t— 1, z= 1 —t , (14) —0.5 0 05

zwe mustassumec

1 = c2= — 1 /,.j~. Fig. 1. Thelogarithmof theenergydensityp andof thep~com-

21, 22~23 correspond,respectively,to a time- ponentof thepressureareplottedfor selectedvaluesof time, in

like vectore0 andto threespacelikevectorse1, e2, e3 thecorrespondingintervalof z, in theregionof interaction.The

provided O;o;o+0; 3.3>0. This inequality is always superscribed+ refersto i’~thecontinuousline to ~satisfiedin the region definedby eqs.(10). Hence _________________________________________we canidentify the energydensity,as measuredby

i0.(~ -

t 01e0, andthe pressurein thethreespacelikedirectionsea as 7.5

~u=—A0, P1P2

21, p3=13. (15)

With this identification,the strongenergyconditions ~°

anobserverwhoseworld-linehasunit tangentvector~t+p1>O, u+~p1>O, 1=1,3,

are satisfied.Plots of —2~and 23, andA~=22 are 2.5 -

It shouldbenotedthatthevelocityfield associated -givenin figs. 1 and2, for differentvaluesof tin thecorrespondingz-interval.to the eigenvalueA~is timelike, andthat the source ‘-0.5 0 0.5

in the region of interactionthereforebehavesas a Z

fluid whosestreamlinesare tirnelike trajectories.The Fig.2. The logarithmof thePi=P2 componentsofthepressureis

fluid hasan anisotropicdistribution of pressure,as plottedforthesamevaluesoft andzusedin fig. 1.

shownby eqs. (15), and the componentalong thedirectionof propagationP3 is alwaysgreaterthanthecomponentsparallel to the wavefront Pi=P2. The Penrosealgorithm, we introducea pair of null co-hydrodynamicalequations,~ ~.= 0, aresatisfiedas ordinatesu andv, relatedto t andzby thefollowinga consequenceof eqs. (4) andthe Ricci identities. equations:

Whent—~0thereis a physicalsingularity: the en- t=1—u4—v4, z=u4—v4. (17)

ergydensityand the pressuredivergelikeThis choiceensuresthe functionsf be finite on the

~,p1,p3~t’

313. (16) boundaries(14), which correspondto the hyper-In order to extend the metric acrossthe null surfacesu=0 andv=0. In termsof u andv, the met-

boundaries(14), by using the standard Khan— ric becomes

235

Volume141, number5,6 PHYSICSLETTERSA 6 November1989

~2 _eFdu dv +t213e’~[(cjv1)2+ (dx2)2], (18) ~ , (21)

whereeF= 64u3v3 e-’i bothin the interactionregionandin the precollisionTheonly non-vanishingWeyl scalarin the region region.Thefunction 0 is continuouseverywherewith

definedby eqs. (10) is ~P2: its first derivatives,and all the requiredjunction

conditionsaresatisfiedon thenull boundariesby the

= — 1 6u6v~e!( stress—energytensorandby the Ricci tensor.This

(1 — u — v4)2 means,inparticular,that~ and~ arecontinuouson (v=0, 0<u< 1), and that in the entire region

+l_~2_q~2)+~), (19) (v~0,0<u<l)theequation

thereforenogravitationalradiationis presentin this ~ = ~ = 0— 0; U;

region. ~P2is singularwhen u

4+v4= 1, which cor- holds, with 0=0(u). This occurrenceallows us torespondsto t = 0. identify the natureof thecolliding shellsofnull dust

Whenu—U0 ± (v—~0~), ~ andP2 tend to zero,~ in termsof scalarwaves.We concludethat our so-andp

3 tendto the samefinite value,and q12 tendsto lution representsthe collision, andthe following in-zero. teraction,of two masslessscalarfields, which pro-

Becauseall the metric functions,andall thephys- duce,in theregionofinteraction,a scalarfield whichical quantitiesare finite on the null boundaries,we behaves,at a macroscopicallevel,as a fluid. Thein-canextendthe metric by the substitution teractionis not affectedby any impulsive phenom-

u—~uH(u), v—~vH(v) , enon.As in the collision of planefronted gravita-tionalwaves,a singularitydevelopsafterthecollision

where H is the Heavisidestep function,which en- of the scalarwaves.suresthe continuity of the metric. In addition, the It shouldbenotedthatthe stress—energytensorwefirst derivativesof the metric are continuous,and use (eq. (12)) is notin the form which is standardtherefore,no impulsivewavesare present, for a scalarfield. However, it has beenshown [81

Theextendedmetric in the regionV~<0, 0 ~ U < 1 that it canbe reducedto that form by a conformalis transformationof themetric, accompaniedby adef-

ds2——8(1—u4)1/12 du dv mite transformationof the scalarpotential.

It is believed [91that in the very earlyuniverse,+ (1 —u4)213~‘~(1+u2) ~ at the timewhenthe variousinteractionswereuni-

x [(~ 1)2+ (~2) 2] . (20) fled, theuniversecontainedpredominantlymasslessparticles. At that time phasetransitionsare sug-Theonly non-zerocomponentofthe Ricci tensoris gestedto occurwhich would generatebubblesof the

~ andit is always positive in the region we are new broken-symmetryfase.Thesebubbles,acceler-considering.Thereforethe solution (20) allows an atedby the energyreleasedin the transition,wouldinterpretationin termsof a distributionof null dust possiblycollide andgeneratepairsof scalarwaves,movingin the v-direction.The situationis symmet- whosefollowinggravitationalinteractionisgenerallync in the region (u~ 0, 0 ~ v< 1). When u <0 and negiectedin thesemodels.v< 0 the spacetimeis flat. Theresultspresentedin this papersuggestthat,due

At thisstagewecansaythatthesolutionpresented to its possiblerelevanceto the physicsof the earlyin this Letter representsthecollisionandthe follow- universe,the gravitationalinteractionof null dusting gravitational interaction of two planar sand- deservesa deeperinvestigation.wiches of massless particles propagatingin flatspacetime.

However,somefurtherinformationscanbe added.The sourceof the EinsteinequationssatisfiestheKlein—Gordonequationsfor a masslessscalarfield

236

Volume 141,number5,6 PHYSICSLETTERSA 6 November1989

References [5] A.H. Taub, Collision of impulsive gravitational wavesfollowed by dust clouds, preprint.

[1] S. ChandrasekharandB.C. Xanthopoulos, Proc.R. Soc. A [6] Th.Kaluza,Sitzungsber.Preuss.Akad.Wiss. Phys.Math.K1.403 (1986) 189. 54 (1921)966.

[7]0. Klein,Z. Phys.37 (1926)875.[2J S. ChandrasekharandB.C. Xanthopoulos, Proc.R. Soc.A[8] V.A. Belinskii andI.M. Khalaktnikov,Sov. Phys.JETP 36402 (1985) 37.

(1972) 591.[3] T. Drayand G. ‘t Hooft, Class.QuantumGray. 3 (1986)

825. [9] S.W. Hawking, I.G. MossandJ.M.Stewart,Phys.Rev.D 26(1982)2681.[4] A.H. Taub, J. Math. Phys. 29 (3) (1988) 690.

237